Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability

Abstract

This work is concerned with Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods to find the breather-wave, 1-Soliton, 2-Soliton, three-wave, and new periodic-wave results and some exact solitons of the special (1 + 1)-dimensional Korteweg–de Vries (KdV) equation. The model of concern is a partial differential equation that is used as a mathematical model of waves on shallow water surfaces. The results are attained as well as verified by Mathematica and Maple softwares. Some of the obtained solutions are represented in three-dimensional (3-D) and contour plots through the Mathematica tool. A stability analysis is performed to verify that the results are precise as well as accurate. Modulation instability is also performed for the steady-state solutions to the governing equation. The solutions are useful for the development of corresponding equations. This work shows that the methods used are simple and fruitful for investigating the results for other nonlinear partial differential models.

1. Introduction

Partial differential equations (PDEs) are a form of natural phenomena. PDEs are models used in the various areas of science and engineering, for example, the extended KdV equation [1], generalized fifth-order KdV equation [2], and unsteady KdV equation [3]. Many methods have been created to obtain results for PDEs, like the two-variable expansion method [4], the extended trial function method [5], the trial equation method [6], and the variational method [7].

In this work, three different methods are utilized: the Hirota bilinear method, the ${exp}_a$ function method, and the Sardar subequation method. The techniques of concern have been used to obtain results for the various models. For example, the Hirota bilinear technique is applied to achieve a new periodic wave, a three wave, and some kinds of solutions for new (2 + 1)-dimensional KdV equations [8]. Furthermore, 1-soliton, 2-soliton, and 3-soliton solutions of (2 + 1)-dimensional variable-coefficient Toda lattice equations are achieved by the Hirota bilinear scheme [9]; soliton and periodic-wave solutions of the two nonlinear evolution equations are gained by applying the Hirota bilinear scheme [10]; 1-soliton, 2-soliton solutions of the (2 + 1)-dimensional variable-coefficient Boussinesq equation are obtained by using the Hirota bilinear scheme along with Bäcklund transformations [11]; and one-soliton, two-soliton solutions and the lump solutions of the (2 + 1)-dimensional generalized variable-coefficient Hirota–Satsuma–Ito equations are gained by utilizing the Hirota bilinear scheme [12]. Some kinds of soliton results of the Westervelt equation are gained by using the ${exp}_a$ function method [13]; various analytical results for the Shynaray–IIA equation are achieved [14], and exact wave results for the Paraxial nonlinear Schrödinger model are achieved [15]. Various kinds of wave results for complex three-coupled Maccari’s systems are gained by using the Sardar sub-equation method [16], and some types of results for generalized fractional Tzitzéica-kinds evolution models are attained in [17].

Consider the (1 + 1)-dimensional Korteweg–de Vries (KdV) equation given as follows:

$$g_t+g{g}_x+\delta g_{xxx}=0,$$

(1)

where $\delta$ is a nonzero constant.

The Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) that serves as a mathematical model of waves on shallow water surfaces. A model equation for any physical system, the Korteweg–de Vries (KdV) equation, commonly credited to Korteweg and de Vries (1895), governs the propagation of weakly dispersive, weakly nonlinear water waves and serves as a model equation for any physical system for which the dispersion relation for frequency versus wave number.

In our research, we consider the special case of the above Equation (1). We consider the special (1 + 1)-D Korteweg–de Vries (KdV) equation [18] given as follows:

$$g_t+6g{g}_x+g_{xxx}=0,$$

(2)

where $g=g(x,t)$ indicates a wave function that depends on the components x and t. Equation (2) has many features; for example, it is originally modeled shallow water waves, describing the balance between nonlinearity and dispersion. This equation supports soliton solutions, which are stable, localized waves that maintain their shape. The KdV equation also describes ion-acoustic waves in plasmas. The KdV equation has several connections to physical problems, including shallow-water waves with weakly non-linear restoring forces, long internal waves in a density-stratified ocean, ion-acoustic waves in a plasma, acoustic waves on a crystal lattice, and many others. There are different types of KdV equations, including the modified KdV equation [19], the generalized KdV equation [20], the damped KdV equation [21], and the generalized perturbed KdV equation [22].

Basically, Equation (2) is obtained from the general form of the KdV equation, given as

$$g_t+\alpha g{g}_x+\beta g_{xxx}=0,$$

(3)

by considering $\alpha =6$ and $\beta =1$. As we consider the specific values of $\alpha$ and $\beta$, we can say that Equation (2) is a special form of the general KdV equation. For Equation (2), breather-wave, 1-soliton, 2-soliton, new three waves, new periodic waves, and some analytical wave solutions based on the Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods do not exist in the literature.

The fundamental aim of this research is to determine the breather-wave, 1-soliton, 2-soliton, new three wave, new periodic-wave, and some analytical wave solutions based on the Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods. The used methods may applied to other nonlinear partial differential equations. Furthermore, dynamical analyses are discussed in the form of stability analysis and modulation instability.

The motivation of our work is to explore the breather-wave, 1-soliton, 2-soliton, new three wave, new periodic-wave, and some analytical wave solutions based on the Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods. Hirota bilinear method provides different types of soliton solutions, including breather-wave, 1-soliton, 2-soliton, new wave, and new periodic wave. The ${exp}_a$ function method provides rational wave solutions, while Sardar sub-equation method provides various types of analytical solutions, including dark, bright, singular, dark-bright, and periodic. These three methods are also reliable, effective, and fruitful for other nonlinear partial differential equations in science and engineering. Additionally, stability analysis and modulation instability analysis are performed to check the stability and accuracy of the obtained solutions.

This paper has different sections. Section 2 presents a description of methodologies; Section 3 presents results of the Hirota bilinear method; Section 4 provides a mathematical analysis of the equation and its exact soliton solutions, Section 5 provides a graphical representation; Section 6 presents stability analysis; Section 7 shows modulation instability; and Section 8 is the conclusion.

