Integrable Kuralay Equations: Geometry, Solutions and Generalizations

Abstract

In this paper, we study the Kuralay equations, namely the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. The nonlocal and dispersionless versions of the Kuralay equations are considered. Some integrable generalizations and other related nonlinear differential equations are presented.

1. Introduction

In this paper, we continue our programme of construction of integrable spin systems (or Heisenberg ferromagnet-type equations) in 1 + 1 and 2 + 1 dimensions (see, e.g., [1,2,3,4] and references therein). We present some such type-integrable systems and their gauge-equivalent counterparts. Generally speaking, soliton equations (integrable equations) are the most important class of nonlinear differential equations (NDE) in mathematics and physics. They have remarkable applications to many physical systems, such as hydrodynamics, nonlinear optics, plasma physics, field theories and so on. The nonlinear integrable system is characterized by several features: solitons, Lax pairs, Painleve tests and so on. Searching for such integrable NDE is an extremely important task in modern mathematical physics and its applications. Another very important actual problem is the construction of the exact solutions of such integrable systems. At present, to find exact solutions of integrable nonlinear equations, there exist several powerful mathematical tools, such as the inverse scattering transform, the Hirota bilinear method, the Wronskian and pfaffian technique, the Bell polynomial approach, the Darboux and Bäcklund transformations, Painleve analysis, etc. Among these methods for the construction of exact solutions, the Hirota bilinear method is most efficient for the construction of exact solutions and multiple collisions of solitons. Note that soliton solutions have a wide range of applications in nonlinear physics and other branches of science. For example, such nonlinear solutions arise in different areas, such as fluid mechanics, nonlinear optics, atomic physics, biophysics, biology, field theory, in plasma physics and Bose–Einstein condensates and so on. The main subject of this work is the following Kuralay-II equation (K-IIE) [5,6,7,8,9]

$$\begin{array}{ccc}\hfill i{q}_t+q_{xt}-vq& =& 0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill v_x-{2ϵ\left(\right|q|}^2{)}_t& =& 0,\hfill \end{array}$$

(2)

where $q(x,t)$ is a complex function, $\overline{q}$ is the complex conjugate of q, $v(x,t)$ is a real function (potential), $ϵ=\pm 1$, x and t are independent real variables. A subscript denotes a partial derivative with respect to x and t. In the soliton theory, the gauge equivalence and geometrical equivalence between integrable equations play an important role [10,11]. In this paper, we prove that the gauge and geometrical equivalent counterpart of the K-IIE (1) and (2) is the following Kuralay-I equation (K-IE) [5,6,7,8,9]

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(4)

where $\mathbf{S}=(S_1,S_2,S_3)$ is the unit spin vector, ${\mathbf{S}}^2=S_1^2+S_2^2+S_3^2=1$, ${\mathbf{S}}_x^2=S_{1x}^2+S_{2x}^2+S_{3x}^2$ and u is the real scalar function (potential). This K-IE is one of the examples of integrable spin systems (see, e.g., [1,2,3,4] and references therein).

The paper is organized as follows. In Section 2, we consider the Kuralay-II equation. The traveling wave solutions and the simplest soliton solution of the K-IIE are considered in Section 3. The integrable motion of the space curves induced by the K-IIE is presented in Section 4. In the following Section 5, the gauge equivalence between the K-IE and the K-IIE is established. The Hirota bilinear form and soliton solutions of the K-IE are considered in Section 6. The nonlocal and dispersionless versions of the Kuralay equations are presented in Section 7 and Section 8, respectively. In Section 9 and Section 10, we present some generalizations of the K-IE and K-IIE, respectively. We conclude in Section 11.

2. The Kuralay-II Equation

In this paper, we will study the Kuralay equations (KE). For example, there are two forms, which are the two versions of the Kuralay-II equation (K-IIE). They are the Kuralay-IIA equation (K-IIAE) and the Kuralay-IIB equation (K-IIBE). In this section, we study these two forms of the K-IIE.

2.1. Kuralay-IIA Equation (K-IIAE)

In this paper, we study the following form of the Kuralay-II equation (K-IIE) [5,6,7,8,9]

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}-vq& =& 0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+vr& =& 0,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill v_x+2d^2{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(7)

which we call the K-IIAE. It is integrable by the inverse scattering transform (IST) method. It is well known that the Lax pair plays a key role in the theory of integrable systems. Below, we write that the nonlinear equation has the Lax representation if it can be written as the compatibility condition of two linear equations. Thus, in this paper, the Lax pair and the Lax representation are equivalent synonyms. In our case, for the K-IIAE, the corresponding Lax representation has the form

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_2\Phi ,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_2\Phi ,\hfill \end{array}$$

(9)

with

$$\begin{array}{ccc}\hfill U_2& =& id\lambda {\sigma }_3+dQ,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill V_2& =& \frac{1}{1-2d\lambda }B.\hfill \end{array}$$

(11)

Here,

$$\begin{array}{c}\hfill B=-0.5iv{\sigma }_3-di{\sigma }_3{Q}_t\end{array}$$

(12)

and

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}0& q\\ r& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).\end{array}$$

(13)

The compatibility condition

$$\begin{array}{c}\hfill U_{2t}-V_{2x}+[U_2,V_2]=0\end{array}$$

(14)

is equivalent to the q-form of the KE (qKE) [5,6,7,8,9], i.e., to the Kuralay-IIA equation (K-IIAE) (5)–(7), where $U_{jt}=\frac{\partial U_j}{\partial t},V_{jx}=\frac{\partial V_j}{\partial x}$ and so on. As $r=ϵ\overline{q},d=1$, from these equations, we obtain the K-IIAE of the form (1) and (1).

2.2. Kuralay-IIB Equation (K-IIBE)

The second form of the K-IIE is given by

$$\begin{array}{ccc}\hfill i{q}_x+q_{xt}-vq& =& 0,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill i{r}_x-r_{xt}+vr& =& 0,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill v_t-2{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(17)

which we call the K-IIBE. It is the second form of the K-IIE. It is natural that this K-IIBE is also integrable in the sense that it admits the Lax representation of the form

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_3\Phi ,\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_3\Phi ,\hfill \end{array}$$

(19)

where

$$\begin{array}{c}\hfill U_3=-i\lambda {\sigma }_3+Q,V_3=\frac{1}{1-2\lambda }B,B=-0.5iv{\sigma }_3-i{\sigma }_3{Q}_x.\end{array}$$

(20)

3. Soliton Solutions

Let us demonstrate that the K-IIE admits some exact solutions, e.g., the simplest traveling wave solutions, including the 1-soliton solution. As an example, here, we consider the K-IIAE. Let $d=1,r=ϵ\overline{q}$. Then, the K-IIAE takes the form

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}-vq& =& 0,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill v_x-{2ϵ\left(\right|q|}^2{)}_t& =& 0.\hfill \end{array}$$

(22)

3.1. Traveling Wave Solutions

Let us assume that $q(x,t)$ has the form

$$\begin{array}{c}\hfill q=\chi (x,t)e^{i(ax+bt+\delta )},\end{array}$$

(23)

where $\chi (x,t)$ is a real function and $a,b,\delta$ are some real constants. Then, the K-IIAE takes the form

$$\begin{array}{ccc}\hfill i({\chi }_t+ib\chi )-[{\chi }_{xt}+ia{\chi }_t+ib{\chi }_x-ab\chi ]-v\chi & =& 0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill v_x-2ϵ{\left({\chi }^2\right)}_t& =& 0.\hfill \end{array}$$

(25)

Hence, we obtain

$$\begin{array}{ccc}\hfill {\chi }_t-a{\chi }_t-b{\chi }_x& =& 0,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill -b\chi -{\chi }_{xt}+ab\chi -v\chi & =& 0,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill v_x-2ϵ{\left({\chi }^2\right)}_t& =& 0,\hfill \end{array}$$

(28)

or

$$\begin{array}{ccc}\hfill {\chi }_t-a{\chi }_t-b{\chi }_x& =& 0,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\chi }_{xt}-b(a-1)\chi +v\chi & =& 0,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill v_x-2ϵ{\left({\chi }^2\right)}_t& =& 0.\hfill \end{array}$$

(31)

Let us now introduce the new independent variable $\xi =mx+ct$, where $m,c$ are some real constants. Then, we have

$$\begin{array}{ccc}\hfill (c-ac-bm){\chi }_{\xi }& =& 0,\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill cm{\chi }_{\xi \xi }-[b(a-1)-c_1]\chi +2c{m}^{-1}{\chi }^3& =& 0,\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill mv-2c{\chi }^2-m{c}_1& =& 0.\hfill \end{array}$$

(34)

Hence, we obtain

$$\begin{array}{ccc}\hfill m& =& \frac{c(1-a)}{b},\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\chi }_{\xi \xi }& =& \frac{b(a-1)-n}{cm}\chi -2m^{-2}{\chi }^3,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill v& =& 2m^{-1}c{\chi }^2+c_1.\hfill \end{array}$$

(37)

It is well known that the solutions of Equation (36) are provided by the Jacobi elliptic functions $cn$ and $dn$. It is well known from the literature that these functions ($cn$ and $dn$) satisfy the following equations [12]:

$$\begin{array}{ccc}\hfill {\chi }_{\xi \xi }+(1-2k^2)\chi +2k^2{\chi }^3& =& 0,\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill {\chi }_{\xi \xi }-(2-k^2)\chi +2{\chi }^3& =& 0,\hfill \end{array}$$

(39)

respectively. The corresponding two solutions of the K-IIE are given by

$$\begin{array}{ccc}\hfill q_1& =& cn\left(\xi \right|k)e^{i(ax+bt+\delta )},\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill v_1& =& 2c{d}^{-1}c{n}^2\left(\xi \right|k)+c_1,\hfill \end{array}$$

(41)

and

$$\begin{array}{ccc}\hfill q_2& =& dn\left(\xi \right|k)e^{i(ax+bt+\delta )},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill v_2& =& 2c{m}^{-1}d{n}^2(\xi ,k)+c_1,\hfill \end{array}$$

(43)

respectively. If $k=1$, from these solutions, we obtain the following 1-soliton solution of the K-IIE

$$\begin{array}{ccc}\hfill q& =& \frac{\alpha }{cosh\xi }{e}^{i(ax+bt+\delta )}\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill v& =& \frac{2cϵ}{mcos{h}^2\xi }+c_1,\hfill \end{array}$$

(45)

where

$$\begin{array}{c}\hfill \alpha =\pm \frac{m}{\sqrt{ϵ}},c_1=-c{m}^{-1}[{(a-1)}^2+m^2],b=c{m}^{-1}(1-a).\end{array}$$

(46)

This 1-soliton solution represents a wave traveling that propagates with constant speed and shape [13].

3.2. Hirota Bilinear Form

One of the most powerful methods of the construction of exact and explicit solutions of the nonlinear differential equations is the famous Hirota bilinear method [14]. In this subsection, we apply it to solve the K-IIAE.

3.2.1. K-IIAE

To construct the N-soliton solution, we can use the Hirota bilinear form of the K-IIAE. It can be obtained by using the following transformation

$$\begin{array}{c}\hfill q=\frac{h}{\varphi },v=2{(ln\varphi )}_{xt},\end{array}$$

(47)

where h is a complex function and $\varphi$ is a real function. Then, we obtain the following Hirota bilinear equations

$$\begin{array}{ccc}\hfill [i{D}_t+D_x{D}_t](h\circ \varphi )& =& 0,\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill D_x^2(\varphi \circ \varphi )-2ϵ\overline{h}h& =& 0,\hfill \end{array}$$

(49)

where the Hirota D-operators are defined as

$$\begin{array}{c}\hfill D_x^{n}f\left(x\right)\circ g\left(x\right)={\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial x^{\prime }}\right)}^{n}f\left(x\right)g\left(x^{\prime }\right){|}_{x=x^{\prime }}.\end{array}$$

(50)

The 1-soliton solution that we look for is

$$\begin{array}{c}\hfill h=e^{\chi },\varphi =1+{\varphi }_2=1+\frac{e^{(\chi +\overline{\chi })}}{2b},\end{array}$$

(51)

where $\chi =i(ax+bt+\delta )$, $(a=const,b=const,\delta =const)$. Finally, we obtain the 1-soliton solution of the form (44)–(45). Proceeding in the standard way, we can construct the N-soliton solutions of the K-IIAE.

