Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

Abstract

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra $Z_2$. A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

1. Introduction

Integrable equations have wide applications in the field of nonlinear science, such as plasma physics [1,2,3,4], hydrodynamics [5,6,7], nonlinear optics [8,9], and so on [10,11,12,13,14,15]. The Painlevé analysis is an effective way to study the integrability of nonlinear systems [16,17]. In [18,19,20], the authors proposed a residual symmetry in the process of the residue of the truncated Painlevé expansion for the bosonized super symmetric KdV equation which is a nonlocal symmetry.

The Boussinesq–Burgers (B–B) system arises in the study of shallow water waves for two-layered fluid flow and describes propagation of shallow water waves in lake and ocean beaches [21,22,23,24,25,26,27]. In [28,29], the authors were concerned with the application of the nonlocal residual symmetry analysis to the Boussinesq–Burgers (B–B) system, which has the form as follows:

$$\begin{array}{cc}\hfill & p_t-\frac{1}{2}(\beta -1)p_{xx}-2p{p}_x-\frac{1}{2}{r}_x=0,\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill & r_t+\frac{1}{2}(\beta -1)r_{xx}+\beta (\frac{\beta }{2}-1)p_{xxx}-2{\left(pr\right)}_x=0,\hfill \end{array}$$

(1b)

where $v=v(x,t)$ is the height deviating from the equilibrium position of water, $u=u(x,t)$ is the field of horizontal velocity, $\beta$ is a constant representing different dispersive power.

In [30,31], a hierarchy called the Frobenius-valued Kakomtsev–Petviashvili hierarchy which takes values in a maximal commutative subalgebra of $gl(m,\mathbb{C})$ was constructed. In [32], the authors considered the Hirota quadratic equation of the commutative version of extended multi-component Toda hierarchy, which should be useful to Frobenius manifold theory [33]. In addition, we studied the $Z_n$-Painlevé IV equation and Frobenius Painlevé equations [34]. In this paper, we consider a new strongly coupled B–B system which is defined by us through taking values in a commutative subalgebra $Z_2=\mathbb{C}\left[\mathsf{\Gamma }\right]/\left({\mathsf{\Gamma }}^2\right)$. The form of the strongly coupled B–B system is as follows:

$$\begin{array}{cc}\hfill & p_t-\frac{1}{2}(\beta -1)p_{xx}-2p{p}_x-2q{q}_x-\frac{1}{2}{r}_x=0,\hfill \\ \hfill & q_t-\frac{1}{2}(\beta -1)q_{xx}-2p{q}_x-2q{p}_x-\frac{1}{2}{s}_x=0,\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill & r_t+\frac{1}{2}(\beta -1)r_{xx}+\beta (\frac{\beta }{2}-1)p_{xxx}-2{(pr+qs)}_x=0,\hfill \\ \hfill & s_t+\frac{1}{2}(\beta -1)s_{xx}+\beta (\frac{\beta }{2}-1)q_{xxx}-2{(ps+qr)}_x=0.\hfill \end{array}$$

(2b)

Then, the residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved. In order to further explore the residual symmetry, we use Lie’s first theorem: Every local analytical group determines on its tangent vector space at the identity the structure of a unique Lie algebra with the Lie bracket given by the formula

$$[x,y]=\underset{t\to 0}{lim}{t}^{-2}\left(\frac{tx\circ ty}{ty\circ tx}\right),$$

where the local group operation ∘ is transported into the tangent space, and where $g/h$ is written for $g\circ h^{-1}$.

The aim of this paper is to promote a Boussinesq–Burgers system to a Frobenius integrable system, which is called a strongly coupled Boussinesq–Burgers system. We apply the nonlocal residual symmetry analysis to the strongly coupled Boussinesq–Burgers system.