2. Presentation of Methods

2.1. Hirota Bilinear Method

$$(i)D_x^p{D}_t^{q}u.v={({\partial }_x-{\partial }_{x^{\prime }})}^p{({\partial }_t-{\partial }_{t^{\prime }})}^{q}u(x,t)v(x^{\prime },t^{\prime }){|}_{x^{\prime }=x,t^{\prime }=t},$$

(4)

where $p,q\ge 0$ and $p+q\ge 1$.

$$D_{x}u.v=u_{x}v-u{v}_x,$$

(5)

$$D_x^{2}u.v=u_{xx}v-2u_x{v}_x+u{v}_{xx},$$

(6)

when $u=v$, one gains the Hirota bilinear formula, given as

$$D_x^p{D}_t^{q}u.u={({\partial }_x-{\partial }_{x^{\prime }})}^p{({\partial }_t-{\partial }_{t^{\prime }})}^{q}u(x,t)u(x^{\prime },t^{\prime }){|}_{x^{\prime }=x,t^{\prime }=t},$$

(7)

where $p,q\ge 0$ and $p+q\ge 1$.

$$D_{x}u.u=0.$$

(8)

$$D_x^{2}u.u=2(u_{xx}u-u_x^2).$$

(9)

If $n=0$, Hirota bilinear derivatives with respect to x, y, and t may be defined completely in the same way. Let us use a transformation given as

$$g(x,t)=2(ln{(u(x,t))}_{xx}.$$

(10)

By using Equation (10), we obtain a bilinear form of Equation (2).

$$(D_x^4+D_x{D}_t)u.u=2(u_{xxxx}u-4u_{xxx}{u}_x+3u_{xx}^2+u_{xt}u-u_x{u}_t)=0.$$

(11)

2.2. Summary of $Ex{p}_a$ Function Method

In this subsection, we will explain the method of concern [23,24,25,26,27,28,29].

Assuming a nonlinear PDE,

$$S(w,w^2{w}_x,w_t,w_{xx},w_{tt},w_{xt},\dots )=0.$$

(12)

Equation (12) changes into a nonlinear ordinary differential equation (ODE):

$$T(W,W^{\prime },W^{″},\dots ,)=0.$$

(13)

Using a traveling wave transformation,

$$h(x,t)=W(\xi ),\xi =\delta x+\lambda t.$$

(14)

Supposing the solutions for Equation (13) are

$$W(\xi )={\displaystyle \frac{{\alpha }_0+{\alpha }_1{d}^{\xi }+\dots +{\alpha }_m{d}^{m\xi }}{{\beta }_0+{\beta }_1{d}^{\xi }+\dots +{\beta }_m{d}^{m\xi }}},d\ne 0,1.$$

(15)

Here, ${\alpha }_i$ and ${\beta }_i(0\le i\le m)$ are unknowns. The value of m is determined by a homogeneous balance approach in Equation (13). Putting Equation (15) into Equation (13) yields

$$\wp (d^{\xi })={\ell }_0+{\ell }_1{d}^{\xi }+\dots +{\ell }_t{d}^{t\xi }=0.$$

(16)

By putting ${\ell }_i(0\le i\le t)$ into Equation (16) equal to zero, a set of equations is obtained:

$${\ell }_i=0,wherei=0,\dots ,t.$$

(17)

In this way, we can attain the results for Equation (12).

2.3. Description of the Sardar Sub-Equation Method

Here, we will mention the main steps of this method [30] by assuming a nonlinear PDE:

$$J(w,w_x,w_{xx},w_{xt},w{w}_{tt},w_{xxt},\dots )=0,$$

(18)

where $w=w(x,t)$ denotes the wave function. Assume the transformation given as

$$w(x,t)=W(\zeta ),\zeta =\lambda x+\mu t.$$

(19)

This results in the non-linear ODE:

$$Y(W,W^{″},W{W}^{″},W^{\prime }{W}^2,\dots )=0.$$

(20)

Consider a result of Equation (20), given as

$$W(\zeta )=\sum _{i=0}^m{b}_i{\psi }^i(\zeta ),$$

(21)

where $\psi (\zeta )$ satisfies the ODE:

$${\psi }^{\prime }(\zeta )=\sqrt{\sigma +\kappa {\psi }^2(\zeta )+{\psi }^4(\zeta )},$$

(22)

where $\sigma$ and $\kappa$ are the parameters.

Next, we proceed by first putting Equations (21) and (22) into Equation (20) and collecting the coefficients of each term of ${\psi }^i$. As a result, we obtain a system of equations. By solving the system, we obtain the given sets. Type 1: if $\kappa >0$ and $\sigma =0$, then

$${\psi }_1^{\pm }=\pm \sqrt{\kappa ab}sec{h}_{ab}(\sqrt{\kappa }\zeta ),$$

(23)

$${\psi }_2^{\pm }=\pm \sqrt{\kappa ab}csc{h}_{ab}(\sqrt{\kappa }\zeta ),$$

(24)

where, $sec{h}_{ab}(\zeta )={\displaystyle \frac{2}{a{e}^{\zeta }+b{e}^{-\zeta }}}$, $csc{h}_{ab}(\zeta )={\displaystyle \frac{2}{a{e}^{\zeta }-b{e}^{-\zeta }}}$

Type 2: if $\kappa <0$ and $\sigma =0$, we have

$${\psi }_3^{\pm }=\pm \sqrt{-\kappa ab}{sec}_{ab}(\sqrt{-\kappa }\zeta ),$$

(25)

$${\psi }_4^{\pm }=\pm \sqrt{-\kappa ab}{csc}_{ab}(\sqrt{-\kappa }\zeta ),$$

(26)

where, ${sec}_{ab}(\zeta )={\displaystyle \frac{2}{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}}$, ${csc}_{ab}(\zeta )={\displaystyle \frac{2\iota }{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}}$