3.2.2. K-IIBE

Similarly, we can construct the soliton solutions of the K-IIBE via the Hirota bilinear method. The corresponding bilinear equations read as

$$\begin{array}{ccc}\hfill [i{D}_x+D_x{D}_t](h\circ \varphi )& =& 0,\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill D_t^2(\varphi \circ \varphi )-2ϵ\overline{h}h& =& 0.\hfill \end{array}$$

(53)

4. Integrable Motion of Space Curves Induced by the K-IIE

It is well known that in 1 + 1 and 2 + 1 dimensions, there exists geometrical equivalence between spin systems and nonlinear Schrödinger-type equations [1,2,3,4,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37], which we call the Lakshmanan equivalence or, in short, the L-equivalence. In this section, we find the L-equivalent counterpart of the K-IIAE (5)–(7). For this purpose, in this section, we want to study the integrable motion of space curves induced by the K-IIAE (5)–(7). For this purpose, consider a moving space curve in $R^3$ parametrized by the arclength x. It is well known that such a space curve is governed by the following spatial and temporal Serret–Frenet equations (SFE)

$$\begin{array}{c}\hfill {\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right)}_x=C\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right),{\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right)}_t=D\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right),\end{array}$$

(54)

where

$$C=\left(\begin{array}{ccc}0& \kappa & \sigma \\ -\kappa & 0& \tau \\ -\sigma & -\tau & 0\end{array}\right),D=\left(\begin{array}{ccc}0& {\omega }_3& {\omega }_2\\ -{\omega }_3& 0& {\omega }_1\\ -{\omega }_2& -{\omega }_1& 0\end{array}\right).$$

(55)

Here, $\kappa$ and $\sigma$ are the geodesic and normal curvatures of the space curve, $\tau$ is its torsion, and ${\omega }_j(j=1,2,3)$ are some real functions describing the motion of space curves. The latter functions must be expressed in terms of $\kappa ,\sigma ,\tau$ and their derivatives. Note that the SFE can be rewritten as

$$\begin{array}{c}\hfill {\mathbf{e}}_{ix}=\mathbf{C}\wedge {\mathbf{e}}_i,{\mathbf{e}}_{it}=\mathbf{D}\wedge {\mathbf{e}}_i,\end{array}$$

(56)

where

$$\begin{array}{c}\hfill \mathbf{C}=\tau {\mathbf{e}}_1+\sigma {\mathbf{e}}_2+\kappa {\mathbf{e}}_3,\mathbf{D}=({\omega }_1,{\omega }_2,{\omega }_3)\end{array}$$

(57)

and ${\mathbf{e}}_i$s, $i=1,2,3,$ form the orthogonal trihedral. The compatibility condition of the linear Equation (54) reads as

$$C_t-D_x+[C,D]=0$$

(58)

or

$$\begin{array}{ccc}\hfill {\kappa }_t& =& {\omega }_{3x}-\tau {\omega }_2+\sigma {\omega }_1,\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill {\sigma }_t& =& {\omega }_{2x}-\kappa {\omega }_1+\tau {\omega }_3,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill {\tau }_t& =& {\omega }_{1x}-\sigma {\omega }_3+\kappa {\omega }_2.\hfill \end{array}$$

(61)

Let us now assume that functions $\tau ,\sigma ,\kappa$ have the following forms

$$\tau =-id(r+q),\sigma =d(r-q),\kappa =2d\lambda ,$$

(62)

$${\omega }_1=\frac{d}{1-2d\lambda }(r_t-q_t),{\omega }_2=\frac{di}{1-2d\lambda }(r_t+q_t),{\omega }_3=-v,$$

(63)

where $r,q$ are some complex functions, v is a real function and $d=const$. Substituting these expressions into the set (59)–(61), we obtain the following equations for the functions $q,r,v$:

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}-vq& =& 0,\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+vr& =& 0,\hfill \end{array}$$

(65)

$$\begin{array}{ccc}\hfill v_x+2d^2{\left(rq\right)}_t& =& 0.\hfill \end{array}$$

(66)

It is nothing but the K-IIAE (5)–(7). Therefore, we have constructed the integrable motion of the space curves induced by the K-IIAE. In this case, it is not difficult to verify that the unit vector ${\mathbf{e}}_3$ satisfies the following set of equations

$$\begin{array}{ccc}\hfill {\mathbf{e}}_{3t}-{\mathbf{e}}_3\wedge {\mathbf{e}}_{3xt}-u{\mathbf{e}}_{3x}& =& 0,\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{e}}_{3x}^2\right)}_t& =& 0.\hfill \end{array}$$

(68)

This set of equations is the geometrical or Lakshmanan equivalent counterpart of the K-IIAE (5)–(7). Note that after the identification ${\mathbf{e}}_3\equiv \mathbf{S}$, Equations (67) and (68) take the form of the K-IAE (3) and (4). Thus, this result proves that the K-IAE and the K-IIAE are geometrically equivalent to each other.

5. Gauge Equivalent Counterpart of the K-IIE

In the previous section, we obtained the geometrical equivalent of the K-IIAE, which has the form (67) and (68). In this section, our aim is to find the gauge-equivalent counterpart of the K-IIAE.

5.1. Derivation of the K-IAE

Thus, in this section, we want to find the gauge-equivalent partner of the K-IIAE. To do this, we consider the following gauge transformation

$$\begin{array}{c}\hfill \Psi =g^{-1}\Phi ,\end{array}$$

(69)

where $\Phi$ is the solution of Equations (8) and (9) and ${g(x,t)=\Phi |}_{\lambda =0}$. After some standard algebra [11], we obtain the following equations for the new function $\Psi$:

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_1\Psi ,\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_1\Psi ,\hfill \end{array}$$

(71)

where

$$\begin{array}{c}\hfill U_1=-i\lambda S,V_1=\frac{2\lambda }{1-2\lambda }Z,Z=0.25([S,S_t]+2iuS).\end{array}$$

(72)

Here,

$$\begin{array}{c}\hfill S=g^{-1}{\sigma }_{3}g.\end{array}$$

(73)

The compatibility condition

$$\begin{array}{c}\hfill U_{1t}-V_{1x}+[U_1,V_1]=0\end{array}$$

(74)

is equivalent to the following Kuralay-IA equation (K-IAE):

$$\begin{array}{ccc}\hfill i{S}_t& =& \frac{1}{2}[S,S_{xt}]+iu{S}_x,\hfill \end{array}$$

(75)

$$\begin{array}{ccc}\hfill u_x& =& \frac{i}{4}tr(S·[S_x,S_t]),\hfill \end{array}$$

(76)

or

$$\begin{array}{ccc}\hfill i{S}_t& =& \frac{1}{2}[S,S_{xt}]+iu{S}_x,\hfill \end{array}$$

(77)

$$\begin{array}{ccc}\hfill u_x& =& -\frac{1}{4}tr\left({\left(S_x^2\right)}_t)\right),\hfill \end{array}$$

(78)

where

$$\begin{array}{c}\hfill S=\left(\begin{array}{cc}{S}_3& S^{-}\\ S^{+}& -S_3\end{array}\right),S^2=I,S^{\pm }=S_1\pm i{S}_2.\end{array}$$

(79)

This K-IAE is one of examples of an integrable spin system (see, e.g., [1,2,3,4] and references therein). The solutions of the K-IE and the K-IIE are related by the following formulas:

$$\begin{array}{c}\hfill tr\left(S_x^2\right)=8{\left|q\right|}^2=2{\mathbf{S}}_x^2.\end{array}$$

(80)

and

$$\begin{array}{c}\hfill -2i\mathbf{S}·({\mathbf{S}}_x\wedge {\mathbf{S}}_{xx})=tr\left(S{S}_x{S}_{xx}\right)=4(\overline{q}{q}_x-{\overline{q}}_{x}q).\end{array}$$

(81)

The K-IAE can be written in the vector form as [5]

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(83)

where $\mathbf{S}=(S_1,S_2,S_3)$ is the unit spin vector, ${\mathbf{S}}^2=S_1^2+S_2^2+S_3^2=1$, ${\mathbf{S}}_x^2=S_{1x}^2+S_{2x}^2+S_{3x}^2$ and u is the real scalar function (potential). Using the stereographic projection, one can obtain the following new form of the K-IAE:

$$\begin{array}{ccc}\hfill i{w}_t+{\omega }_{xt}-u{w}_x-\frac{2\overline{w}{w}_x{w}_t}{1+{\left|w\right|}^2}& =& 0,\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill u_x+\frac{2i(w_x{\overline{w}}_t-{\overline{w}}_x{w}_t)}{{(1+|w|}^2{)}^2}& =& 0.\hfill \end{array}$$

(85)

Here,

$$\begin{array}{c}\hfill S^{+}=S_1+i{S}_2=\frac{2w}{1+{\left|w\right|}^2},S_3=\frac{1-{\left|w\right|}^2}{1+{\left|w\right|}^2},\end{array}$$

(86)

and

$$\begin{array}{c}\hfill w=\frac{S^{+}}{1+S_3}.\end{array}$$

(87)

5.2. Derivation of the K-IBE

Analogically, we can derive the K-IBE. It has the form

$$\begin{array}{ccc}\hfill i{S}_x& =& \frac{1}{2}[S,S_{xt}]+iu{S}_t,\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill u_t& =& -\frac{1}{4}tr\left({\left(S_t^2\right)}_x)\right),\hfill \end{array}$$

(89)

or

$$\begin{array}{ccc}\hfill i{w}_x+{\omega }_{xt}-u{w}_t-\frac{2\overline{w}{w}_x{w}_t}{1+{\left|w\right|}^2}& =& 0,\hfill \end{array}$$

(90)

$$\begin{array}{ccc}\hfill u_t+\frac{2i(w_t{\overline{w}}_x-{\overline{w}}_t{w}_x)}{{(1+|w|}^2{)}^2}& =& 0.\hfill \end{array}$$

(91)

6. Soliton Solutions of the K-IE

There are two methods for the construction the exact solutions of a spin system such as the K-IE. One is to find solutions of the spin system using the corresponding solutions of its gauge-equivalent counterpart. Another method is finding the solutions of the spin system directly.

6.1. Solutions from Gauge Equivalence

The gauge equivalence between two equations allows us to construct the solutions of one equation using the solutions of the other gauge-equivalent equation. Here, we use this approach to find solutions of the K-IAE. Let the seed solution of the K-IIAE have the form $r=q=0,v=2c$. Then, the associated linear system (8) and (9) takes the form

$$\begin{array}{ccc}\hfill {\Phi }_{0x}& =& id\lambda {\sigma }_3{\Phi }_0,\hfill \end{array}$$

(92)

$$\begin{array}{ccc}\hfill {\Phi }_{0t}& =& -\frac{ic}{1-2d\lambda }{\sigma }_3{\Phi }_0,\hfill \end{array}$$

(93)

where

$${\Phi }_0=\left(\begin{array}{cc}{\varphi }_{01}& -{\overline{\varphi }}_{02}\\ {\varphi }_{02}& {\overline{\varphi }}_{01}\end{array}\right),{\Phi }_0^{-1}=\frac{1}{det{\Phi }_0}\left(\begin{array}{cc}{\overline{\varphi }}_{01}& {\overline{\varphi }}_{02}\\ -{\varphi }_{02}& {\varphi }_{01}\end{array}\right),det{\Phi }_0=|{\varphi }_{01}{|}^2+{\left|{\varphi }_{02}\right|}^2.$$

(94)

The corresponding solution of the linear Equations (92) and (93) has the form

$$\begin{array}{c}\hfill {\varphi }_{01}=c_1{e}^{-\chi },{\varphi }_{02}=c_2{e}^{\chi +i{\delta }_{21}},\end{array}$$

(95)

where $c_j$ are complex constants, $\chi ={\chi }_1+i{\chi }_2=i(d\lambda -\frac{c}{1-2d\lambda }t+{\delta }_1),{\delta }_{21}={\delta }_2-{\delta }_1,\lambda =\alpha +i\beta$ and ${\delta }_j,\alpha ,\beta$ are real constants. For the spin matrix S, we have

$$S=\left(\begin{array}{cc}{S}_3& S^{-}\\ S^{+}& -S_3\end{array}\right)={\Phi }_0^{-1}{\sigma }_3{\Phi }_0=\left(\begin{array}{cc}|{\varphi }_{01}{|}^2-{\left|{\varphi }_{02}\right|}^2& -2{\overline{\varphi }}_{01}{\overline{\varphi }}_{02}\\ -2{\varphi }_{01}{\varphi }_{02}& |{\varphi }_{02}{|}^2-{\left|{\varphi }_{01}\right|}^2\end{array}\right).$$

(96)

For the components of the spin matrix S, we obtain the following expressions

$$S_3=\frac{|{\varphi }_{01}{|}^2-{\left|{\varphi }_{02}\right|}^2}{det{\Phi }_0},S^{+}=-\frac{2{\varphi }_{01}{\varphi }_{02}}{det{\Phi }_0}.$$

(97)

Substituting the expressions for the functions ${\varphi }_{oj}$ into the formula (97), we obtain the following 1-soliton solution of the K-IE as

$$S_3=\frac{|c_1{|}^2{e}^{-2{\chi }_1}-{\left|c_2\right|}^2{e}^{2{\chi }_1}}{|c_1{|}^2{e}^{-2{\chi }_1}+{\left|c_2\right|}^2{e}^{2{\chi }_1}},S^{+}=-\frac{2c_1{c}_2{e}^{i{\delta }_{21}}}{|c_1{|}^2{e}^{-2{\chi }_1}+{\left|c_2\right|}^2{e}^{2{\chi }_1}}.$$

(98)

or

$$S_3=-tanh\left(2{\chi }_1\right)=1-\frac{e^{2{\chi }_1}}{|c_1|cosh\left(2{\chi }_1\right)},S^{+}=-\frac{e^{i({\delta }_{21}+{ϵ}_1+{ϵ}_2)}}{cosh\left(2{\chi }_1\right)},S^{-}=\overline{S^{+}},$$

(99)

where $c_j=\left|c_j\right|e^{i{ϵ}_j}$. Thus, using the gauge equivalence between two Kuralay equations, we have constructed the 1-soliton solution of the K-IE. Finally, we note that the functions ${\varphi }_{oj}$ are the analogies of eigenvectors. As it is well known, these eigenvectors are closely related to the soliton problems (see, e.g., [38,39,40] and references therein).