2. Residual Symmetries of (2+1)-Dimensional Strongly Coupled Burgers System

We first introduce the truncated Painlevé expansion:

$$\begin{array}{c}\hfill p=\sum _{i=0}^{{\alpha }_0}(p_i{(\psi +\varphi )}^{i-{\alpha }_0}+p_i{(\psi -\varphi )}^{i-{\alpha }_0}-q_i{(\psi -\varphi )}^{i-{\alpha }_0}+q_i{(\psi +\varphi )}^{i-{\alpha }_0}),\\ \hfill q=\sum _{i=0}^{{\alpha }_1}(q_i{(\psi +\varphi )}^{i-{\alpha }_1}+q_i{(\psi -\varphi )}^{i-{\alpha }_1}-p_i{(\psi -\varphi )}^{i-{\alpha }_1}+p_i{(\psi +\varphi )}^{i-{\alpha }_1}),\\ \hfill r=\sum _{i=0}^{{\alpha }_2}(r_i{(\psi +\varphi )}^{i-{\alpha }_2}+r_i{(\psi -\varphi )}^{i-{\alpha }_2}-s_i{(\psi -\varphi )}^{i-{\alpha }_2}+s_i{(\psi +\varphi )}^{i-{\alpha }_2}),\\ \hfill s=\sum _{i=0}^{{\alpha }_3}(s_i{(\psi +\varphi )}^{i-{\alpha }_3}+s_i{(\psi -\varphi )}^{i-{\alpha }_3}-r_i{(\psi -\varphi )}^{i-{\alpha }_3}+r_i{(\psi +\varphi )}^{i-{\alpha }_3}),\end{array}$$

(3)

where $p_{\alpha }$, $q_{\alpha }$ are a set of arbitrary solutions of the equation, and $p_{\alpha -1}$, $p_{\alpha -2}$, $\cdots ,$ $p_0$, $q_{\alpha -1}$, $q_{\alpha -2}$,$\cdots ,$$q_0$ to be expressed by derivatives of $\psi$ and $\varphi$. By balancing the dispersion and nonlinear terms according to the leader order analysis to the system Equation (2), the truncated Painlevé expansion has the following form:

$$\begin{array}{c}\hfill p=\frac{p_0\psi -q_0\varphi }{{\psi }^2-{\varphi }^2}+p_1,\\ \hfill q=\frac{q_0\psi -p_0\varphi }{{\psi }^2-{\varphi }^2}+q_1,\end{array}$$

(4a)

$$\begin{array}{c}\hfill r=\frac{r_0({\psi }^2+{\varphi }^2)-2s_0\psi \varphi }{{({\psi }^2-{\varphi }^2)}^2}+\frac{r_1\psi -s_1\varphi }{{\psi }^2-{\varphi }^2}+r_2,\\ \hfill s=\frac{s_0({\psi }^2+{\varphi }^2)-2r_0\psi \varphi }{{({\psi }^2-{\varphi }^2)}^2}+\frac{s_1\psi -r_1\varphi }{{\psi }^2-{\varphi }^2}+s_2.\end{array}$$

(4b)

Then, plugging Equation (4) into Equation (2), and vanishing all the coefficients of each power of ${(\psi +\varphi )}^{-1}+{(\psi -\varphi )}^{-1}$ and ${(\psi -\varphi )}^{-1}-{(\psi +\varphi )}^{-1}$, we obtain

$$\begin{array}{c}\hfill p_0=-\frac{1}{2}{\psi }_x,q_0=-\frac{1}{2}{\varphi }_x,\end{array}$$

(5a)

$$\begin{array}{c}\hfill p_1=\frac{{\psi }_x({\psi }_{xx}+2{\psi }_t)-{\varphi }_x({\varphi }_{xx}+2{\varphi }_t)}{4({\psi }_x^2-{\varphi }_x^2)},q_1=\frac{{\psi }_x({\varphi }_{xx}+2{\varphi }_t)-{\varphi }_x({\psi }_{xx}+2{\psi }_t)}{4({\psi }_x^2-{\varphi }_x^2)},\end{array}$$

(5b)

$$\begin{array}{c}\hfill r_0=-\frac{\beta }{2}({\psi }_x^2+{\varphi }_x^2),s_0=-\beta {\psi }_x{\varphi }_x,\end{array}$$