Type 3: if $\kappa <0$ and $\sigma ={\displaystyle \frac{{\kappa }^2}{4}}$, then

$${\psi }_5^{\pm }=\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}{tanh}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{2}}}\zeta ),$$

(27)

$${\psi }_6^{\pm }=\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}{coth}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{2}}}\zeta ),$$

(28)

$${\psi }_7^{\pm }=\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}({tanh}_{ab}(\sqrt{-2\kappa }\zeta )\pm \iota \sqrt{ab}sec{h}_{ab}(\sqrt{-2\kappa }\zeta )),$$

(29)

$${\psi }_8^{\pm }=\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}({coth}_{ab}(\sqrt{-2\kappa }\zeta )\pm \sqrt{ab}csc{h}_{ab}(\sqrt{-2\kappa }\zeta )),$$

(30)

$${\psi }_9^{\pm }=\pm \sqrt{-{\displaystyle \frac{\kappa }{8}}}({tanh}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{8}}}\zeta )+{coth}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{8}}}\zeta )),$$

(31)

where, ${tanh}_{ab}(\zeta )={\displaystyle \frac{a{e}^{\zeta }-b{e}^{-\zeta }}{a{e}^{\zeta }+b{e}^{-\zeta }}}$, ${coth}_{ab}(\zeta )={\displaystyle \frac{a{e}^{\zeta }+b{e}^{-\zeta }}{a{e}^{\zeta }-b{e}^{-\zeta }}}$

Type 4: if $\kappa >0$ and $\sigma ={\displaystyle \frac{{\kappa }^2}{4}}$, then

$${\psi }_{10}^{\pm }=\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}{tan}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{2}}}\zeta ),$$

(32)

$${\psi }_{11}^{\pm }=\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}{cot}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{2}}}\zeta ),$$

(33)

$${\psi }_{12}^{\pm }=\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}({tan}_{ab}(\sqrt{2\kappa }\zeta )\pm \sqrt{ab}{sec}_{ab}(\sqrt{2\kappa }\zeta )),$$

(34)

$${\psi }_{13}^{\pm }=\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}({cot}_{ab}(\sqrt{2\kappa }\zeta )\pm \sqrt{ab}{csc}_{ab}(\sqrt{2\kappa }\zeta )),$$

(35)

$${\psi }_{14}^{\pm }=\pm \sqrt{{\displaystyle \frac{\kappa }{8}}}({tan}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{8}}}\zeta )+{cot}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{8}}}\zeta )),$$

(36)

where, ${tan}_{ab}(\zeta )=-\iota {\displaystyle \frac{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}}$, ${cot}_{ab}(\zeta )=\iota {\displaystyle \frac{a{e}^{\iota \zeta }+b{e}^{-\iota \zeta }}{a{e}^{\iota \zeta }-b{e}^{-\iota \zeta }}}$

3. Breather Wave Soliton

Consider the given relation to gain breather wave results [31].

$$u(x,t)={\kappa }_2{e}^{p_1\left(c_{1}t+x\right)}+{\kappa }_{1}cos\left(p\left(c_{2}t+x\right)\right)+e^{-p_1\left(c_{1}t+x\right)}.$$

(37)

Put Equation (37) into Equation (11) and sum up coefficients of every order of x,t,$e^{-p_1\left(c_{1}t+x\right)}$, $e^{p_1\left(c_{1}t+x\right)}$ equal to 0. By solving the system, we obtain

Set 1:

$$\left\{p={\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1}{{\kappa }_1}},c_1={\displaystyle \frac{\left(-{\kappa }_1^2-12{\kappa }_2\right)p_1^2}{{\kappa }_1^2}},c_2={\displaystyle \frac{\left(-{\kappa }_1^2-4{\kappa }_2\right)p_1^2}{{\kappa }_1^2}}\right\}.$$

(38)

$$\begin{array}{c}{g}_1(x,t)=2[(p_1^2{e}^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}(4{\kappa }_1({\kappa }_2(2e^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}+\sqrt{-{\kappa }_2}{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}\hfill \\ sin({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-1)p_1^{2}t+x)}{{\kappa }_1}}))-\sqrt{-{\kappa }_2}sin({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-1)p_1^{2}t+x)}{{\kappa }_1}}))\\ +{\kappa }_1^2({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+1)cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-1)p_1^{2}t+x)}{{\kappa }_1}})\\ +4{\kappa }_2({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+1)cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-1)p_1^{2}t+x)}{{\kappa }_1}})))/\\ \hfill ({\kappa }_1({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+{\kappa }_1{e}^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-1)p_1^{2}t+x)}{{\kappa }_1}})+1{)}^2)].\end{array}$$

(39)

Set 2:

$$\left\{p=-{\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1}{{\kappa }_1}},c_1={\displaystyle \frac{\left(-{\kappa }_1^2-12{\kappa }_2\right)p_1^2}{{\kappa }_1^2}},c_2={\displaystyle \frac{\left(-3{\kappa }_1^2-4{\kappa }_2\right)p_1^2}{{\kappa }_1^2}}\right\}.$$

(40)

$$\begin{array}{c}{g}_2(x,t)=2[(p_1^2{e}^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}(4{\kappa }_1({\kappa }_2(2e^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}+\sqrt{-{\kappa }_2}{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}\hfill \\ sin({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-3)p_1^{2}t+x)}{{\kappa }_1}}))-\sqrt{-{\kappa }_2}sin({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-3)p_1^{2}t+x)}{{\kappa }_1}}))\\ +{\kappa }_1^2({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+1)cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-3)p_1^{2}t+x)}{{\kappa }_1}})+4{\kappa }_2({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+1)\\ cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-3)p_1^{2}t+x)}{{\kappa }_1}})))/({\kappa }_1({\kappa }_2{e}^{2p_1((-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{2}t+x)}+{\kappa }_1{e}^{(-{\displaystyle \frac{12{\kappa }_2}{{\kappa }_1^2}}-1)p_1^{3}t+p_{1}x}\\ \hfill cos({\displaystyle \frac{2\sqrt{-{\kappa }_2}{p}_1((-\frac{4{\kappa }_2}{{\kappa }_1^2}-3)p_1^{2}t+x)}{{\kappa }_1}})+1{)}^2)].\end{array}$$