6.2. Hirota Bilinear Form of the K-IE

To construct the N-soliton solution of the K-IE, we can use the Hirota bilinear method. For this purpose, we consider the w-form of the K-IAE. Consider the transformation

$$\begin{array}{c}\hfill \omega =\frac{g}{f},\end{array}$$

(100)

where f and g are some complex valued functions. Substituting this expression into the Kuralay-I equation, after some algebra, we obtain the following bilinear form

$$\begin{array}{ccc}\hfill (i{D}_t-D_x{D}_t)(\overline{f}\circ g)& =& 0,\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill (i{D}_t-D_x{D}_t)(\overline{f}\circ f-\overline{g}\circ g)& =& 0,\hfill \end{array}$$

(102)

$$\begin{array}{ccc}\hfill D_x(\overline{f}\circ f+\overline{g}\circ g)& =& 0,\hfill \end{array}$$

(103)

and

$$\begin{array}{c}\hfill u=-\frac{i{D}_t(\overline{f}\circ f+\overline{g}\circ g)}{\overline{f}\circ f+\overline{g}\circ g}.\end{array}$$

(104)

Here, $D_x$ is the Hirota bilinear operator, defined by

$$\begin{array}{c}\hfill D_x^k{D}_t^n(f\circ g)={({\partial }_x-{\partial }_{x^{\prime }})}^k{({\partial }_t-{\partial }_{t^{\prime }})}^{n}f(x,t)g(x,t){\parallel }_{x=x^{\prime },t=t^{\prime }}.\end{array}$$

(105)

Note that from the definition of the D-operator, it follows that

$$\begin{array}{c}\hfill u_x=-2i\left[D_t(f\circ g)D_x(\overline{f}\circ \overline{g})-c·c\right].\end{array}$$

(106)

On the other hand, the spin field takes the form

$$\begin{array}{c}\hfill S^{+}=\frac{2\overline{f}g}{\mid f{\mid }^2+\mid g{\mid }^2},S_3=\frac{\mid f{\mid }^2-\mid g{\mid }^2}{\mid f{\mid }^2+\mid g{\mid }^2}.\end{array}$$

(107)

The bilinear form of the K-IE represents the starting point to obtain interesting classes of its solutions. The construction of the solutions is standard. One expands the functions g and f as a series

$$\begin{array}{ccc}\hfill g& =& ϵg_1+{ϵ}^3{g}_3+{ϵ}^5{g}_5+\cdots \cdots \hfill \end{array}$$

(108)

$$\begin{array}{ccc}\hfill f& =& 1+{ϵ}^2{f}_2+{ϵ}^4{f}_4+{ϵ}^6{f}_6+\cdots \cdots \hfill \end{array}$$

(109)

Substituting these expansions into (101)–(103) and equating the coefficients of $ϵ$, one obtains the following system of equations from (101):

$$\begin{array}{ccc}\hfill {ϵ}^1& :& i{g}_{1t}+g_{1xt}=0,\hfill \end{array}$$

(110)

$$\begin{array}{ccc}\hfill {ϵ}^3& :& \left[i{\partial }_t+{\partial }_x{\partial }_t\right]g_3=\left[i{D}_t-D_x{D}_t\right]({\overline{f}}_2.g_1),\hfill \end{array}$$

(111)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(112)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(113)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(114)

$$\begin{array}{ccc}\hfill {ϵ}^{2n+1}& :& \left[i{\partial }_t+{\partial }_x{\partial }_t\right]g_{2n+1}=\sum _{k+m=n}\left[i{D}_t-D_x{D}_t\right]({\overline{f}}_{2k}.g_{2m+1}),\hfill \end{array}$$

(115)

and from (102):

$$\begin{array}{ccc}\hfill {ϵ}^2& :& i{\partial }_t({\overline{f}}_2-f_2)-{\partial }_x{\partial }_t({\overline{f}}_2+f_2)=\left[i{D}_t-D_x{D}_t\right]({\overline{g}}_1.g_1),\hfill \end{array}$$

(116)

$$\begin{array}{ccc}\hfill {ϵ}^4& :& i{\partial }_t({\overline{f}}_4-f_4)-{\partial }_x{\partial }_t({\overline{f}}_4+f_4)=\left[i{D}_t-D_x{D}_t\right]({\overline{g}}_1.g_3+{\overline{g}}_3.g_1-{\overline{f}}_2.f_2),\hfill \end{array}$$

(117)

$$\begin{array}{c}\hfill ⋮\end{array}$$

(118)

$$\begin{array}{ccc}\hfill {ϵ}^{2n}& :& i{\partial }_t({\overline{f}}_{2n}-f_{2n})-{\partial }_x{\partial }_t({\overline{f}}_{2n}+f_{2n})=\hfill \end{array}$$

(119)

$$\begin{array}{ccc}& & (i{D}_t-D_x{D}_t)\left(\sum _{n_1+n_2=n-1}{\overline{g}}_{2n_1+1}.g_{2n_2+1}\right)-(i{D}_t-D_x{D}_t)\left(\sum _{m_1+m_2=n}{\overline{f}}_{2m_1}.f_{2m_2}\right).\hfill \end{array}$$

(120)

Further from (103), we have the following:

$$\begin{array}{ccc}\hfill {ϵ}^2& :& {\partial }_x({\overline{f}}_2-f_2)=-D_x({\overline{g}}_1.g_1),\hfill \end{array}$$

(121)

$$\begin{array}{ccc}\hfill {ϵ}^4& :& {\partial }_x({\overline{f}}_4-f_4)=-D_x({\overline{g}}_1.g_3+{\overline{g}}_3.g_1+{\overline{f}}_2.f_2),\hfill \end{array}$$

(122)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(123)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(124)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(125)

$$\begin{array}{ccc}\hfill {ϵ}^{2n}& :& {\partial }_x({\overline{f}}_{2n}-f_{2n})=-D_x\left[\sum _{n_1+n_2=n-1}({\overline{g}}_{2n_1+1}.g_{2n_{(}2+1)}+\sum _{n_1+n_2=n}{\overline{f}}_{2n_1}.f_{2n_2})\right].\hfill \end{array}$$

(126)

Solving recursively the above equations, we obtain many interesting classes of solutions to the K-IE. At last, we note that the functions $g,f$ are the analogies of eigenvectors. As it is well known, these eigenvectors are closely related to the soliton problems (see, e.g., [38,39,40] and references therein). However, in this paper, to construct the exact and explicit solutions of the Kuralay equations, we use a more modern and effective tool, namely the Hirota bilinear method.

7. Nonlocal KE

Recently, there has been significant interest in studying the nonlocal integrable NDE [41,42,43]. In the previous sections, we have considered the local Kuralay equations. In this section, let us we present some main results about the nonlocal Kuralay equations. In particular, the nonlocal K-IIE has the form

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}-vq& =& 0,\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xx}+vr& =& 0,\hfill \end{array}$$

(128)

$$\begin{array}{ccc}\hfill v_x-2ϵ{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(129)

where

$$\begin{array}{c}\hfill r=k\overline{q}({ϵ}_{1}x,{ϵ}_{2}t),r=kq({ϵ}_{1}x,{ϵ}_{2}t),k=\pm 1,{ϵ}_j^2=1\end{array}$$

(130)

or

$$\begin{array}{ccc}\hfill r& =& k\overline{q}(-x,t),r=k\overline{q}(x,-t),r=k\overline{q}(-x,-t),\hfill \end{array}$$

(131)

$$\begin{array}{ccc}\hfill r& =& kq(-x,t),r=kq(x,-t),r=kq(-x,-t).\hfill \end{array}$$

(132)

The gauge equivalent spin system corresponding to the K-IIE is given by (3) and (4). However, here, we must note that in contrast to the local case, in our nonlocal case, in the Serret–Frenet Equation (54), the curvatures $\kappa (t,x)$ and $\sigma (t,x)$, the torsion $\tau (t,x)$ and ${\omega }_j(t,x)$ are complex-valued functions. As a result, in the nonlocal case, the spin matrix S is not Hermitian and has $PT$ symmetry $S(t,x)={\sigma }_3{S}^{+}(t,-x){\sigma }_3$. The corresponding spin vector $\mathbf{S}(t,x)=(S_1(t,x),S_2(t,x),S_3(t,x))$ is a complex-valued vector. As mentioned above, in the nonlocal case, the spin matrix $S(t,x)$ is not Hermitian. However, we can decompose it as the sum of a Hermitian matrix and a skew-Hermitian matrix as [44]

$$S=M+iL,$$

(133)

where

$$M=\frac{1}{2}(S^{+}+S),L=\frac{i}{2}(S^{+}-S).$$

(134)

Next, we use the standard Pauli matrix representations of these matrices: $M=\mathbf{m}·\sigma ,L=\mathbf{l}·\sigma$, where $\mathbf{m}$ and $\mathbf{l}$ are real-valued vector functions. From $\mathbf{S}=\mathbf{m}+i\mathbf{l}$ and ${\mathbf{S}}^2=1$, we obtain

$${\mathbf{m}}^2-{\mathbf{l}}^2=1,\mathbf{m}·\mathbf{l}=0.$$

(135)

Finally, we obtain the following nonlocal Kuralay-I equation

$$\begin{array}{ccc}\hfill {\mathbf{m}}_t-\mathbf{m}\wedge {\mathbf{m}}_{xt}+\mathbf{l}\wedge {\mathbf{l}}_{tx}-(u_1{\mathbf{m}}_x-u_2{\mathbf{l}}_x)& =& 0,\hfill \end{array}$$

(136)

$$\begin{array}{ccc}\hfill {\mathbf{l}}_t-\mathbf{m}\wedge {\mathbf{l}}_{xt}-\mathbf{l}\wedge {\mathbf{m}}_{xt}-(u_1{\mathbf{l}}_x+u_2{\mathbf{m}}_x)& =& 0,\hfill \end{array}$$

(137)

$$\begin{array}{ccc}\hfill u_{1x}-\frac{1}{2}({\mathbf{m}}_x^2-{\mathbf{l}}_x^2)& =& 0,\hfill \end{array}$$

(138)

$$\begin{array}{ccc}\hfill u_{2x}-{\mathbf{m}}_x·{\mathbf{l}}_x& =& 0,\hfill \end{array}$$

(139)

where $u_j$ are real functions and $u=u_1+i{u}_2$. This nonlocal K-IE is integrable. Its Lax representation is given by

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_4\Psi ,\hfill \end{array}$$

(140)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_4\Psi .\hfill \end{array}$$

(141)

Here,

$$\begin{array}{c}\hfill U_4=-i\lambda (M+iL),V_4=\frac{2\lambda }{1-2\lambda }Z,\end{array}$$

(142)

where

$$\begin{array}{c}\hfill Z=0.25\left(([M,M_t]-[L,L_t])+i([M,L_t]+[L_t,M])+2iu(M+iL)\right).\end{array}$$

(143)

8. Dispersionless KE

To find the dispersionless limit of the Kuralay-II equation, we consider the following representation of the function $q(x,t)$:

$$\begin{array}{c}\hfill q=\sqrt{f}{e}^{\frac{is}{ϵ}},\end{array}$$

(144)

where $f,s$ are some functions, and $ϵ$ is a real parameter. Substituting this expression into the K-IIE, we obtain the following set of equations

$$\begin{array}{ccc}\hfill s_t-s_x{s}_t+v& =& 0,\hfill \end{array}$$

(145)

$$\begin{array}{ccc}\hfill f_t-s_t{f}_x-s_x{f}_t& =& 0,\hfill \end{array}$$

(146)

$$\begin{array}{ccc}\hfill v_x-2\delta f_t& =& 0.\hfill \end{array}$$

(147)

This is the desired dispersionless Kuralay-II equation. It is integrable.

9. Some Integrable Generalizations of the K-IIE

The Kuralay equations admit several generalizations. As examples, here, we present some of them: the Zhaidary equation, the two-component Kuralay-II equation, multicomponent generalization and so on.