(5c)

$$r_1=\frac{\beta }{2}{\psi }_{xx},s_1=\frac{\beta }{2}{\varphi }_{xx},$$

(5d)

$$\begin{array}{cc}\hfill r_2=& \frac{\beta }{4}\frac{({\psi }_{xx}^2+{\varphi }_{xx}^2)({\psi }_x^2+{\varphi }_x^2)-4{\psi }_x{\varphi }_x{\psi }_{xx}{\varphi }_{xx}}{{({\psi }_x^2-{\varphi }_x^2)}^2}-\frac{\beta }{4}\frac{{\psi }_{xxx}{\psi }_x-{\varphi }_{xxx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2}\hfill \\ \hfill & +\frac{\beta }{2}\frac{({\psi }_{xx}{\psi }_t+{\varphi }_{xx}{\varphi }_t)({\psi }_x^2+{\varphi }_x^2)-2{\psi }_x{\varphi }_x({\psi }_{xx}{\varphi }_t+{\varphi }_{xx}{\psi }_t)}{{({\psi }_x^2-{\varphi }_x^2)}^2}-\frac{\beta }{2}\frac{{\psi }_{tx}{\psi }_x-{\varphi }_{tx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2},\hfill \\ \hfill s_2=& \frac{\beta }{2}\frac{{\psi }_{xx}{\varphi }_{xx}({\psi }_x^2+{\varphi }_x^2)-{\psi }_x{\varphi }_x({\psi }_{xx}^2+{\varphi }_{xx}^2)}{{({\psi }_x^2-{\varphi }_x^2)}^2}-\frac{\beta }{4}\frac{{\varphi }_{xxx}{\psi }_x-{\psi }_{xxx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2}\hfill \\ \hfill & +\frac{\beta }{2}\frac{({\psi }_{xx}{\varphi }_t+{\varphi }_{xx}{\psi }_t)({\psi }_x^2+{\varphi }_x^2)-2{\psi }_x{\varphi }_x({\psi }_{xx}{\psi }_t+{\varphi }_{xx}{\varphi }_t)}{{({\psi }_x^2-{\varphi }_x^2)}^2}-\frac{\beta }{2}\frac{{\varphi }_{tx}{\psi }_x-{\psi }_{tx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2}.\hfill \end{array}$$

(5e)

Further, the Schwarzian form of the field $\psi$ and $\varphi$ is obtained as follows:

$$\begin{array}{c}\hfill C_t=C{C}_x+D{D}_x-\frac{1}{4}{S}_x-C_{xx},\\ \hfill D_t=C{D}_x+D{C}_x-\frac{1}{4}{T}_x-D_{xx},\end{array}$$

(6)

where

$$\begin{array}{cc}\hfill & C=\frac{{\psi }_t{\psi }_x-{\varphi }_t{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2},D=\frac{{\varphi }_t{\psi }_x-{\psi }_t{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2},\hfill \\ \hfill & S=\frac{{\psi }_{xxx}{\psi }_x-{\varphi }_{xxx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2}-\frac{3}{2}\frac{({\psi }_{xx}^2+{\varphi }_{xx}^2)({\psi }_x^2+{\varphi }_x^2)-4{\psi }_x{\varphi }_x{\psi }_{xx}{\varphi }_{xx}}{{({\psi }_x^2-{\varphi }_x^2)}^2},\hfill \\ \hfill & T=\frac{{\varphi }_{xxx}{\psi }_x-{\psi }_{xxx}{\varphi }_x}{{\psi }_x^2-{\varphi }_x^2}-3\frac{{\psi }_{xx}{\varphi }_{xx}({\psi }_x^2+{\varphi }_x^2)-{\psi }_x{\varphi }_x({\psi }_{xx}^2+{\varphi }_{xx}^2)}{{({\psi }_x^2-{\varphi }_x^2)}^2}.\hfill \end{array}$$