(41)

Set 3:

$$\left\{p=p_1\pm \iota ,c_1=-4p_1^2,c_2=-4p_1^2\right\}.$$

(42)

$$\begin{array}{c}{g}_3(x,t)=2[(p_1^2{e}^{p_1(x-4p_1^{2}t)}({\kappa }_1{e}^{p_1(x-4p_1^{2}t)}(\iota sin(\iota p_1(x-4p_1^{2}t))+cos(\iota p_1(x-4p_1^{2}t)))+2)\hfill \\ ({\kappa }_1(cos(\iota p_1(x-4p_1^{2}t))-\iota sin(\iota p_1(x-4p_1^{2}t)))+2{\kappa }_2{e}^{p_1(x-4p_1^{2}t)}))/(({\kappa }_1{e}^{p_1(x-4p_1^{2}t)}\\ \hfill cos(\iota p_1(x-4p_1^{2}t))+{\kappa }_2{e}^{2p_1(x-4p_1^{2}t)}+1{)}^2)].\end{array}$$

(43)

4. Soliton Solutions

In this part, we study the (1 + 1)-dimensional KdV equation of soliton solutions in Equation (11). The 1-soliton, 2-soliton, and three-wave solutions will be achieved.

4.1. 1-Soliton Solutions

For the 1-soliton result, we assume the transformation given as [32].

$$u(x,t)=e^{\kappa \left(a_{1}x+b_{1}t\right)+\delta }+1.$$

(44)

Putting Equation (44) into Equation (11), we obtain a system by collecting coefficients of $e^{\kappa \left(a_{1}x+b_{1}t\right)+\delta }$ equal to 0. By solving the system, we achieve

Set:

$$\left\{b_1=-a_1^3{\kappa }^2\right\}.$$

(45)

$$g(x,t)=2[{\displaystyle \frac{a_1^2{\kappa }^2{e}^{\kappa \left(a_{1}x-a_1^3{\kappa }^{2}t\right)+\delta }}{e^{\kappa \left(a_{1}x-a_1^3{\kappa }^{2}t\right)+\delta }+1}}-{\displaystyle \frac{a_1^2{\kappa }^2{e}^{2\kappa \left(a_{1}x-a_1^3{\kappa }^{2}t\right)+2\delta }}{{\left(e^{\kappa \left(a_{1}x-a_1^3{\kappa }^{2}t\right)+\delta }+1\right)}^2}}].$$

(46)

4.2. 2-Soliton Solutions

For 2-soliton solutions, we consider the relation given as [32].

$$\begin{array}{c}g(x,t)=A_{12}exp\left({\kappa }_1\left(a_{1}x+d_{1}t\right)+{\kappa }_2\left(a_{2}x+d_{2}t\right)+{\delta }_1+{\delta }_2\right)+e^{{\kappa }_1\left(a_{1}x+d_{1}t\right)+{\delta }_1}\hfill \\ \hfill +e^{{\kappa }_2\left(a_{2}x+d_{2}t\right)+{\delta }_2}+1.\end{array}$$

(47)

Put Equation (47) into Equation (11), and collect coefficients of each order of $exp\left({\kappa }_1\left(a_{1}x+d_{1}t\right)+{\delta }_1\right)$, $exp\left({\kappa }_2\left(a_{2}x+d_{2}t\right)+{\delta }_2\right)$, and $exp\left({\kappa }_1\left(a_{1}x+d_{1}t\right)+{\kappa }_2\left(a_{2}x+d_{2}t\right)+{\delta }_1+{\delta }_2\right)$, taking it as equal to 0. By solving the system, we achieve

Set:

$$\left\{d_1=-a_1^3{\kappa }_1^2,d_2=-a_2^3{\kappa }_2^2,A_{12}={\displaystyle \frac{{\left(a_1{\kappa }_1-a_2{\kappa }_2\right)}^2}{{\left(a_1{\kappa }_1+a_2{\kappa }_2\right)}^2}}\right\}.$$

(48)

$$\begin{array}{c}g(x,t)=2[(({(a_1{\kappa }_1-a_2{\kappa }_2)}^{2}exp(-a_1^3{\kappa }_1^{3}t-a_2^3{\kappa }_2^{3}t+a_1{\kappa }_{1}x+a_2{\kappa }_{2}x+{\delta }_1+{\delta }_2)+a_1^2{\kappa }_1^2{e}^{-a_1^3{\kappa }_1^{3}t+a_1{\kappa }_{1}x+{\delta }_1}\hfill \\ +a_2^2{\kappa }_2^2{e}^{a_2^3{\kappa }_2^3(-t)+a_2{\kappa }_{2}x+{\delta }_2})({\displaystyle \frac{{(a_1{\kappa }_1-a_2{\kappa }_2)}^{2}exp(a_1^3{\kappa }_1^3(-t)-a_2^3{\kappa }_2^{3}t+a_1{\kappa }_{1}x+a_2{\kappa }_{2}x+{\delta }_1+{\delta }_2)}{{(a_1{\kappa }_1+a_2{\kappa }_2)}^2}}\\ +e^{a_1^3{\kappa }_1^3(-t)+a_1{\kappa }_{1}x+{\delta }_1}+e^{a_2^3{\kappa }_2^3(-t)+a_2{\kappa }_{2}x+{\delta }_2}+1)-(({(a_1{\kappa }_1-a_2{\kappa }_2)}^{2}exp(-a_1^3{\kappa }_1^{3}t\\ -a_2^3{\kappa }_2^{3}t+a_1{\kappa }_{1}x+a_2{\kappa }_{2}x+{\delta }_1+{\delta }_2))/(a_1{\kappa }_1+a_2{\kappa }_2)+a_1{\kappa }_1{e}^{-a_1^3{\kappa }_1^{3}t+a_1{\kappa }_{1}x+{\delta }_1}+a_2{\kappa }_2\\ e^{-a_2^3{\kappa }_2^{3}t+a_2{\kappa }_{2}x+{\delta }_2}{)}^2)/((({(a_1{\kappa }_1-a_2{\kappa }_2)}^{2}exp(a_1^3{\kappa }_1^3(-t)-a_2^3{\kappa }_2^{3}t+a_1{\kappa }_{1}x+a_2{\kappa }_{2}x+{\delta }_1+{\delta }_2))/\\ \hfill ({(a_1{\kappa }_1+a_2{\kappa }_2)}^2)+e^{a_1^3{\kappa }_1^3(-t)+a_1{\kappa }_{1}x+{\delta }_1}+e^{a_2^3{\kappa }_2^3(-t)+a_2{\kappa }_{2}x+{\delta }_2}+1{)}^2)].\end{array}$$