9.1. Integrable Zhaidary Equation

9.1.1. Case 1: Z-IIAE

One of integrable generalizations of the K-IIE is the following Zhaidary-IIA equation (Z-IIAE) [5,6,7,8,9]:

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}+4ic{\left(vq\right)}_x-2d^{2}vq& =& 0,\hfill \end{array}$$

(148)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+4ic{\left(vr\right)}_x+2d^{2}vr& =& 0,\hfill \end{array}$$

(149)

$$\begin{array}{ccc}\hfill v_x-{\left(rq\right)}_t& =& 0.\hfill \end{array}$$

(150)

Hence, as $c=0$, we have the K-IIAE

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}-2d^{2}vq& =& 0,\hfill \end{array}$$

(151)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+2d^{2}vr& =& 0,\hfill \end{array}$$

(152)

$$\begin{array}{ccc}\hfill v_x-{\left(rq\right)}_t& =& 0.\hfill \end{array}$$

(153)

Thus, the K-IIAE is the particular reduction of the Z-IIAE. Note that the ZE (148)–(150) is integrable with the following LR:

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_5\Phi ,\hfill \end{array}$$

(154)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_5\Phi ,\hfill \end{array}$$

(155)

where

$$\begin{array}{ccc}\hfill U_5& =& [i(c{\lambda }^2+d\lambda ){\sigma }_3+(2c\lambda +d)Q,\hfill \end{array}$$

(156)

$$\begin{array}{ccc}\hfill V_5& =& \frac{1}{1-2c{\lambda }^2-2d\lambda }({\lambda }^2{B}_2+\lambda B_1+B_0).\hfill \end{array}$$

(157)

Here,

$$\begin{array}{c}\hfill B_2=-4ic{\sigma }_3,B_1=-4icdv{\sigma }_3-2ic{\sigma }_3{Q}_t-8c^{2}vQ,B_0=\frac{d}{2c}{B}_1-\frac{d^2}{4c^2}{B}_2,\end{array}$$

(158)

and

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}0& q\\ r& 0\end{array}\right),r=ϵ\overline{q},ϵ=\pm 1.\end{array}$$

(159)

The compatibility condition

$$\begin{array}{c}\hfill U_{5t}-V_{5x}+[U_5,V_5]=0\end{array}$$

(160)

gives the ZE (148)–(150). Thus, we have proven that as $c=0$, the Zhaidary equation reduces to the KE, so that the ZE is one of the integrable generalizations of the KE.

9.1.2. Case 2: Z-IIBE

The second form of the ZE (148)–(150) can be written as

$$\begin{array}{ccc}\hfill i{q}_x-q_{xt}+4ic{\left(vq\right)}_t-2d^{2}vq& =& 0,\hfill \end{array}$$

(161)

$$\begin{array}{ccc}\hfill i{r}_x+r_{xt}+4ic{\left(vr\right)}_t+2d^{2}vr& =& 0,\hfill \end{array}$$

(162)

$$\begin{array}{ccc}\hfill v_t-{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(163)

which is the Zhaidary-IIB equation (Z-IIBE). Hence, as $c=0$, we have the following K-IIBE (15)–(17):

$$\begin{array}{ccc}\hfill i{q}_x-q_{xt}-2d^{2}vq& =& 0,\hfill \end{array}$$

(164)

$$\begin{array}{ccc}\hfill i{r}_x+r_{xt}+2d^{2}vr& =& 0,\hfill \end{array}$$

(165)

$$\begin{array}{ccc}\hfill v_t-{\left(rq\right)}_x& =& 0.\hfill \end{array}$$

(166)

As in Case 1, the Z-IIBE (161)–(163) is also integrable with the following LR:

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_6\Phi ,\hfill \end{array}$$

(167)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_6\Phi ,\hfill \end{array}$$

(168)

where

$$\begin{array}{ccc}\hfill U_6& =& [i(c{\lambda }^2+d\lambda ){\sigma }_3+(2c\lambda +d)Q,\hfill \end{array}$$

(169)

$$\begin{array}{ccc}\hfill V_6& =& \frac{1}{1-2c{\lambda }^2-2d\lambda }({\lambda }^2{B}_2+\lambda B_1+B_0).\hfill \end{array}$$

(170)

Here,

$$\begin{array}{c}\hfill B_2=-4ic{\sigma }_3,B_1=-4icdv{\sigma }_3-2ic{\sigma }_3{Q}_x-8c^{2}vQ,B_0=\frac{d}{2c}{B}_1-\frac{d^2}{4c^2}{B}_2,\end{array}$$

(171)

and

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}0& q\\ r& 0\end{array}\right),r=ϵ\overline{q},ϵ=\pm 1.\end{array}$$

(172)

The compatibility condition

$$\begin{array}{c}\hfill U_{6x}-V_{6t}+[U_6,V_6]=0\end{array}$$

(173)

gives the Z-IIBE (161)–(163). Thus, we have proven that as $c=0$, the Zhaidary-I equation reduces to the K-IE in Case 1 and in Case 2.

9.2. Integrable Two-Component K-IIE

The KE admits the vector generalizations. As an example, here, we present the integrable two-component Kuralay-IIA equation (K-IIAE).

9.2.1. Integrable Two-Component K-IIAE

The integrable two-component K-IIAE has the form [5,6,7,8,9]

$$\begin{array}{ccc}\hfill i{q}_{1t}+q_{1xt}-(v_1+0.5v_2)q_1-w_1{q}_2& =& 0,\hfill \end{array}$$

(174)

$$\begin{array}{ccc}\hfill i{q}_{2t}+q_{2xt}-(v_1+0.5v_2)q_2-w_2{q}_1& =& 0,\hfill \end{array}$$

(175)

$$\begin{array}{ccc}\hfill i{r}_{1t}-r_{1xt}+(v_1+0.5v_2)r_1+w_2{r}_2& =& 0,\hfill \end{array}$$

(176)

$$\begin{array}{ccc}\hfill i{r}_{2t}-r_{2xt}+(v_1+0.5v_2)r_2+w_1{r}_1& =& 0,\hfill \end{array}$$

(177)

$$\begin{array}{ccc}\hfill v_{1x}-2b^2{\left(r_1{q}_1\right)}_t& =& 0,\hfill \end{array}$$

(178)

$$\begin{array}{ccc}\hfill v_{2x}-2b^2{\left(r_2{q}_2\right)}_t& =& 0,\hfill \end{array}$$

(179)

$$\begin{array}{ccc}\hfill w_{1x}-b^2{\left(r_2{q}_1\right)}_t& =& 0,\hfill \end{array}$$

(180)

$$\begin{array}{ccc}\hfill w_{2x}-b^2{\left(r_1{q}_2\right)}_t& =& 0.\hfill \end{array}$$

(181)

The LR of this two-component K-IIAE is given by

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_7\Phi ,\hfill \end{array}$$

(182)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_7\Phi ,\hfill \end{array}$$

(183)

with

$$\begin{array}{ccc}\hfill U_7& =& -ia\lambda \Sigma +bQ,\hfill \end{array}$$

(184)

$$\begin{array}{ccc}\hfill V_7& =& \frac{1}{1-2d\lambda }B.\hfill \end{array}$$

(185)

Here,

$$\begin{array}{c}\hfill B=\left(\begin{array}{ccc}0.5i(v_1+v_2)& ib{q}_{1t}& ib{q}_{2t}\\ -ib{r}_{1t}& 0.5i{v}_1& i{w}_2\\ -ib{r}_{2t}& i{w}_1& 0.5i{v}_2\end{array}\right),Q=\left(\begin{array}{ccc}0& q_1& q_2\\ r_1& 0& 0\\ r_2& 0& 0\end{array}\right),\Sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right).\end{array}$$

(186)

The compatibility condition

$$\begin{array}{c}\hfill U_{7t}-V_{7x}+[U_7,V_7]=0\end{array}$$

(187)

gives the two-component K-IIAE (174)–(181).

9.2.2. Integrable Two-Component K-IIBE

Similarly, the integrable two-component K-IIBE is given by [5,6,7,8,9]

$$\begin{array}{ccc}\hfill i{q}_{1x}+q_{1xt}-(v_1+0.5v_2)q_1-w_1{q}_2& =& 0,\hfill \end{array}$$

(188)

$$\begin{array}{ccc}\hfill i{q}_{2x}+q_{2xt}-(v_1+0.5v_2)q_2-w_2{q}_1& =& 0,\hfill \end{array}$$

(189)

$$\begin{array}{ccc}\hfill i{r}_{1x}-r_{1xt}+(v_1+0.5v_2)r_1+w_2{r}_2& =& 0,\hfill \end{array}$$

(190)

$$\begin{array}{ccc}\hfill i{r}_{2x}-r_{2xt}+(v_1+0.5v_2)r_2+w_1{r}_1& =& 0,\hfill \end{array}$$

(191)

$$\begin{array}{ccc}\hfill v_{1t}-2b^2{\left(r_1{q}_1\right)}_x& =& 0,\hfill \end{array}$$

(192)

$$\begin{array}{ccc}\hfill v_{2t}-2b^2{\left(r_2{q}_2\right)}_x& =& 0,\hfill \end{array}$$

(193)

$$\begin{array}{ccc}\hfill w_{1t}-b^2{\left(r_2{q}_1\right)}_x& =& 0,\hfill \end{array}$$

(194)

$$\begin{array}{ccc}\hfill w_{2t}-b^2{\left(r_1{q}_2\right)}_x& =& 0.\hfill \end{array}$$

(195)

The corresponding LR of this two-component K-IIBE reads as

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_7\Phi ,\hfill \end{array}$$

(196)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_7\Phi ,\hfill \end{array}$$

(197)

with

$$\begin{array}{ccc}\hfill U_7& =& -ia\lambda \Sigma +bQ,\hfill \end{array}$$

(198)

$$\begin{array}{ccc}\hfill V_7& =& \frac{1}{1-2d\lambda }B.\hfill \end{array}$$

(199)

Here,

$$\begin{array}{c}\hfill B=\left(\begin{array}{ccc}0.5i(v_1+v_2)& ib{q}_{1x}& ib{q}_{2x}\\ -ib{r}_{1x}& 0.5i{v}_1& i{w}_2\\ -ib{r}_{2x}& i{w}_1& 0.5i{v}_2\end{array}\right),Q=\left(\begin{array}{ccc}0& q_1& q_2\\ r_1& 0& 0\\ r_2& 0& 0\end{array}\right),\Sigma =\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right).\end{array}$$

(200)

Then, from the compatibility condition

$$\begin{array}{c}\hfill U_{7x}-V_{7t}+[U_7,V_7]=0\end{array}$$

(201)

we obtain the two-component K-IIBE (188)–(195).

9.3. Multicomponent KE

One of the multicomponent generalizations of the K-IIAE has the form

$$\begin{array}{ccc}\hfill i{q}_{kt}+q_{kxt}-v{q}_k& =& 0,\hfill \end{array}$$

(202)

$$\begin{array}{ccc}\hfill i{r}_{kt}-r_{kxt}+v{r}_k& =& 0,\hfill \end{array}$$

(203)

$$\begin{array}{ccc}\hfill v_x-2b^2\sum _{k=1}^N{\left(r_k{q}_k\right)}_t& =& 0,\hfill \end{array}$$

(204)

or

$$\begin{array}{ccc}\hfill i{q}_{kx}+q_{kxt}-v{q}_k& =& 0,\hfill \end{array}$$

(205)

$$\begin{array}{ccc}\hfill i{r}_{kx}-r_{kxt}+v{r}_k& =& 0,\hfill \end{array}$$

(206)

$$\begin{array}{ccc}\hfill v_t-2b^2\sum _{k=1}^N{\left(r_k{q}_k\right)}_x& =& 0,\hfill \end{array}$$

(207)

where $k=1,2,\dots ,N$. In particular, the two-component versions of these equations read as

$$\begin{array}{ccc}\hfill i{q}_{1t}+q_{1xt}-v{q}_1& =& 0,\hfill \end{array}$$

(208)

$$\begin{array}{ccc}\hfill i{q}_{2t}+q_{2xt}-v{q}_2& =& 0,\hfill \end{array}$$

(209)

$$\begin{array}{ccc}\hfill i{r}_{1t}-r_{1xt}+v{r}_1& =& 0,\hfill \end{array}$$

(210)

$$\begin{array}{ccc}\hfill i{r}_{2t}-r_{2xt}+v{r}_2& =& 0,\hfill \end{array}$$

(211)

$$\begin{array}{ccc}\hfill v_x-2b^2{(r_1{q}_1+r_2{q}_2)}_t& =& 0,\hfill \end{array}$$

(212)

or

$$\begin{array}{ccc}\hfill i{q}_{1x}+q_{1xt}-v{q}_1& =& 0,\hfill \end{array}$$

(213)

$$\begin{array}{ccc}\hfill i{q}_{2x}+q_{2xt}-v{q}_2& =& 0,\hfill \end{array}$$

(214)

$$\begin{array}{ccc}\hfill i{r}_{1x}-r_{1xt}+v{r}_1& =& 0,\hfill \end{array}$$

(215)

$$\begin{array}{ccc}\hfill i{r}_{2x}-r_{2xt}+v{r}_2& =& 0,\hfill \end{array}$$

(216)

$$\begin{array}{ccc}\hfill v_t-2b^2{(r_1{q}_1+r_2{q}_2)}_x& =& 0.\hfill \end{array}$$

(217)

9.4. Integrable Akbota Equation

One of the interesting integrable generalizations of the KE is the following Akbota equation (AE) [5,6,7,8,9]

$$\begin{array}{ccc}\hfill i{q}_t+\alpha q_{xx}+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(218)

$$\begin{array}{ccc}\hfill v_x-{2\left[\alpha \right(\left|q\right|}^2{)}_x+{\beta \left(\right|q|}^2{)}_t]& =& 0,\hfill \end{array}$$