The Schwarzian form Equation (6) is invariant under the Möbius transformation:

$$\psi \to \frac{(a+b\psi )(c+d\psi )-bd{\varphi }^2}{{(c+d\psi )}^2-{\left(d\varphi \right)}^2},\varphi \to \frac{b\varphi (c+d\psi )-d\varphi (a+b\psi )}{{(c+d\psi )}^2-{\left(d\varphi \right)}^2},(a,b,c,d\in \mathbb{C}),$$

which means the field $\psi$ and $\varphi$ possesses six point symmetries, ${\sigma }^{\psi }=a_1$, ${\sigma }^{\varphi }=a_2$, ${\sigma }^{\psi }=b_1\psi$, ${\sigma }^{\varphi }=b_1\varphi$, ${\sigma }^{\psi }=c_1({\psi }^2+{\varphi }^2)$, ${\sigma }^{\varphi }=2c_1\psi \varphi$, where $a_1$, $a_2$, $b_1$ and $c_1$ are arbitrary constants.

Actually, the residuals $p_0$, $q_0$, $r_1$, and $s_1$ are nonlocal symmetries of Equation (2) with $p_1$, $q_1$, $r_2$, and $s_2$ as solutions. Then, we introduce dependent variables f, g, h, and k with the relations $f={\psi }_x$, $g={\varphi }_x$, $h=f_x$, and $k=g_x$ to obtain a local symmetry.

Substituting Equation (5) into Equation (4), one has

$$\begin{array}{c}\hfill p=-\frac{{\psi }_x\psi -{\varphi }_x\varphi }{2({\psi }^2-{\varphi }^2)}+\frac{{\psi }_x({\psi }_{xx}+2{\psi }_t)-{\varphi }_x({\varphi }_{xx}+2{\varphi }_t)}{4({\psi }_x^2-{\varphi }_x^2)},\\ \hfill q=-\frac{{\varphi }_x\psi -{\psi }_x\varphi }{2({\psi }^2-{\varphi }^2)}+\frac{{\psi }_x({\varphi }_{xx}+2{\varphi }_t)-{\varphi }_x({\psi }_{xx}+2{\psi }_t)}{4({\psi }_x^2-{\varphi }_x^2)},\end{array}$$

(7)

$$\begin{array}{c}\hfill r=-\frac{\beta }{2}\frac{({\psi }_x^2+{\varphi }_x^2)({\psi }^2+{\varphi }^2)-4{\psi }_x{\varphi }_x}{{({\psi }^2-{\varphi }^2)}^2}+\frac{\beta }{2}\frac{{\psi }_{xx}\psi -{\varphi }_{xx}\varphi }{{\psi }^2-{\varphi }^2}+r_2,\\ \hfill s=-\beta \frac{{\psi }_x{\varphi }_x({\psi }^2+{\varphi }^2)-\psi \varphi ({\psi }_x^2+{\varphi }_x^2)}{{({\psi }^2-{\varphi }^2)}^2}+\frac{\beta }{2}\frac{{\varphi }_{xx}\psi -{\psi }_{xx}\varphi }{{\psi }^2-{\varphi }^2}+s_2.\end{array}$$

(8)

Then, one can get the localized Lie point symmetry

$$\begin{array}{c}\hfill {\sigma }^p=-\frac{f}{2},{\sigma }^q=-\frac{g}{2},\end{array}$$

(9a)

$$\begin{array}{c}\hfill {\sigma }^r=\frac{\beta h}{2},{\sigma }^s=\frac{\beta k}{2},\end{array}$$

(9b)

$$\begin{array}{c}\hfill {\sigma }^f=-2(f\psi +g\varphi ),{\sigma }^g=-2(g\psi +f\varphi ),\end{array}$$

(9c)

$${\sigma }^h=-2(f^2+g^2)-2(h\psi +k\varphi ),{\sigma }^k=-4fg-2(k\psi +h\varphi ),$$

(9d)

$${\sigma }^{\psi }=-{\psi }^2-{\varphi }^2,{\sigma }^{\varphi }=-2\psi \varphi ,$$