(49)

4.3. New Three-Wave Soliton

For new three-wave solutions, we assume a relation [33].

$$g(x,t)={\kappa }_2{e}^{a_{1}x+d_{1}t}+{\kappa }_{3}sin\left(a_{3}x+d_{3}t\right)+{\kappa }_{1}cos\left(a_{2}x+d_{2}t\right)+e^{-\left(a_{1}x+d_{1}t\right)}.$$

(50)

Put Equation (50) into Equation (11), collect coefficients of each order of $e^{a_{1}x+d_{1}t}$, $e^{-\left(a_{1}x+d_{1}t\right)}$,$sin\left(a_{3}x+d_{3}t\right)$ and $cos\left(a_{2}x+d_{2}t\right)$, and take it as equal to 0. Solving the system yields

Set 1:

$$\left\{a_1=-i{a}_2,a_3=-a_2,d_1=-4i{a}_2^3,d_2=4a_2^3,d_3=-4a_2^3\right\}.$$

(51)

$$g_1(x,t)=2[-{\displaystyle \frac{4a_2^2\left({\kappa }_1+2{\kappa }_2-i{\kappa }_3\right)\left({\kappa }_1+i{\kappa }_3+2\right)e^{2i{a}_2\left(4a_2^{2}t+x\right)}}{{\left({\kappa }_1\left(1+e^{2i{a}_2\left(4a_2^{2}t+x\right)}\right)+i{\kappa }_3{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}+2e^{2i{a}_2\left(4a_2^{2}t+x\right)}+2{\kappa }_2-i{\kappa }_3\right)}^2}}].$$

(52)

Set 2:

$$\left\{a_1=i{a}_2,a_3=-a_2,d_1=4i{a}_2^3,d_2=4a_2^3,d_3=-4a_2^3\right\}.$$

(53)

$$g_2(x,t)=2[-{\displaystyle \frac{4a_2^2\left({\kappa }_1-i{\kappa }_3+2\right)\left({\kappa }_1+2{\kappa }_2+i{\kappa }_3\right)e^{2i{a}_2\left(4a_2^{2}t+x\right)}}{{\left({\kappa }_1\left(1+e^{2i{a}_2\left(4a_2^{2}t+x\right)}\right)+2{\kappa }_2{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}+i{\kappa }_3{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}-i{\kappa }_3+2\right)}^2}}].$$

(54)

Set 3:

$$\left\{a_1=-i{a}_2,a_3=a_2,d_1=-4i{a}_2^3,d_2=4a_2^3,d_3=4a_2^3\right\}.$$

(55)

$$g_3(x,t)=2[-{\displaystyle \frac{4a_2^2\left({\kappa }_1-i{\kappa }_3+2\right)\left({\kappa }_1+2{\kappa }_2+i{\kappa }_3\right)e^{2i{a}_2\left(4a_2^{2}t+x\right)}}{{\left({\kappa }_1\left(1+e^{2i{a}_2\left(4a_2^{2}t+x\right)}\right)-i{\kappa }_3{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}+2e^{2i{a}_2\left(4a_2^{2}t+x\right)}+2{\kappa }_2+i{\kappa }_3\right)}^2}}].$$

(56)

Set 4:

$$\left\{a_1=i{a}_2,a_3=a_2,d_1=4i{a}_2^3,d_2=4a_2^3,d_3=4a_2^3\right\}.$$

(57)

$$g_4(x,t)=2[-{\displaystyle \frac{4a_2^2\left({\kappa }_1+2{\kappa }_2-i{\kappa }_3\right)\left({\kappa }_1+i{\kappa }_3+2\right)e^{2i{a}_2\left(4a_2^{2}t+x\right)}}{{\left({\kappa }_1\left(1+e^{2i{a}_2\left(4a_2^{2}t+x\right)}\right)+2{\kappa }_2{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}-i{\kappa }_3{e}^{2i{a}_2\left(4a_2^{2}t+x\right)}+i{\kappa }_3+2\right)}^2}}].$$

(58)

4.4. New Periodic-Wave

For new periodic wave results, consider a relation [34]:

$$g(x,t)={\kappa }_1{e}^{a_{2}t+a_{1}x+a_3}+e^{-(a_{2}t+a_{1}x+a_3)}+{\kappa }_{2}cos(p(b_{2}t+b_{1}x+b_3))+{\kappa }_{3}cosh(c_{2}t+c_{1}x+c_3)+{\kappa }_4.$$

(59)

Insert Equation (59) into Equation (11), collect coefficients of each order of $e^{a_{2}t+a_{1}x+a_3}$, $e^{-\left(a_{2}t+a_{1}x+a_3\right)}$,$cos\left(p\left(b_{2}t+b_{1}x+b_3\right)\right)$, $cosh\left(c_{2}t+c_{1}x+c_3\right)$, and take it as equal to 0. By solving the system, we obtain