(219)

or

$$\begin{array}{ccc}\hfill i{q}_t+\alpha q_{xx}+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(220)

$$\begin{array}{ccc}\hfill v_x-{2\left[\alpha \right(\left|q\right|}^2{)}_x+{\beta \left(\right|q|}^2{)}_t]& =& 0.\hfill \end{array}$$

(221)

In fact, as $\alpha =0$, these AEs become

$$\begin{array}{ccc}\hfill i{q}_t+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(222)

$$\begin{array}{ccc}\hfill v_x-{2\beta \left(\right|q|}^2{)}_t& =& 0.\hfill \end{array}$$

(223)

and

$$\begin{array}{ccc}\hfill i{q}_x+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(224)

$$\begin{array}{ccc}\hfill v_t-{2\beta \left(\right|q|}^2{)}_x& =& 0,\hfill \end{array}$$

(225)

respectively. These equations, up to the simple scale transformations, coincide with the K-IIAE and the K-IIBE, respectively. The Lax representation of the A-IIAE is given by

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{14}\Phi ,\hfill \end{array}$$

(226)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{14}\Phi ,\hfill \end{array}$$

(227)

where

$$\begin{array}{c}\hfill U_{14}=\frac{i\lambda }{2}{\sigma }_3+Q,Q=\left(\begin{array}{cc}0& \overline{q}\\ q& 0\end{array}\right),V_{14}=\frac{1}{1-\lambda \beta }\{\frac{i{\lambda }^2}{2}\alpha {\sigma }_3+\alpha \lambda Q+V_0\}\end{array}$$

(228)

with

$$\begin{array}{c}\hfill V_0=\left(\begin{array}{cc}{\alpha i\left|q\right|}^2+i\beta {\partial }_x^{-1}{\left|q\right|}_t^2& -i\beta {\overline{q}}_t-i\alpha {\overline{q}}_x\\ i\beta q_t+\alpha i{q}_x& -\left[\alpha i\right|q{|}^2+i\beta {\partial }_x^{-1}\left|q{|}_t^2\right]\end{array}\right).\end{array}$$

(229)

At the same time, the Lax representation of the A-IIBE is given by

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_{14}\Phi ,\hfill \end{array}$$

(230)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_{14}\Phi ,\hfill \end{array}$$

(231)

where

$$\begin{array}{c}\hfill U_{14}=\frac{i\lambda }{2}{\sigma }_3+Q,Q=\left(\begin{array}{cc}0& \overline{q}\\ q& 0\end{array}\right),V_{14}=\frac{1}{1-\lambda \beta }\{\frac{i{\lambda }^2}{2}\alpha {\sigma }_3+\alpha \lambda Q+V_0\}\end{array}$$

(232)

with

$$\begin{array}{c}\hfill V_0=\left(\begin{array}{cc}{\alpha i\left|q\right|}^2+i\beta {\partial }_t^{-1}{\left|q\right|}_x^2& -i\beta {\overline{q}}_x-i\alpha {\overline{q}}_t\\ i\beta q_x+\alpha i{q}_t& -\left[\alpha i\right|q{|}^2+i\beta {\partial }_t^{-1}\left|q{|}_x^2\right]\end{array}\right).\end{array}$$

(233)

9.5. Integrable Zhanbota Equation

9.5.1. Zhanbota-IIA Equation

Another integrable generalization of the K-IIAE is the following Zhanbota-IIA equation [5,6,7,8,9]:

$$\begin{array}{ccc}\hfill i{q}_t+q_{xt}-vq-2ip& =& 0,\hfill \end{array}$$

(234)

$$\begin{array}{ccc}\hfill v_x+2{\delta }_1{\left(\right|q|}^2{)}_t& =& 0,\hfill \end{array}$$

(235)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(236)

$$\begin{array}{ccc}\hfill {\eta }_x+({\delta }_1\overline{q}p+{\delta }_2\overline{p}q)& =& 0.\hfill \end{array}$$

(237)

This Zhanbota-IIA equation, as $p=\eta =0$, takes the form

$$\begin{array}{ccc}\hfill i{q}_t+q_{xt}-vq& =& 0,\hfill \end{array}$$

(238)

$$\begin{array}{ccc}\hfill v_x+2{\delta }_1{\left(\right|q|}^2{)}_t& =& 0,\hfill \end{array}$$

(239)

which is nothing but the K-IIAE. Note that the Lax representation of the Zhanbota-IIA equation reads

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{12}\Phi ,\hfill \end{array}$$

(240)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{12}\Phi ,\hfill \end{array}$$

(241)

where

$$\begin{array}{ccc}\hfill U_{12}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(242)

$$\begin{array}{ccc}\hfill V_{12}& =& \frac{1}{1-\kappa \lambda }\{B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(243)

Here,

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(244)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3-\frac{\kappa }{2i}\left(\begin{array}{cc}0& q_t\\ r_t& 0\end{array}\right),\hfill \end{array}$$

(245)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(246)

9.5.2. Zhanbota-IIB Equation

One of the integrable generalizations of the K-IIBE is the following Zhanbota-IIB equation [5,6,7,8,9]:

$$\begin{array}{ccc}\hfill i{q}_x+q_{xt}-vq-2ip& =& 0,\hfill \end{array}$$

(247)

$$\begin{array}{ccc}\hfill v_t+2{\delta }_1{\left(\right|q|}^2{)}_x& =& 0,\hfill \end{array}$$

(248)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(249)

$$\begin{array}{ccc}\hfill {\eta }_t+({\delta }_1\overline{q}p+{\delta }_2\overline{p}q)& =& 0.\hfill \end{array}$$

(250)

This Zhanbota-IIB equation, as $p=\eta =0$, takes the form

$$\begin{array}{ccc}\hfill i{q}_x+q_{xt}-vq& =& 0,\hfill \end{array}$$

(251)

$$\begin{array}{ccc}\hfill v_t+2{\delta }_1{\left(\right|q|}^2{)}_x& =& 0,\hfill \end{array}$$

(252)

which is nothing but the K-IIBE. The Lax representation of the Zhanbota-IIB equation reads

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_{12}\Phi ,\hfill \end{array}$$

(253)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_{12}\Phi ,\hfill \end{array}$$

(254)

where

$$\begin{array}{ccc}\hfill U_{12}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(255)

$$\begin{array}{ccc}\hfill V_{12}& =& \frac{1}{1-\kappa \lambda }\{B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(256)

Here,

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(257)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3-\frac{\kappa }{2i}\left(\begin{array}{cc}0& q_x\\ r_x& 0\end{array}\right),\hfill \end{array}$$

(258)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(259)

9.6. Integrable Nurshuak Equation

Let us present two more examples of the integrable generalizations of the KE.

9.6.1. N-IIAE

First, let us consider the following Nurshuak-IIA equation (N-IIAE) [5,6,7,8,9]:

$$\begin{array}{ccc}\hfill i{q}_t+{ϵ}_1{q}_{xt}+i{ϵ}_2{q}_{xxt}-vq+{\left(wq\right)}_x-2ip& =& 0,\hfill \end{array}$$

(260)

$$\begin{array}{ccc}\hfill i{r}_t-{ϵ}_1{r}_{xt}+i{ϵ}_2{r}_{xxt}+vr+{\left(wr\right)}_x-2ik& =& 0,\hfill \end{array}$$

(261)

$$\begin{array}{ccc}\hfill v_x+2{ϵ}_1{\left(rq\right)}_t-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(262)

$$\begin{array}{ccc}\hfill w_x-2i{ϵ}_2{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(263)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(264)

$$\begin{array}{ccc}\hfill k_x+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(265)

$$\begin{array}{ccc}\hfill {\eta }_x+rp+kq& =& 0.\hfill \end{array}$$

(266)

From this N-IIAE, we obtain the K-IIAE as ${ϵ}_2=w=p=k=\eta =0,{ϵ}_1=1$. Note that the N-IIAE is integrable. Its Lax representation reads as

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_8\Phi ,\hfill \end{array}$$

(267)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_8\Phi .\hfill \end{array}$$

(268)

Here,

$$\begin{array}{ccc}\hfill U_8& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(269)

$$\begin{array}{ccc}\hfill V_8& =& \frac{1}{1-(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\},\hfill \end{array}$$

(270)

where

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0t},A_0=\left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(271)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& i{ϵ}_1{q}_t-{ϵ}_2{q}_{xt}+iwq\\ i{ϵ}_1{r}_t+{ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(272)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(273)

9.6.2. N-IIBE

Here, let us consider the following Nurshuak-IIB equation (N-IIBE) [5,6,7,8,9]:

$$\begin{array}{ccc}\hfill i{q}_x+{ϵ}_1{q}_{xt}+i{ϵ}_2{q}_{ttx}-vq+{\left(wq\right)}_t-2ip& =& 0,\hfill \end{array}$$

(274)

$$\begin{array}{ccc}\hfill i{r}_x-{ϵ}_1{r}_{xt}+i{ϵ}_2{r}_{ttx}+vr+{\left(wr\right)}_t-2ik& =& 0,\hfill \end{array}$$

(275)

$$\begin{array}{ccc}\hfill v_t+2{ϵ}_1{\left(rq\right)}_x-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(276)

$$\begin{array}{ccc}\hfill w_t-2i{ϵ}_2{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(277)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(278)

$$\begin{array}{ccc}\hfill k_t+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(279)

$$\begin{array}{ccc}\hfill {\eta }_t+rp+kq& =& 0.\hfill \end{array}$$

(280)

From this N-IIBE, we obtain the K-IIBE as ${ϵ}_2=w=p=k=\eta =0,{ϵ}_1=1$. Note that the N-IIBE is integrable. Its Lax representation reads as

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_8\Phi ,\hfill \end{array}$$

(281)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_8\Phi .\hfill \end{array}$$

(282)

Here,

$$\begin{array}{ccc}\hfill U_8& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(283)

$$\begin{array}{ccc}\hfill V_8& =& \frac{1}{1-(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\},\hfill \end{array}$$

(284)

where

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0x},A_0=\left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(285)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& i{ϵ}_1{q}_x-{ϵ}_2{q}_{xt}+iwq\\ i{ϵ}_1{r}_x+{ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(286)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(287)

10. Some Integrable Generalizations of the K-IE

In the previous section, we have presented some generalizations of the K-II equation. In this section, we consider the integrable generalizations of the K-IE. In fact, the Kuralay-I equation also admits several integrable generalizations. As examples, here, we present some of them.

10.1. Zhaidary-I Equation

The Zhaidary-I equations are integrable generalizations of the Kuralay-I equations. In this subsection, we consider these integrable generalizations.

10.1.1. Z-IAE

The Zhaidary-IA equation (Z-IAE) is given by

$$\begin{array}{ccc}\hfill (1+2\beta (c\beta +d)){\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x+4cw{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(288)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(289)

$$\begin{array}{ccc}\hfill w_x+\frac{1}{4{(2\beta c+d)}^2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0.\hfill \end{array}$$

(290)

This Z-IAE is completely integrable, i.e., it can be solved by the inverse scattering transformation method (IST). It possesses all the basic characteristics of integrable equations. The corresponding Lax representation has the form

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_1\Psi ,\hfill \end{array}$$

(291)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_1\Psi .\hfill \end{array}$$

(292)

Here,

$$\begin{array}{ccc}\hfill U_1& =& [ic({\lambda }^2-{\beta }^2)+id(\lambda -\beta )]S+\frac{c(\lambda -\beta )}{2c\beta +d}S{S}_x,\hfill \end{array}$$

(293)

$$\begin{array}{ccc}\hfill V_1& =& \frac{1}{1-2c{\lambda }^2-2d\lambda }\{[2c({\lambda }^2-{\beta }^2)+2d(\lambda -\beta )]B+{\lambda }^2{F}_2+\lambda F_1+F_0\},\hfill \end{array}$$

(294)

where

$$\begin{array}{ccc}\hfill F_2& =& -4i{c}^{2}wS,F_1=-4icdwS-\frac{4c^2}{2c\beta +d}wS{S}_x-\frac{ic}{2c\beta +d}S\{{\left(S{S}_x\right)}_t-[S{S}_x,B]\},\hfill \end{array}$$

(295)

$$\begin{array}{ccc}\hfill F_0& =& -\beta F_1-{\beta }^2{F}_2,B=0.25([S,S_t]+2iuS),S=\mathbf{S}·\sigma .\hfill \end{array}$$

(296)

If $\beta =0$, the ZE takes the form

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x+4cw{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(297)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(298)

$$\begin{array}{ccc}\hfill w_x+\frac{1}{4d^2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(299)

or

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}+(2c{d}^{-2}-1)u{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(300)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0.\hfill \end{array}$$

(301)

Note that the gauge-equivalent counterpart of the Z-IAE reads as

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}+4ic{\left(vq\right)}_x-2d^{2}vq& =& 0,\hfill \end{array}$$

(302)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+4ic{\left(vr\right)}_x+2d^{2}vr& =& 0,\hfill \end{array}$$

(303)

$$\begin{array}{ccc}\hfill v_x-{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(304)

which is, in fact, the Z-IIAE.