(9e)

and the corresponding Lie point symmetry vector takes the following form:

$$\begin{array}{cc}\hfill \overrightarrow{V}=& -\frac{f}{2}{\partial }_p+\frac{\beta h}{2}{\partial }_r-2(f\psi +g\varphi ){\partial }_f-2(f^2+g^2+h\psi +k\varphi ){\partial }_h-({\psi }^2+{\varphi }^2){\partial }_{\psi },\hfill \\ \hfill \overrightarrow{W}=& -\frac{g}{2}{\partial }_q+\frac{\beta k}{2}{\partial }_s-2(g\psi +f\varphi ){\partial }_g-2(2fg+k\psi +h\varphi ){\partial }_k-2\psi \varphi {\partial }_{\varphi }.\hfill \end{array}$$

(10)

Next we will give the Bäcklund symmetry theorem, which is obtained by using a finite transformation of the Lie point symmetry Equation (10).

Theorem 1.

If $\{p,q,r,s,f,g,h,k,\psi ,\varphi \}$ is a solution of the extended system Equation (9), then so is$\{\overline{p},\overline{q},\overline{r},\overline{s},\overline{f},\overline{g},\overline{h},\overline{k},\overline{\psi },\overline{\varphi }\}$ with

$$\begin{array}{c}\hfill \overline{p}=p-\frac{ϵ}{2}\frac{f(ϵ\psi +1)-ϵg\varphi }{{(ϵ\psi +1)}^2-{ϵ}^2{\varphi }^2},\overline{q}=q-\frac{ϵ}{2}\frac{g(ϵ\psi +1)-ϵf\varphi }{{(ϵ\psi +1)}^2-{ϵ}^2{\varphi }^2},\end{array}$$

(11a)

$$\begin{array}{c}\hfill \overline{r}=r+\frac{A({(ϵ\psi +1)}^2+{ϵ}^2{\varphi }^2)-2B(ϵ\psi +1)ϵ\varphi }{2{(ϵ\psi +ϵ\varphi +1)}^2{(ϵ\psi -ϵ\varphi +1)}^2},\overline{s}=s+\frac{B({(ϵ\psi +1)}^2+{ϵ}^2{\varphi }^2)-2A(ϵ\psi +1)ϵ\varphi }{2{(ϵ\psi +ϵ\varphi +1)}^2{(ϵ\psi -ϵ\varphi +1)}^2},\end{array}$$

(11b)

$$\begin{array}{c}\hfill \overline{\psi }=\frac{\psi (ϵ\psi +1)-ϵ{\varphi }^2}{{(ϵ\psi +1)}^2-{ϵ}^2{\varphi }^2},\overline{\varphi }=\frac{\varphi (ϵ\psi +1)-ϵ\psi \varphi }{{(ϵ\psi +1)}^2-{ϵ}^2{\varphi }^2},\end{array}$$

(11c)

$$\overline{f}=\frac{f({(ϵ\psi +1)}^2+{ϵ}^2{\varphi }^2)-2g(ϵ\psi +1)ϵ\varphi }{{(ϵ\psi +ϵ\varphi +1)}^2{(ϵ\psi -ϵ\varphi +1)}^2},\overline{g}=\frac{g({(ϵ\psi +1)}^2+{ϵ}^2{\varphi }^2)-2f(ϵ\psi +1)ϵ\varphi }{{(ϵ\psi +ϵ\varphi +1)}^2{(ϵ\psi -ϵ\varphi +1)}^2},$$

(11d)

$$\begin{array}{c}\hfill \overline{h}=\frac{C_0({(ϵ\psi +1)}^3+3{ϵ}^2{\varphi }^2(ϵ\psi +1))-D_0(3ϵ\varphi {(ϵ\psi +1)}^2+{ϵ}^3{\varphi }^3)}{{(ϵ\psi +ϵ\varphi +1)}^3{(ϵ\psi -ϵ\varphi +1)}^3},\\ \hfill \overline{k}=\frac{D_0({(ϵ\psi +1)}^3+3{ϵ}^2{\varphi }^2(ϵ\psi +1))-C_0(3ϵ\varphi {(ϵ\psi +1)}^2+{ϵ}^3{\varphi }^3)}{{(ϵ\psi +ϵ\varphi +1)}^3{(ϵ\psi -ϵ\varphi +1)}^3},\end{array}$$