Set 1:

$$\left\{a_1=-c_1,a_2=4c_1^3,b_2=-4b_1{c}_1^2,c_2=-4c_1^3,p=\mp {\displaystyle \frac{i{c}_1}{b_1}},{\kappa }_4=0\right\}.$$

(60)

$$\begin{array}{c}g(x,t)=2[{\displaystyle \frac{c_1^2{\kappa }_1{e}^{a_3+4c_1^{3}t-c_{1}x}+c_1^2{e}^{-a_3-4c_1^{3}t+c_{1}x}+c_1^2{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+c_1^2{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3)}{{\kappa }_1{e}^{a_3+4c_1^{3}t-c_{1}x}+e^{-a_3-4c_1^{3}t+c_{1}x}+{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3)}}\hfill \\ \hfill -{\displaystyle \frac{{(-c_1{\kappa }_1{e}^{a_3+4c_1^{3}t-c_{1}x}+c_1{e}^{-a_3-4c_1^{3}t+c_{1}x}+c_1{\kappa }_{2}sinh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+c_1{\kappa }_{3}sinh(-4c_1^{3}t+c_{1}x+c_3))}^2}{{({\kappa }_1{e}^{a_3+4c_1^{3}t-c_{1}x}+e^{-a_3-4c_1^{3}t+c_{1}x}+{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3))}^2}}].\end{array}$$

(61)

Set 2:

$$\left\{a_1=c_1,a_2=-4c_1^3,b_2=-4b_1{c}_1^2,c_2=-4c_1^3,p=\mp {\displaystyle \frac{i{c}_1}{b_1}},{\kappa }_4=0\right\}.$$

(62)

$$\begin{array}{c}g(x,t)=2[{\displaystyle \frac{c_1^2{\kappa }_1{e}^{a_3-4c_1^{3}t+c_{1}x}+c_1^2{e}^{-a_3+4c_1^{3}t-c_{1}x}+c_1^2{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+c_1^2{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3)}{{\kappa }_1{e}^{a_3-4c_1^{3}t+c_{1}x}+e^{-a_3+4c_1^{3}t-c_{1}x}+{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3)}}\hfill \\ \hfill -{\displaystyle \frac{{(c_1{\kappa }_1{e}^{a_3-4c_1^{3}t+c_{1}x}+c_1(-e^{-a_3+4c_1^{3}t-c_{1}x})+c_1{\kappa }_{2}sinh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+c_1{\kappa }_{3}sinh(-4c_1^{3}t+c_{1}x+c_3))}^2}{{({\kappa }_1{e}^{a_3-4c_1^{3}t+c_{1}x}+e^{-a_3+4c_1^{3}t-c_{1}x}+{\kappa }_{2}cosh(\frac{c_1(-4b_1{c}_1^{2}t+b_{1}x+b_3)}{b_1})+{\kappa }_{3}cosh(-4c_1^{3}t+c_{1}x+c_3))}^2}}].\end{array}$$

(63)

5. Exact Wave Solutions

Consider a traveling wave transformation;

$$g(x,t)=G(\zeta ),\zeta =\lambda x+\tau t.$$

(64)

By putting Equation (64) into Equation (2), we gain an ordinary differential equation given as follows:

$${\lambda }^3{G}^{″}+3\lambda G^2+\tau G=0.$$

(65)

Using homogenous balance scheme into Equation (65) yields $m=2$. Now, we will obtain some new exact wave results for Equation (65).

5.1. Exact Siliton Solutions by $Ex{p}_a$ Function Method

For $m=2$, Equation (15) transforms into

$$G(\zeta )={\displaystyle \frac{{\alpha }_0+{\alpha }_1{d}^{\zeta }+{\alpha }_2{d}^{2\zeta }}{{\beta }_0+{\beta }_1{d}^{\zeta }+{\beta }_2{d}^{2\zeta }}}.$$

(66)

Putting Equation (66) into Equation (65) and solving the system of equations yields

Set 1:

$$\left\{{\alpha }_0=0,{\alpha }_1={\beta }_1{\lambda }^2{log}^2(d),{\alpha }_2=0,{\beta }_2={\displaystyle \frac{{\beta }_1^2}{4{\beta }_0}},\tau =-{\lambda }^3{log}^2(d)\right\}.$$

(67)

$$g(x,t)=4{\beta }_0{\beta }_1{\lambda }^2{log}^2(d)\left({\displaystyle \frac{d^{(\lambda x-{\lambda }^3{log}^2(d)t)}}{{\left(2{\beta }_0+{\beta }_1{d}^{(\lambda x-{\lambda }^3{log}^2(d)t)}\right)}^2}}\right).$$

(68)

Set 2:

$$\left\{{\alpha }_0=-{\displaystyle \frac{1}{3}}{\beta }_0{\lambda }^2{log}^2(d),{\alpha }_1={\displaystyle \frac{2}{3}}{\beta }_1{\lambda }^2{log}^2(d),{\alpha }_2=-{\displaystyle \frac{{\beta }_1^2{\lambda }^2{log}^2(d)}{12{\beta }_0}},{\beta }_2={\displaystyle \frac{{\beta }_1^2}{4{\beta }_0}},\tau ={\lambda }^3{log}^2(d)\right\}.$$

(69)

$$g(x,t)=-{\displaystyle \frac{{\lambda }^2{log}^2(d)}{3}}\left({\displaystyle \frac{4{\beta }_0^2+{\beta }_1^2{d}^{2(\lambda x+{\lambda }^3{log}^2(d)t)}-8{\beta }_0{\beta }_1{d}^{(\lambda x+{\lambda }^3{log}^2(d)t)}}{{\left(2{\beta }_0+{\beta }_1{d}^{(\lambda x+{\lambda }^3{log}^2(d)t)}\right)}^2}}\right).$$