10.1.2. Z-IBE

The Zhaidary-IB equation (Z-IBE) has the form

$$\begin{array}{ccc}\hfill (1-2\beta (c\beta +d)){\mathbf{S}}_x-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_t+4cw{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(305)

$$\begin{array}{ccc}\hfill u_t+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_x& =& 0,\hfill \end{array}$$

(306)

$$\begin{array}{ccc}\hfill w_t+\frac{1}{4{(2\beta c+d)}^2}{\left({\mathbf{S}}_x^2\right)}_x& =& 0.\hfill \end{array}$$

(307)

This Z-IAE is completely integrable, i.e., it can be solved by the inverse scattering transformation method (IST). It possesses all the basic characteristics of integrable equations. The corresponding Lax representation has the form

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_1\Psi ,\hfill \end{array}$$

(308)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_1\Psi .\hfill \end{array}$$

(309)

Here,

$$\begin{array}{ccc}\hfill U_1& =& [ic({\lambda }^2-{\beta }^2)+id(\lambda -\beta )]S+\frac{c(\lambda -\beta )}{2c\beta +d}S{S}_t,\hfill \end{array}$$

(310)

$$\begin{array}{ccc}\hfill V_1& =& \frac{1}{1-2c{\lambda }^2-2d\lambda }\{[2c({\lambda }^2-{\beta }^2)+2d(\lambda -\beta )]B+{\lambda }^2{F}_2+\lambda F_1+F_0\},\hfill \end{array}$$

(311)

where

$$\begin{array}{ccc}\hfill F_2& =& -4i{c}^{2}wS,F_1=-4icdwS-\frac{4c^2}{2c\beta +d}wS{S}_t-\frac{ic}{2c\beta +d}S\{{\left(S{S}_t\right)}_t-[S{S}_t,B]\},\hfill \end{array}$$

(312)

$$\begin{array}{ccc}\hfill F_0& =& -\beta F_1-{\beta }^2{F}_2,B=0.25([S,S_x]+2iuS),S=\mathbf{S}·\sigma .\hfill \end{array}$$

(313)

If $\beta =0$, the ZE takes the form

$$\begin{array}{ccc}\hfill {\mathbf{S}}_x-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_t+4cw{\mathbf{S}}_t& =& 0,\hfill \end{array}$$

(314)

$$\begin{array}{ccc}\hfill u_t+\frac{1}{2}{\left({\mathbf{S}}_t^2\right)}_x& =& 0,\hfill \end{array}$$

(315)

$$\begin{array}{ccc}\hfill w_t+\frac{1}{4d^2}{\left({\mathbf{S}}_t^2\right)}_x& =& 0,\hfill \end{array}$$

(316)

or

$$\begin{array}{ccc}\hfill {\mathbf{S}}_x-\mathbf{S}\wedge {\mathbf{S}}_{xt}+(2c{d}^{-2}-1)u{\mathbf{S}}_t& =& 0,\hfill \end{array}$$

(317)

$$\begin{array}{ccc}\hfill u_t+\frac{1}{2}{\left({\mathbf{S}}_t^2\right)}_x& =& 0.\hfill \end{array}$$

(318)

Finally, let us present the following gauge-equivalent counterpart of the Z-IBE:

$$\begin{array}{ccc}\hfill i{q}_x-q_{xt}+4ic{\left(vq\right)}_t-2d^{2}vq& =& 0,\hfill \end{array}$$

(319)

$$\begin{array}{ccc}\hfill i{r}_x+r_{xt}+4ic{\left(vr\right)}_t+2d^{2}vr& =& 0,\hfill \end{array}$$

(320)

$$\begin{array}{ccc}\hfill v_t-{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(321)

which is the Z-IIBE.

10.2. Shynaray Equation

10.2.1. S-IAE

Our next example is the Shynaray-IA equation (S-IAE). It has the form

$$\begin{array}{ccc}\hfill (1+2c{\beta }^2){\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x+4cw{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(322)

$$\begin{array}{ccc}\hfill u_x+\frac{1}{2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0,\hfill \end{array}$$

(323)

$$\begin{array}{ccc}\hfill w_x+\frac{1}{16c^2{\beta }^2}{\left({\mathbf{S}}_x^2\right)}_t& =& 0.\hfill \end{array}$$

(324)

This S-IAE is integrable, in the sense that it has the following Lax representation

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_5\Psi ,\hfill \end{array}$$

(325)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_5\Psi .\hfill \end{array}$$

(326)

Here,

$$\begin{array}{ccc}\hfill U_5& =& ic({\lambda }^2-{\beta }^2)S+\frac{\lambda -\beta }{2\beta }S{S}_x,\hfill \end{array}$$

(327)

$$\begin{array}{ccc}\hfill V_5& =& \frac{1}{1-2c{\lambda }^2}\{2c({\lambda }^2-{\beta }^2)B+{\lambda }^2{F}_2+\lambda F_1+F_0\},\hfill \end{array}$$

(328)

where

$$\begin{array}{ccc}\hfill F_2& =& -4i{c}^{2}wS,F_1=-\frac{4c^2}{2c\beta }wS{S}_x-\frac{ic}{2c\beta }S\{{\left(S{S}_x\right)}_t-[S{S}_x,B]\},\hfill \end{array}$$

(329)

$$\begin{array}{ccc}\hfill F_0& =& -\beta F_1-{\beta }^2{F}_2,B=0.25([S,S_t]+2iuS),S=\mathbf{S}·\sigma .\hfill \end{array}$$

(330)

The gauge equivalent counterpart of the S-IAE is given by

$$\begin{array}{ccc}\hfill i{q}_t-q_{xt}+4ic{\left(vq\right)}_x& =& 0,\hfill \end{array}$$

(331)

$$\begin{array}{ccc}\hfill i{r}_t+r_{xt}+4ic{\left(vr\right)}_x& =& 0,\hfill \end{array}$$

(332)

$$\begin{array}{ccc}\hfill v_x-{\left(rq\right)}_t& =& 0.\hfill \end{array}$$

(333)

10.2.2. S-IBE

The Shynaray-IB equation (S-IBE) has the form:

$$\begin{array}{ccc}\hfill (1+2c{\beta }^2){\mathbf{S}}_x-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_t+4cw{\mathbf{S}}_t& =& 0,\hfill \end{array}$$

(334)

$$\begin{array}{ccc}\hfill u_t+\frac{1}{2}{\left({\mathbf{S}}_t^2\right)}_x& =& 0,\hfill \end{array}$$

(335)

$$\begin{array}{ccc}\hfill w_t+\frac{1}{16c^2{\beta }^2}{\left({\mathbf{S}}_t^2\right)}_x& =& 0.\hfill \end{array}$$

(336)

This S-IBE is integrable, in the sense that it admits the following Lax representation

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_5\Psi ,\hfill \end{array}$$

(337)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_5\Psi .\hfill \end{array}$$

(338)

Here,

$$\begin{array}{ccc}\hfill U_5& =& ic({\lambda }^2-{\beta }^2)S+\frac{\lambda -\beta }{2\beta }S{S}_t,\hfill \end{array}$$

(339)

$$\begin{array}{ccc}\hfill V_5& =& \frac{1}{1-2c{\lambda }^2}\{2c({\lambda }^2-{\beta }^2)B+{\lambda }^2{F}_2+\lambda F_1+F_0\},\hfill \end{array}$$

(340)

where

$$\begin{array}{ccc}\hfill F_2& =& -4i{c}^{2}wS,F_1=-\frac{4c^2}{2c\beta }wS{S}_x-\frac{ic}{2c\beta }S\{{\left(S{S}_t\right)}_x-[S{S}_t,B]\},\hfill \end{array}$$

(341)

$$\begin{array}{ccc}\hfill F_0& =& -\beta F_1-{\beta }^2{F}_2,B=0.25([S,S_x]+2iuS),S=\mathbf{S}·\sigma .\hfill \end{array}$$

(342)

The gauge-equivalent equation for the S-IBE reads as

$$\begin{array}{ccc}\hfill i{q}_x-q_{xt}+4ic{\left(vq\right)}_t& =& 0,\hfill \end{array}$$

(343)

$$\begin{array}{ccc}\hfill i{r}_x+r_{xt}+4ic{\left(vr\right)}_t& =& 0,\hfill \end{array}$$

(344)

$$\begin{array}{ccc}\hfill v_t-{\left(rq\right)}_x& =& 0.\hfill \end{array}$$

(345)

10.3. Nurshuak Equation

In this subsection, we would like to demonstrate another generalization of the K-IE. Consider the so-called Nurshuak-I equation. It has two forms: the Nurshuak-IA equation and the Nurshuak-IB equation.

10.3.1. N-IAE

Let us start from the Nurshuak-IA equation (N-IAE), which has the form

$$\begin{array}{ccc}\hfill i{S}_t+2{ϵ}_1{Z}_x+i{ϵ}_2{(S_{xt}+[S_x,Z])}_x+{\left(wS\right)}_x+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(346)

$$\begin{array}{ccc}\hfill u_x-\frac{i}{4}tr(S\times [S_x,S_t])& =& 0,\hfill \end{array}$$

(347)

$$\begin{array}{ccc}\hfill w_x-\frac{i}{4}{ϵ}_2{\left[tr\left(S_x^2\right)\right]}_t& =& 0,\hfill \end{array}$$

(348)

$$\begin{array}{ccc}\hfill i{W}_x+\omega [S,W]& =& 0,\hfill \end{array}$$

(349)

where

$$\begin{array}{c}\hfill Z=\frac{1}{4}([S,S_t]+2iuS).\end{array}$$

(350)

If ${ϵ}_2=0$, from this N-IAE, we obtain the K-IAE. As an integrable equation, the N-IAE admits the LR of the form

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_7\Psi ,\hfill \end{array}$$

(351)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_7\Psi .\hfill \end{array}$$

(352)

Here,

$$\begin{array}{ccc}\hfill U_7& =& -i\lambda S,\hfill \end{array}$$

(353)

$$\begin{array}{ccc}\hfill V_7& =& \frac{1}{1-2{ϵ}_1\lambda -4{ϵ}_2{\lambda }^2}\{(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)Z+\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\},\hfill \end{array}$$

(354)

where

$$\begin{array}{ccc}\hfill V_1& =& wS+i{ϵ}_2(S_{xt}+[S_x,Z]),\hfill \end{array}$$

(355)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right),S=\left(\begin{array}{cc}{S}_3& S^{-}\\ S^{+}& -S_3\end{array}\right),S^{\pm }=S_1\pm i{S}_2.\hfill \end{array}$$

(356)

The compatibility condition ${\Psi }_{xt}={\Psi }_{tx}$ gives the N-IAE. The gauge partner of the N-IAE is

$$\begin{array}{ccc}\hfill i{q}_t+{ϵ}_1{q}_{xt}+i{ϵ}_2{q}_{xxt}-vq+{\left(wq\right)}_x-2ip& =& 0,\hfill \end{array}$$

(357)

$$\begin{array}{ccc}\hfill i{r}_t-{ϵ}_1{r}_{xt}+i{ϵ}_2{r}_{xxt}+vr+{\left(wr\right)}_x-2ik& =& 0,\hfill \end{array}$$

(358)

$$\begin{array}{ccc}\hfill v_x+2{ϵ}_1{\left(rq\right)}_t-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(359)

$$\begin{array}{ccc}\hfill w_x-2i{ϵ}_2{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(360)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(361)

$$\begin{array}{ccc}\hfill k_x+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(362)

$$\begin{array}{ccc}\hfill {\eta }_x+rp+kq& =& 0,\hfill \end{array}$$

(363)

where $r={\delta }_1\overline{q},k={\delta }_2\overline{p},{\delta }_j=\pm 1$. This is the N-IIBE. Its Lax representation reads as

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_8\Psi ,\hfill \end{array}$$

(364)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_8\Psi .\hfill \end{array}$$

(365)

Here,

$$\begin{array}{ccc}\hfill U_8& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(366)

$$\begin{array}{ccc}\hfill V_8& =& \frac{1}{1-(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\},\hfill \end{array}$$

(367)

where

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0t},A_0=\left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(368)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& i{ϵ}_1{q}_t-{ϵ}_2{q}_{xt}+iwq\\ i{ϵ}_1{r}_t+{ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(369)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(370)

10.3.2. N-IBE

Now, we consider the Nurshuak-IB equation (N-IBE). It has the form

$$\begin{array}{ccc}\hfill i{S}_x+2{ϵ}_1{Z}_t+i{ϵ}_2{(S_{xt}+[S_t,Z])}_t+{\left(wS\right)}_t+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(371)

$$\begin{array}{ccc}\hfill u_t-\frac{i}{4}tr(S\times [S_t,S_x])& =& 0,\hfill \end{array}$$

(372)

$$\begin{array}{ccc}\hfill w_t-\frac{i}{4}{ϵ}_2{\left[tr\left(S_t^2\right)\right]}_x& =& 0,\hfill \end{array}$$

(373)

$$\begin{array}{ccc}\hfill i{W}_t+\omega [S,W]& =& 0,\hfill \end{array}$$

(374)

where

$$\begin{array}{c}\hfill Z=\frac{1}{4}([S,S_x]+2iuS).\end{array}$$

(375)