(11e)

where

$$\begin{array}{cc}\hfill & A=\beta hϵ-\beta {ϵ}^2(f^2+g^2)+\beta {ϵ}^2(h\psi +k\varphi ),B=\beta kϵ-2\beta {ϵ}^{2}fg+\beta {ϵ}^2(h\varphi +k\psi ),\hfill \\ \hfill & C_0=-2ϵ(f^2+g^2)+ϵ(h\psi +k\varphi )+h,D_0=-4ϵfg+ϵ(k\psi +h\varphi )+k,\hfill \end{array}$$

and ϵ is an arbitrary group parameter.

Proof. 

According to Lie’s first theorem on vector Equation (10) with the corresponding initial value problem:

$$\begin{array}{c}\hfill \frac{d\overline{p}}{dϵ}=-\frac{\overline{f}}{2},\frac{d\overline{q}}{dϵ}=-\frac{\overline{g}}{2},\end{array}$$

(12a)

$$\begin{array}{c}\hfill \frac{d\overline{r}}{dϵ}=\frac{\beta \overline{h}}{2},\frac{d\overline{s}}{dϵ}=\frac{\beta \overline{k}}{2},\end{array}$$

(12b)

$$\begin{array}{c}\hfill \frac{d\overline{f}}{dϵ}=-2(\overline{\psi }\overline{f}+\overline{\varphi }\overline{g}),\frac{d\overline{g}}{dϵ}=-2(\overline{\psi }\overline{g}+\overline{\varphi }\overline{f}),\end{array}$$

(12c)

$$\frac{d\overline{h}}{dϵ}=-2({\overline{f}}^2+{\overline{g}}^2+\overline{\psi }\overline{h}+\overline{\varphi }\overline{k}),\frac{d\overline{k}}{dϵ}=-2(2\overline{f}\overline{g}+\overline{\psi }\overline{k}+\overline{\varphi }\overline{h}),$$

(12d)

$$\frac{d\overline{\psi }}{dϵ}=-({\overline{\psi }}^2+{\overline{\varphi }}^2),\frac{d\overline{\varphi }}{dϵ}=-2\overline{\psi }\overline{\varphi },$$

(12e)

$$\begin{array}{cc}\hfill & \overline{p}\left(0\right)=p,\overline{q}\left(0\right)=q,\overline{r}\left(0\right)=r,\overline{s}\left(0\right)=s,\overline{f}\left(0\right)=f,\hfill \\ \hfill & \overline{g}\left(0\right)=g,\overline{h}\left(0\right)=h,\overline{k}\left(0\right)=k,\overline{\psi }\left(0\right)=\psi ,\overline{\varphi }\left(0\right)=\varphi ,\hfill \end{array}$$

(12f)

one can find the solutions of the above system is Equation (11). Therefore, we have completed the proof of Theorem 1. □

3. Bäcklund Transformations of Strongly Coupled Burgers System Related to Multiple Residual Symmetries

In the strongly coupled B–B system Equation (2), the original residual symmetry have forms:

$${\sigma }^p=-\frac{1}{2}{\psi }_x,{\sigma }^q=2b{\varphi }_x,{\sigma }^r=\frac{\beta }{2}{\psi }_{xx},{\sigma }^s=\frac{\beta }{2}{\varphi }_{xx},$$

which is related to the solution of the Equation (6). Then, according to the linear property of symmetry equations, the multiple residual symmetries are expressed in terms of any linear superposition of symmetries

$$\begin{array}{cc}\hfill & {\sigma }_n^p=\sum _{i=1}^n{c}_i{\psi }_{i,x},{\sigma }_n^q=\sum _{i=1}^n{c}_i{\varphi }_{i,x},\hfill \\ \hfill & {\sigma }_n^r=\sum _{i=1}^n{d}_i{\psi }_{i,xx},{\sigma }_n^s=\sum _{i=1}^n{d}_i{\varphi }_{i,xx},(n=1,2,3,\dots ),\hfill \end{array}$$

(13)

where ${\psi }_i$ and ${\varphi }_i$$i=1,2,3,\dots$, are different solutions of Equation (6). The symmetry Equation (13) should be localized to a Lie point symmetry by introducing more variables. In fact, one can find the finite transformation group of the symmetry Equation (13).