(70)

5.2. Exact Soliton Results via the Sardar Sub-Equation Method

For m = 2, Equation (21) changes to

$$G(\zeta )=b_0+b_1\psi (\zeta )+b_2{\psi }^2.$$

(71)

Put Equation (71) into Equation (65) along with Equation (22). Collect coefficients of every order of $\psi (\zeta )$ equal to 0. Solving the system yields

Set 1:

$$\left\{b_0={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp \sqrt{{\lambda }^4\left({\kappa }^2-3\sigma \right)}\right),b_1=0,b_2=-2{\lambda }^2,\tau =(\pm 4\lambda \sqrt{{\lambda }^4\left({\kappa }^2-3\sigma \right)})\right\}.$$

(72)

Type 1: if $\kappa >0$ and $\sigma =0$, we have

$$g(x,t)=-{\displaystyle \frac{4\kappa {\lambda }^2}{3}}-2{\lambda }^2{(\pm \sqrt{-\kappa ab}sec{h}_{ab}(\sqrt{\kappa }(\lambda x\pm 4{\lambda }^3\kappa t)))}^2.$$

(73)

$$g(x,t)=-{\displaystyle \frac{4\kappa {\lambda }^2}{3}}-2{\lambda }^2{(\pm \sqrt{\kappa ab}csc{h}_{ab}(\sqrt{\kappa }(\lambda x\pm 4{\lambda }^3\kappa t)))}^2.$$

(74)

Type 2: if $\kappa <0$ and $\sigma =0$, we have

$$g(x,t)=-{\displaystyle \frac{4\kappa {\lambda }^2}{3}}-2{\lambda }^2{(\pm \sqrt{-\kappa ab}{sec}_{ab}(\sqrt{-\kappa }(\lambda x\pm 4{\lambda }^3\kappa t)))}^2.$$

(75)

$$g(x,t)=-{\displaystyle \frac{4\kappa {\lambda }^2}{3}}-2{\lambda }^2{(\pm \sqrt{-\kappa ab}{csc}_{ab}(\sqrt{-\kappa }(\lambda x\pm 4{\lambda }^3\kappa t)))}^2.$$

(76)

Type 3: if $\kappa <0$ and $\sigma ={\displaystyle \frac{{\kappa }^2}{4}}$, we have

$$g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2{(\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}{tanh}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{2}}}(\lambda x\pm 2{\lambda }^3\kappa t)))}^2.$$

(77)

$$g(x,t)={\displaystyle \frac{2}{3}}(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}})-2{\lambda }^2{(\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}{coth}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{2}}}(\lambda x\pm 2{\lambda }^3\kappa t)))}^2.$$

(78)

$$\begin{array}{c}g(x,t)={\displaystyle \frac{2}{3}}(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}})-2{\lambda }^2(\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}({tanh}_{ab}(\sqrt{-2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t))\hfill \\ \hfill \pm \iota \sqrt{ab}sec{h}_{ab}(\sqrt{-2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t)){))}^2.\end{array}$$

(79)

$$\begin{array}{c}g(x,t)={\displaystyle \frac{2}{3}}(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}})-2{\lambda }^2(\pm \sqrt{-{\displaystyle \frac{\kappa }{2}}}({coth}_{ab}(\sqrt{-2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t))\hfill \\ \hfill \pm \sqrt{ab}csc{h}_{ab}(\sqrt{-2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t)){))}^2.\end{array}$$

(80)

$$\begin{array}{c}g(x,t)={\displaystyle \frac{2}{3}}(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}})-2{\lambda }^2(\pm \sqrt{-{\displaystyle \frac{\kappa }{8}}}({tanh}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{8}}}(\lambda x\pm 2{\lambda }^3\kappa t))+\hfill \\ \hfill {coth}_{ab}(\sqrt{-{\displaystyle \frac{\kappa }{8}}}(\lambda x\pm 2{\lambda }^3\kappa t)){))}^2.\end{array}$$

(81)

Type 4: if $\kappa >0$ and $\sigma ={\displaystyle \frac{{\kappa }^2}{4}}$, we have

$$g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2{(\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}{tan}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{2}}}(\lambda x\pm 2{\lambda }^3\kappa t)))}^2.$$

(82)

$$g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2{(\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}{cot}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{2}}}(\lambda x\pm 2{\lambda }^3\kappa t)))}^2.$$

(83)

$$\begin{array}{c}g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2(\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}({tan}_{ab}(\sqrt{2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t))\hfill \\ \hfill \pm \sqrt{ab}{sec}_{ab}(\sqrt{2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t)){))}^2.\end{array}$$

(84)

$$\begin{array}{c}g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2(\pm \sqrt{{\displaystyle \frac{\kappa }{2}}}({cot}_{ab}(\sqrt{2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t))\hfill \\ \hfill \pm \sqrt{ab}{csc}_{ab}(\sqrt{2\kappa }(\lambda x\pm 2{\lambda }^3\kappa t)){))}^2.\end{array}$$

(85)

$$g(x,t)={\displaystyle \frac{2}{3}}\left(-\kappa {\lambda }^2\mp {\displaystyle \frac{\kappa {\lambda }^2}{2}}\right)-2{\lambda }^2{(\pm \sqrt{{\displaystyle \frac{\kappa }{8}}}({tan}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{8}}}(\lambda x\pm 2{\lambda }^3\kappa t))+{cot}_{ab}(\sqrt{{\displaystyle \frac{\kappa }{8}}}(\lambda x\pm 2{\lambda }^3\kappa t))))}^2.$$

(86)

6. Graphical Explanation

Here, we will represent some of the solutions gained graphically in 3D and contour (see Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7).