As an integrable equation, the N-IAE admits the LR of the form

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_7\Psi ,\hfill \end{array}$$

(376)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_7\Psi .\hfill \end{array}$$

(377)

Here,

$$\begin{array}{ccc}\hfill U_7& =& -i\lambda S,\hfill \end{array}$$

(378)

$$\begin{array}{ccc}\hfill V_7& =& \frac{1}{1-2{ϵ}_1\lambda -4{ϵ}_2{\lambda }^2}\{(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)Z+\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\},\hfill \end{array}$$

(379)

where

$$\begin{array}{ccc}\hfill V_1& =& wS+i{ϵ}_2(S_{xt}+[S_t,Z]),\hfill \end{array}$$

(380)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right),S=\left(\begin{array}{cc}{S}_3& S^{-}\\ S^{+}& -S_3\end{array}\right),S^{\pm }=S_1\pm i{S}_2.\hfill \end{array}$$

(381)

The gauge equivalent partner of the N-IBE is given by

$$\begin{array}{ccc}\hfill i{q}_x+{ϵ}_1{q}_{xt}+i{ϵ}_2{q}_{ttx}-vq+{\left(wq\right)}_t-2ip& =& 0,\hfill \end{array}$$

(382)

$$\begin{array}{ccc}\hfill i{r}_x-{ϵ}_1{r}_{xt}+i{ϵ}_2{r}_{ttx}+vr+{\left(wr\right)}_t-2ik& =& 0,\hfill \end{array}$$

(383)

$$\begin{array}{ccc}\hfill v_t+2{ϵ}_1{\left(rq\right)}_x-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(384)

$$\begin{array}{ccc}\hfill w_t-2i{ϵ}_2{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(385)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(386)

$$\begin{array}{ccc}\hfill k_t+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(387)

$$\begin{array}{ccc}\hfill {\eta }_t+rp+kq& =& 0.\hfill \end{array}$$

(388)

Since this N-IBE is integrable, it has the Lax representation of the form

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_8\Psi ,\hfill \end{array}$$

(389)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_8\Psi .\hfill \end{array}$$

(390)

Here,

$$\begin{array}{ccc}\hfill U_8& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(391)

$$\begin{array}{ccc}\hfill V_8& =& \frac{1}{1-(2{ϵ}_1\lambda +4{ϵ}_2{\lambda }^2)}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\},\hfill \end{array}$$

(392)

where

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0x},A_0=\left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(393)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& i{ϵ}_1{q}_x-{ϵ}_2{q}_{xt}+iwq\\ i{ϵ}_1{r}_x+{ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(394)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(395)

10.4. Aizhan-I Equation

The Aizhan-I equation has two integrable forms, namely, the Aizhan-IA equation and the Aizhan-IB equation.

10.4.1. The Aizhan-IA Equation

The Aizhan-IA equation is given by

$$\begin{array}{ccc}\hfill i{S}_t+i{ϵ}_2{(S_{xt}+[S_x,Z])}_x+{\left(wS\right)}_x+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(396)

$$\begin{array}{ccc}\hfill u_x-\frac{i}{4}tr(S\times [S_x,S_t])& =& 0,\hfill \end{array}$$

(397)

$$\begin{array}{ccc}\hfill w_x-\frac{i}{4}{ϵ}_2{\left[tr\left(S_x^2\right)\right]}_t& =& 0,\hfill \end{array}$$

(398)

$$\begin{array}{ccc}\hfill i{W}_x+\omega [S,W]& =& 0.\hfill \end{array}$$

(399)

The Aizhan-IA equation can be considered as one of the integrable generalizations of the K-IAE. The Lax representation of the Aizhan-IA equation has the form

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_9\Psi ,\hfill \end{array}$$

(400)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_9\Psi ,\hfill \end{array}$$

(401)

where

$$\begin{array}{ccc}\hfill U_9& =& -i\lambda S,\hfill \end{array}$$

(402)

$$\begin{array}{ccc}\hfill V_9& =& \frac{1}{1-4{ϵ}_2{\lambda }^2}\{4{ϵ}_2{\lambda }^{2}Z+\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\}\hfill \end{array}$$

(403)

with

$$\begin{array}{ccc}\hfill V_1& =& wS+i{ϵ}_2(S_{xt}+[S_x,Z]),\hfill \end{array}$$

(404)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right).\hfill \end{array}$$

(405)

The gauge-equivalent equation for the Aizhan-IA equation reads as

$$\begin{array}{ccc}\hfill i{q}_t+i{ϵ}_2{q}_{xxt}-vq+{\left(wq\right)}_x-2ip& =& 0,\hfill \end{array}$$

(406)

$$\begin{array}{ccc}\hfill i{r}_t+i{ϵ}_2{r}_{xxt}+vr+{\left(wr\right)}_x-2ik& =& 0,\hfill \end{array}$$

(407)

$$\begin{array}{ccc}\hfill v_x-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(408)

$$\begin{array}{ccc}\hfill w_x-2i{ϵ}_2{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(409)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(410)

$$\begin{array}{ccc}\hfill k_x+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(411)

$$\begin{array}{ccc}\hfill {\eta }_x+rp+kq& =& 0.\hfill \end{array}$$

(412)

This is the Aizhan-IIA equation. For this equation, the corresponding Lax representation is given by

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{10}\Phi ,\hfill \end{array}$$

(413)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{10}\Phi ,\hfill \end{array}$$

(414)

where

$$\begin{array}{ccc}\hfill U_{10}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(415)

$$\begin{array}{ccc}\hfill V_{10}& =& \frac{1}{1-4{ϵ}_2{\lambda }^2}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(416)

Here,

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0t},\hfill \end{array}$$

(417)

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(418)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& -{ϵ}_2{q}_{xt}+iwq\\ {ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(419)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(420)

10.4.2. The Aizhan-IB Equation

The Aizhan-IB equation has the form

$$\begin{array}{ccc}\hfill i{S}_x+i{ϵ}_2{(S_{xt}+[S_t,Z])}_t+{\left(wS\right)}_t+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(421)

$$\begin{array}{ccc}\hfill u_t-\frac{i}{4}tr(S\times [S_t,S_x])& =& 0,\hfill \end{array}$$

(422)

$$\begin{array}{ccc}\hfill w_t-\frac{i}{4}{ϵ}_2{\left[tr\left(S_t^2\right)\right]}_x& =& 0,\hfill \end{array}$$

(423)

$$\begin{array}{ccc}\hfill i{W}_t+\omega [S,W]& =& 0.\hfill \end{array}$$

(424)

The Aizhan-IB equation can be considered as one of the integrable generalizations of the K-IBE. The Lax representation of the Aizhan-IB equation has the form

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_9\Psi ,\hfill \end{array}$$

(425)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_9\Psi ,\hfill \end{array}$$

(426)

where

$$\begin{array}{ccc}\hfill U_9& =& -i\lambda S,\hfill \end{array}$$

(427)

$$\begin{array}{ccc}\hfill V_9& =& \frac{1}{1-4{ϵ}_2{\lambda }^2}\{4{ϵ}_2{\lambda }^{2}Z+\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\}\hfill \end{array}$$

(428)

with

$$\begin{array}{ccc}\hfill V_1& =& wS+i{ϵ}_2(S_{xt}+[S_t,Z]),\hfill \end{array}$$

(429)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right).\hfill \end{array}$$

(430)

The gauge equivalent counterpart of the Aizhan-IB equation reads as

$$\begin{array}{ccc}\hfill i{q}_x+i{ϵ}_2{q}_{ttx}-vq+{\left(wq\right)}_t-2ip& =& 0,\hfill \end{array}$$

(431)

$$\begin{array}{ccc}\hfill i{r}_x+i{ϵ}_2{r}_{ttx}+vr+{\left(wr\right)}_t-2ik& =& 0,\hfill \end{array}$$

(432)

$$\begin{array}{ccc}\hfill v_t-2i{ϵ}_2(r_{xt}q-r{q}_{xt})& =& 0,\hfill \end{array}$$

(433)

$$\begin{array}{ccc}\hfill w_t-2i{ϵ}_2{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(434)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(435)

$$\begin{array}{ccc}\hfill k_t+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(436)

$$\begin{array}{ccc}\hfill {\eta }_t+rp+kq& =& 0.\hfill \end{array}$$

(437)

This is the Aizhan-IIB equation. Its Lax representation is

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{10}\Phi ,\hfill \end{array}$$

(438)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{10}\Phi ,\hfill \end{array}$$

(439)

where

$$\begin{array}{ccc}\hfill U_{10}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(440)

$$\begin{array}{ccc}\hfill V_{10}& =& \frac{1}{1-4{ϵ}_2{\lambda }^2}\{\lambda B_1+B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(441)

Here,

$$\begin{array}{ccc}\hfill B_1& =& w{\sigma }_3+2i{ϵ}_2{\sigma }_3{A}_{0t},\hfill \end{array}$$

(442)

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(443)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3+\left(\begin{array}{cc}0& -{ϵ}_2{q}_{xt}+iwq\\ {ϵ}_2{r}_{xt}-iwr& 0\end{array}\right),\hfill \end{array}$$

(444)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(445)

10.5. Zhanbota-I Equation

In this section, we consider the Zhanbota-I equation. It has a similar form, namely the Zhanbota-IA equation and the Zhanbota-IB equation.

10.5.1. Zhanbota-IA Equation

The Zhanbota-IA equation reads

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_x-\frac{1}{\omega }\mathbf{S}\wedge \mathbf{W}& =& 0,\hfill \end{array}$$

(446)

$$\begin{array}{ccc}\hfill u_x+\mathbf{S}\times ({\mathbf{S}}_x\wedge {\mathbf{S}}_{\mathbf{t}})& =& 0,\hfill \end{array}$$

(447)

$$\begin{array}{ccc}\hfill {\mathbf{W}}_x-\omega \mathbf{S}\wedge \mathbf{W}& =& 0.\hfill \end{array}$$

(448)

The Zhanbota-IA equation is also one of the integrable generalization of the K-IAE. The matrix form of the Zhanbota-IA equation reads as

$$\begin{array}{ccc}\hfill i{S}_t+\frac{1}{2}[S,S_{xt}]+iu{S}_x+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(449)

$$\begin{array}{ccc}\hfill u_x-\frac{i}{4}tr\left(S[S_x,S_t]\right)& =& 0,\hfill \end{array}$$

(450)

$$\begin{array}{ccc}\hfill i{W}_x+\omega [S,W]& =& 0,\hfill \end{array}$$

(451)

where $S=S_i{\sigma }_i$, $W=W_i{\sigma }_i$,($i=1,2,3$) and $\omega$ is a constant parameter. The $\mathbf{W}=(W_1,W_2,W_3)$ is the vector potential. The Zhanbota-IA equation possesses the following Lax representation:

$$\begin{array}{ccc}\hfill {\Psi }_x& =& U_{11}\Psi ,\hfill \end{array}$$

(452)

$$\begin{array}{ccc}\hfill {\Psi }_t& =& V_{11}\Phi .\hfill \end{array}$$

(453)

Here, the matrix operators $U_{11}$ and $V_{11}$ have the forms

$$\begin{array}{ccc}\hfill U_{11}& =& -i\lambda S,\hfill \end{array}$$

(454)

$$\begin{array}{ccc}\hfill V_{11}& =& \frac{1}{1-2\lambda }\{\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\},\hfill \end{array}$$

(455)

where

$$\begin{array}{ccc}\hfill V_1& =& 2Z=\frac{1}{2}([S,S_t]+2iuS),\hfill \end{array}$$

(456)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right).\hfill \end{array}$$

(457)

Let us present the gauge-equivalent counterpart of the Zhanbota-IA equation. It is not difficult to verify that the gauge-equivalent counterpart of the Zhanbota-IA equation is given by

$$\begin{array}{ccc}\hfill q_t+\frac{\kappa }{2i}{q}_{xt}+ivq-2p& =& 0,\hfill \end{array}$$

(458)

$$\begin{array}{ccc}\hfill r_t-\frac{\kappa }{2i}{r}_{xt}-ivr-2k& =& 0,\hfill \end{array}$$

(459)

$$\begin{array}{ccc}\hfill v_x+\frac{\kappa }{2}{\left(rq\right)}_t& =& 0,\hfill \end{array}$$

(460)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(461)

$$\begin{array}{ccc}\hfill k_x+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(462)

$$\begin{array}{ccc}\hfill {\eta }_x+rp+kq& =& 0,\hfill \end{array}$$

(463)

where $q,r,p,k$ are some complex functions; $v,\eta$ are real potential functions and $\kappa$ is a constant parameter. The Lax representation for this equation reads as

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{12}\Phi ,\hfill \end{array}$$

(464)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{12}\Phi ,\hfill \end{array}$$

(465)

where

$$\begin{array}{ccc}\hfill U_{12}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(466)

$$\begin{array}{ccc}\hfill V_{12}& =& \frac{1}{1-\kappa \lambda }\{B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(467)

Here,

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(468)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3-\frac{\kappa }{2i}\left(\begin{array}{cc}0& q_y\\ r_y& 0\end{array}\right),\hfill \end{array}$$

(469)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(470)

Let us consider the reduction $r={\delta }_1\overline{q},k={\delta }_2\overline{p}$ with $\kappa =2,{\delta }_j=\pm 1$, where the bar denotes the complex conjugate. Then, the system (487)–(492) takes the following more compact form

$$\begin{array}{ccc}\hfill i{q}_t+q_{xt}-vq-2ip& =& 0,\hfill \end{array}$$

(471)

$$\begin{array}{ccc}\hfill v_x+2{\delta }_1{\left(\right|q|}^2{)}_t& =& 0,\hfill \end{array}$$

(472)

$$\begin{array}{ccc}\hfill p_x-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(473)

$$\begin{array}{ccc}\hfill {\eta }_x+({\delta }_1\overline{q}p+{\delta }_2\overline{p}q)& =& 0.\hfill \end{array}$$

(474)

Finally, note that if $W=0$ and $p=k=\eta =0$, then, from (449)–(451) and (458)–(463), we obtain the K-IAE and the K-IIAE, respectively.