Theorem 2.

The symmetry Equation (13) is localized to the Lie point symmetry

$$\begin{array}{c}\hfill {\sigma }^p=-\frac{1}{2}\sum _{j=1}^n{c}_j{f}_j,{\sigma }^q=-\frac{1}{2}\sum _{j=1}^n{c}_j{g}_j,\end{array}$$

(14a)

$$\begin{array}{c}\hfill {\sigma }^r=\frac{\beta }{2}\sum _{j=1}^n{c}_j{h}_j,{\sigma }^s=\frac{\beta }{2}\sum _{j=1}^n{c}_j{k}_j,\end{array}$$

(14b)

$$\begin{array}{cc}\hfill & {\sigma }^{{\psi }_i}=-c_i({\psi }_i^2+{\varphi }_i^2)-\sum _{j\ne i}^n{c}_j({\psi }_j{\psi }_i+{\varphi }_j{\varphi }_i),\hfill \\ \hfill & {\sigma }^{{\varphi }_i}=-2c_i{\psi }_i{\varphi }_i-\sum _{j\ne i}^n{c}_j({\psi }_j{\varphi }_i+{\varphi }_j{\psi }_i),\hfill \end{array}$$

(14c)

$$\begin{array}{cc}\hfill & {\sigma }^{f_i}=-2c_i(f_i{\psi }_i+g_i{\varphi }_i)-\sum _{j\ne i}^n{c}_j(f_j{\psi }_i+g_j{\varphi }_i+f_i{\psi }_j+g_i{\varphi }_j),\hfill \\ \hfill & {\sigma }^{g_i}=-2c_i(g_i{\psi }_i+f_i{\varphi }_i)-\sum _{j\ne i}^n{c}_j(f_j{\varphi }_i+g_j{\psi }_i+g_i{\psi }_j+f_i{\varphi }_j),\hfill \end{array}$$

(14d)

$$\begin{array}{cc}\hfill & {\sigma }^{h_i}=-2c_i(f_i^2+g_i^2)-2c_i(h_i{\psi }_i+k_i{\varphi }_i)-\sum _{j\ne i}^n{c}_j(h_j{\psi }_i+k_j{\varphi }_i+2f_j{f}_i+2g_j{g}_i+h_i{\psi }_j+k_i{\varphi }_j),\hfill \\ \hfill & {\sigma }^{k_i}=-4c_i{f}_i{g}_i-2c_i(k_i{\psi }_i+h_i{\varphi }_i)-\sum _{j\ne i}^n{c}_j(h_j{\varphi }_i+k_j{\psi }_i+2f_j{g}_i+2g_j{f}_i+k_i{\psi }_j+h_i{\varphi }_j),\hfill \end{array}$$

(14e)

where $\{p,q,r,s,f_i,g_i,h_i,k_i,{\psi }_i,{\varphi }_i\}$$(i=1,2,\dots ,n)$ is a solution of the enlarged system

$$\begin{array}{cc}\hfill & p_t-\frac{1}{2}(\beta -1)p_{xx}-2p{p}_x-2q{q}_x-\frac{1}{2}{r}_x=0,\hfill \\ \hfill & q_t-\frac{1}{2}(\beta -1)q_{xx}-2p{q}_x-2q{p}_x-\frac{1}{2}{s}_x=0,\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & r_t+\frac{1}{2}(\beta -1)r_{xx}+\beta (\frac{\beta }{2}-1)p_{xxx}-2{(pr+qs)}_x=0,\hfill \\ \hfill & s_t+\frac{1}{2}(\beta -1)s_{xx}+\beta (\frac{\beta }{2}-1)q_{xxx}-2{(ps+qr)}_x=0,\hfill \end{array}$$