7. Stability Analysis

In this section, we discuss the stability analysis of the equation of concern. It is applied for many equations, like [35,36]. For the Equation (2) stability analysis, we use the Hamiltonian transformation,

$$\begin{array}{c}\hfill \mathcal{S}={\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }{\mathit{v}}^{\mathit{2}}dx,\end{array}$$

(87)

where $\mathcal{S}$ represents a momentum factor, where $\mathit{v}(\mathit{x},\mathit{t})$ indicates the possibility power. The necessary criterion for the stable solution is shown as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{S}}{\partial \tau }}>0,\end{array}$$

(88)

where $\tau$ denotes a soliton velocity, and substituting Equation (74) in Equation (87) yields

$$\mathcal{S}={\displaystyle \frac{1}{2}}{\int }_{-6}^6{(-{\displaystyle \frac{4\kappa {\lambda }^2}{3}}-2{\lambda }^2{(\pm \sqrt{\kappa ab}csc{h}_{ab}(\sqrt{\kappa }(\lambda x\pm 4{\lambda }^3\kappa t)))}^2)}^{2}dx,$$

(89)

by using the criterion given in Equation (88), we obtain

$$\begin{array}{c}{\displaystyle \frac{1}{2}}({\displaystyle \frac{16}{3}}\kappa {\lambda }^3(-{\displaystyle \frac{1}{2}}\sqrt{\kappa }ttanh({\displaystyle \frac{1}{2}}\sqrt{\kappa }(6\lambda +4\kappa {\lambda }^{3}t))-{\displaystyle \frac{1}{2}}\sqrt{\kappa }ttanh({\displaystyle \frac{1}{2}}\sqrt{\kappa }(6\lambda -4\kappa {\lambda }^{3}t))\hfill \\ +{\displaystyle \frac{1}{2}}\sqrt{\kappa }tcoth({\displaystyle \frac{1}{2}}\sqrt{\kappa }(6\lambda +4\kappa {\lambda }^{3}t))+{\displaystyle \frac{1}{2}}\sqrt{\kappa }tcoth({\displaystyle \frac{1}{2}}\sqrt{\kappa }(6\lambda -4\kappa {\lambda }^{3}t)))\\ \hfill -4\sqrt{\kappa }{\lambda }^3(\sqrt{\kappa }t{\csc \mathrm{h}}^2(\sqrt{\kappa }(6\lambda -4\kappa {\lambda }^{3}t))-\sqrt{\kappa }t{\csc \mathrm{h}}^2(\sqrt{\kappa }(6\lambda +4\kappa {\lambda }^{3}t))))>0.\end{array}$$

(90)

Hence, Equation (2) indicates that a stable nonlinear model provides the condition is fulfilled.

8. Modulation Instability (MI)

Dispersive and unsteady components of nonlinear systems have the ability to reduce the stability of the results. Modulating instabilities, or nonlinear factors interacting via the dispersion notion, are frequently the cause of instabilities for the stable solution of a new equation for partial differential equations fitting the optic transmission concept. In a nonlinear dispersion medium, a continuous surface wave causes modulation instability (MI), which results in the self-modulation of the phase and intensity. Assume the given relation to obtain the steady state solution of the (1 + 1)-dimensional special KdV equation is given as [37]

$$g(x,t)=\left(G(x,t)+\sqrt{\mu }\right)e^{\iota \mu t}.$$

(91)

Here, $\mu$ shows the normalized optical power. Insert Equation (91) into Equation (2). By linearizing, one obtains

$$G^{(0,1)}(x,t)+G^{(3,0)}(x,t)+\iota \mu G(x,t)+\iota {\mu }^{3/2}=0.$$

(92)

Consider the result of Equation (92) is given as

$$G(x,t)=A_1{e}^{\iota (px-qt)}+A_2{e}^{-\iota (px-qt)},$$

(93)

where q and p indicate the perturbation frequency and normalization wave number. Insert the Equation (93) in Equation (92). By collecting each coefficient of $e^{\iota (px-qt)}$ and $e^{-\iota (px-qt)}$, we obtain a dispersion relation by solving the determinant of the coefficient matrix.

$${\mu }^2-p^6-2p^{3}q-q^2=0.$$

(94)

Finding the dispersion result of Equation (94) for q yields

$$q=-p^3\pm \sqrt{{\mu }^2}.$$

(95)

The achieved dispersion relation denotes the stable steady-state solution. When q is not real, then the steady-state result will be unstable because the perturbation grows gradually. When q is real, then the steady state changes to a stable one due to small perturbations. A steady-state solution is unstable if

$${\mu }^2<0.$$

(96)

The modulation instability gain spectrum $G(p)$ is obtained:

$$G(p)=2Im(q)=-p^3\pm \sqrt{{\mu }^2}.$$

(97)

See Figure 8

9. Conclusions

In this paper, successful breather-wave, 1-soliton, 2-soliton, new three-wave, new periodic-wave, and some exact soliton results of special (1 + 1)-dimensional KdV equations are achieved by applying the Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods. Some of the results are shown in 3D and contour plots. All the obtained solutions were verified by Mathematica software. The results are useful for the development of governing equations. This research shows that the methods used are simple and fruitful for other partial differential equations.

Moreover, a stability analysis and the modulation instability of the governing model were conducted to verify the stability and precision of the obtained solutions. The methods utilized are not only straightforward but also exceptionally effective in solving nonlinear partial differential equations (PDEs). Furthermore, these techniques prove to be valuable for addressing higher-order NLPDEs and larger systems of equations. The findings presented here offer substantial insights and potential applications across various scientific and engineering areas. The solutions obtained by these methods are useful for optical fibers, telecommunications, plasma physics, fluid dynamics, and many more. In the end, it is concluded that the techniques used are reliable and provide useful results. Nowadays, exact solutions, especially soliton-like solutions, have gained much importance because this has become a special topic in nonlinear science. Soliton theory has gained importance because of the exceptional properties of solitons. Solitons maintain their shape and velocity after interaction and stability.