10.5.2. Zhanbota-IB Equation

Consider the Zhanbota-IB equation, which is given by

$$\begin{array}{ccc}\hfill {\mathbf{S}}_x-\mathbf{S}\wedge {\mathbf{S}}_{xt}-u{\mathbf{S}}_t-\frac{1}{\omega }\mathbf{S}\wedge \mathbf{W}& =& 0,\hfill \end{array}$$

(475)

$$\begin{array}{ccc}\hfill u_t+\mathbf{S}\times ({\mathbf{S}}_t\wedge {\mathbf{S}}_{\mathbf{x}})& =& 0,\hfill \end{array}$$

(476)

$$\begin{array}{ccc}\hfill {\mathbf{W}}_t-\omega \mathbf{S}\wedge \mathbf{W}& =& 0.\hfill \end{array}$$

(477)

The Zhanbota-IB equation is also one of the integrable generalization of the K-IBE. The matrix form of the Zhanbota-IB equation is given by

$$\begin{array}{ccc}\hfill i{S}_x+\frac{1}{2}[S,S_{xt}]+iu{S}_t+\frac{1}{\omega }[S,W]& =& 0,\hfill \end{array}$$

(478)

$$\begin{array}{ccc}\hfill u_t-\frac{i}{4}tr\left(S[S_t,S_x]\right)& =& 0,\hfill \end{array}$$

(479)

$$\begin{array}{ccc}\hfill i{W}_t+\omega [S,W]& =& 0.\hfill \end{array}$$

(480)

The Zhanbota-IB equation admits the following Lax representation:

$$\begin{array}{ccc}\hfill {\Psi }_t& =& U_{11}\Psi ,\hfill \end{array}$$

(481)

$$\begin{array}{ccc}\hfill {\Psi }_x& =& V_{11}\Phi .\hfill \end{array}$$

(482)

Here, the matrix operators $U_{11}$ and $V_{11}$ have the forms

$$\begin{array}{ccc}\hfill U_{11}& =& -i\lambda S,\hfill \end{array}$$

(483)

$$\begin{array}{ccc}\hfill V_{11}& =& \frac{1}{1-2\lambda }\{\lambda V_1+\frac{i}{\lambda +\omega }W-\frac{i}{\omega }W\},\hfill \end{array}$$

(484)

where

$$\begin{array}{ccc}\hfill V_1& =& 2Z=\frac{1}{2}([S,S_x]+2iuS),\hfill \end{array}$$

(485)

$$\begin{array}{ccc}\hfill W& =& \left(\begin{array}{cc}{W}_3& W^{-}\\ W^{+}& -W_3\end{array}\right).\hfill \end{array}$$

(486)

Finally, let us present the gauge-equivalent counterpart of the Zhanbota-IB equation. It has the form

$$\begin{array}{ccc}\hfill q_x+\frac{\kappa }{2i}{q}_{xt}+ivq-2p& =& 0,\hfill \end{array}$$

(487)

$$\begin{array}{ccc}\hfill r_x-\frac{\kappa }{2i}{r}_{xt}-ivr-2k& =& 0,\hfill \end{array}$$

(488)

$$\begin{array}{ccc}\hfill v_t+\frac{\kappa }{2}{\left(rq\right)}_x& =& 0,\hfill \end{array}$$

(489)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(490)

$$\begin{array}{ccc}\hfill k_t+2i\omega k-2\eta r& =& 0,\hfill \end{array}$$

(491)

$$\begin{array}{ccc}\hfill {\eta }_t+rp+kq& =& 0.\hfill \end{array}$$

(492)

Its Lax representation reads as

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_{12}\Phi ,\hfill \end{array}$$

(493)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_{12}\Phi ,\hfill \end{array}$$

(494)

where

$$\begin{array}{ccc}\hfill U_{12}& =& -i\lambda {\sigma }_3+A_0,\hfill \end{array}$$

(495)

$$\begin{array}{ccc}\hfill V_{12}& =& \frac{1}{1-\kappa \lambda }\{B_0+\frac{i}{\lambda +\omega }{B}_{-1}\}.\hfill \end{array}$$

(496)

Here,

$$\begin{array}{ccc}\hfill A_0& =& \left(\begin{array}{cc}0& q\\ -r& 0\end{array}\right),\hfill \end{array}$$

(497)

$$\begin{array}{ccc}\hfill B_0& =& -\frac{i}{2}v{\sigma }_3-\frac{\kappa }{2i}\left(\begin{array}{cc}0& q_x\\ r_x& 0\end{array}\right),\hfill \end{array}$$

(498)

$$\begin{array}{ccc}\hfill B_{-1}& =& \left(\begin{array}{cc}\eta & -p\\ -k& -\eta \end{array}\right).\hfill \end{array}$$

(499)

For the reduction $r={\delta }_1\overline{q},k={\delta }_2\overline{p}$ with $\kappa =2,{\delta }_j=\pm 1$, the system (487)–(492) takes the following compact form

$$\begin{array}{ccc}\hfill i{q}_x+q_{xt}-vq-2ip& =& 0,\hfill \end{array}$$

(500)

$$\begin{array}{ccc}\hfill v_t+2{\delta }_1{\left(\right|q|}^2{)}_x& =& 0,\hfill \end{array}$$

(501)

$$\begin{array}{ccc}\hfill p_t-2i\omega p-2\eta q& =& 0,\hfill \end{array}$$

(502)

$$\begin{array}{ccc}\hfill {\eta }_t+({\delta }_1\overline{q}p+{\delta }_2\overline{p}q)& =& 0.\hfill \end{array}$$

(503)

10.6. Akbota Equation

Our last example is the Akbota-I equation. As in the previous examples, it admits two cases, namely the Akbota-IA equation and the Akbota-IB equation.

10.6.1. The Akbota-IA Equation

The subject of this subsection is the following Akbota-IA equation

$$\begin{array}{ccc}\hfill {\mathbf{S}}_t-\mathbf{S}\wedge (\alpha S_{xx}+\beta {\mathbf{S}}_{xt})-u{\mathbf{S}}_x& =& 0,\hfill \end{array}$$

(504)

$$\begin{array}{ccc}\hfill u_x+\mathbf{S}\times ({\mathbf{S}}_x\wedge {\mathbf{S}}_t)& =& 0.\hfill \end{array}$$

(505)

This Akbota-IA equation is one of the integrable generalizations of the K-IAE. It has the following Lax representation

$$\begin{array}{ccc}\hfill {\Psi }_x-U_{13}\Psi & =& 0,\hfill \end{array}$$

(506)

$$\begin{array}{ccc}\hfill {\Psi }_t-V_{13}\Psi & =& 0,\hfill \end{array}$$

(507)

where

$$U_{13}=\frac{i}{2}\lambda S,V_{13}=\frac{1}{1-\lambda \beta }\{\alpha (\frac{1}{2}i{\lambda }^{2}S+\frac{1}{4}[S,S_x])+\beta \lambda Z\}.$$

(508)

Note that the gauge-equivalent counterpart of the Akbota-IA equation is the following nonlinear evolution equation

$$\begin{array}{ccc}\hfill i{q}_t+\alpha q_{xx}+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(509)

$$\begin{array}{ccc}\hfill v_x-{2\left[\alpha \right(\left|q\right|}^2{)}_x+{\beta \left(\right|q|}^2{)}_t]& =& 0.\hfill \end{array}$$

(510)

This is the Akbota-IIA equation. Its Lax representation is given by

$$\begin{array}{ccc}\hfill {\Phi }_x& =& U_{14}\Phi ,\hfill \end{array}$$

(511)

$$\begin{array}{ccc}\hfill {\Phi }_t& =& V_{14}\Phi ,\hfill \end{array}$$

(512)

where

$$\begin{array}{c}\hfill U_{14}=\frac{i\lambda }{2}{\sigma }_3+Q,Q=\left(\begin{array}{cc}0& \overline{q}\\ q& 0\end{array}\right),V_{14}=\frac{1}{1-\lambda \beta }\{\frac{i{\lambda }^2}{2}\alpha {\sigma }_3+\alpha \lambda Q+V_0\}\end{array}$$

(513)

with

$$\begin{array}{c}\hfill V_0=\left(\begin{array}{cc}{\alpha i\left|q\right|}^2+i\beta {\partial }_x^{-1}{\left|q\right|}_t^2& -i\beta {\overline{q}}_t-i\alpha {\overline{q}}_x\\ i\beta q_t+\alpha i{q}_x& -\left[\alpha i\right|q{|}^2+i\beta {\partial }_x^{-1}\left|q{|}_t^2\right]\end{array}\right).\end{array}$$

(514)

10.6.2. The Akbota-IB Equation

Consider the following Akbota-IB equation

$$\begin{array}{ccc}\hfill {\mathbf{S}}_x-\mathbf{S}\wedge (\alpha S_{tt}+\beta {\mathbf{S}}_{xt})-u{\mathbf{S}}_t& =& 0,\hfill \end{array}$$

(515)

$$\begin{array}{ccc}\hfill u_t+\mathbf{S}\times ({\mathbf{S}}_t\wedge {\mathbf{S}}_x)& =& 0.\hfill \end{array}$$

(516)

This Akbota-IB equation is an integrable generalization of the K-IBE. Its Lax representation is given by

$$\begin{array}{ccc}\hfill {\Psi }_t-U_{13}\Psi & =& 0,\hfill \end{array}$$

(517)

$$\begin{array}{ccc}\hfill {\Psi }_x-V_{13}\Psi & =& 0,\hfill \end{array}$$

(518)

where

$$U_{13}=\frac{i}{2}\lambda S,V_{13}=\frac{1}{1-\lambda \beta }\{\alpha (\frac{1}{2}i{\lambda }^{2}S+\frac{1}{4}[S,S_t])+\beta \lambda Z\}.$$

(519)

The gauge-equivalent counterpart of the Akbota-IB equation reads as

$$\begin{array}{ccc}\hfill i{q}_x+\alpha q_{tt}+\beta q_{xt}+vq& =& 0,\hfill \end{array}$$

(520)

$$\begin{array}{ccc}\hfill v_t-{2\left[\alpha \right(\left|q\right|}^2{)}_t+{\beta \left(\right|q|}^2{)}_x]& =& 0.\hfill \end{array}$$

(521)

This is the Akbota-IIB equation. Its Lax representation is given by

$$\begin{array}{ccc}\hfill {\Phi }_t& =& U_{14}\Phi ,\hfill \end{array}$$

(522)

$$\begin{array}{ccc}\hfill {\Phi }_x& =& V_{14}\Phi ,\hfill \end{array}$$

(523)

where

$$\begin{array}{c}\hfill U_{14}=\frac{i\lambda }{2}{\sigma }_3+Q,Q=\left(\begin{array}{cc}0& \overline{q}\\ q& 0\end{array}\right),V_{14}=\frac{1}{1-\lambda \beta }\{\frac{i{\lambda }^2}{2}\alpha {\sigma }_3+\alpha \lambda Q+V_0\}\end{array}$$

(524)

with

$$\begin{array}{c}\hfill V_0=\left(\begin{array}{cc}{\alpha i\left|q\right|}^2+i\beta {\partial }_t^{-1}{\left|q\right|}_x^2& -i\beta {\overline{q}}_x-i\alpha {\overline{q}}_t\\ i\beta q_x+\alpha i{q}_t& -\left[\alpha i\right|q{|}^2+i\beta {\partial }_t^{-1}\left|q{|}_x^2\right]\end{array}\right).\end{array}$$

(525)

11. Conclusions

In this paper, the Kuralay equations, namely the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE), have been studied. As is known, these equations are integrable by the inverse scattering transform method. The integrable motion of space curves induced by the K-IE and K-IIE was investigated. The gauge and geometrical equivalences between these two equations were established. The Hirota bilinear form of the KE was constructed. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. Note that these simplest soliton solutions admit generalizations as the traveling wave in terms of Jacobi elliptic functions. For example, we have shown that there are two such generalizations of the 1-soliton solution. The nonlocal and dispersionless versions of the Kuralay equations have been discussed. Finally, some integrable generalizations of the KE have been presented.