(15b)

$$\begin{array}{c}\hfill p=\frac{{\psi }_{i,x}({\psi }_{i,xx}+2{\psi }_{i,t})-{\varphi }_{i,x}({\varphi }_{i,xx}+2{\varphi }_{i,t})}{4({\psi }_{i,x}^2-{\varphi }_{i,x}^2)},\\ \hfill q=\frac{{\psi }_{i,x}({\varphi }_{i,xx}+2{\varphi }_{i,t})-{\varphi }_{i,x}({\psi }_{i,xx}+2{\psi }_{i,t})}{4({\psi }_{i,x}^2-{\varphi }_{i,x}^2)},\end{array}$$

(15c)

$$\begin{array}{cc}r=\hfill & \frac{\beta }{4}\frac{({\psi }_{i,xx}^2+{\varphi }_{i,xx}^2)({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-4{\psi }_{i,x}{\varphi }_{i,x}{\psi }_{i,xx}{\varphi }_{i,xx}}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\frac{\beta }{4}\frac{{\psi }_{i,xxx}{\psi }_{i,x}-{\varphi }_{i,xxx}{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2}\hfill \\ & +\frac{\beta }{2}\frac{({\psi }_{i,xx}{\psi }_{i,t}+{\varphi }_{i,xx}{\varphi }_{i,t})({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2{\psi }_{i,x}{\varphi }_{i,x}({\psi }_{i,xx}{\varphi }_{i,t}+{\varphi }_{i,xx}{\psi }_{i,t})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\frac{\beta }{2}\frac{{\psi }_{i,tx}{\psi }_{i,x}-{\varphi }_{i,tx}{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2},\hfill \\ s=\hfill & \frac{\beta }{2}\frac{{\psi }_{i,xx}{\varphi }_{i,xx}({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-{\psi }_{i,x}{\varphi }_{i,x}({\psi }_{i,xx}^2+{\varphi }_{i,xx}^2)}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\frac{\beta }{4}\frac{{\varphi }_{i,xxx}{\psi }_{i,x}-{\psi }_{i,xxx}{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2}\hfill \\ & +\frac{\beta }{2}\frac{({\psi }_{i,xx}{\varphi }_{i,t}+{\varphi }_{i,xx}{\psi }_{i,t})({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2{\psi }_{i,x}{\varphi }_{i,x}({\psi }_{i,xx}{\psi }_{i,t}+{\varphi }_{i,xx}{\varphi }_{i,t})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\frac{\beta }{2}\frac{{\varphi }_{i,tx}{\psi }_{i,x}-{\psi }_{i,tx}{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2},\hfill \end{array}$$

(15d)

$$f_i={\psi }_{i,x},g_i={\varphi }_{i,x},$$

(15e)

$$h_i=f_{i,x},k_i=g_{i,x},$$

(15f)

$(i=1,2,\dots ,n).$

Proof. 

The extended system Equation (15) has the following linearized form:

$$\begin{array}{c}\hfill {\sigma }_t^p-\frac{1}{2}\beta {\sigma }_{xx}^p+\frac{1}{2}{\sigma }_{xx}^p-2p{\sigma }_x^p-2q{\sigma }_x^q-2{\sigma }^p{p}_x-2{\sigma }^q{q}_x-\frac{1}{2}{\sigma }_x^r=0,\\ \hfill {\sigma }_t^q-\frac{1}{2}\beta {\sigma }_{xx}^q+\frac{1}{2}{\sigma }_{xx}^q-2p{\sigma }_x^q-2q{\sigma }_x^p-2{\sigma }^p{q}_x-2{\sigma }^q{p}_x-\frac{1}{2}{\sigma }_x^s=0,\end{array}$$

(16a)

$$\begin{array}{cc}\hfill & {\sigma }_t^r-\beta {\sigma }_{xxx}^p+\frac{1}{2}{\beta }^2{\sigma }_{xxx}^p-\frac{1}{2}{\sigma }_{xx}^r+\frac{1}{2}\beta {\sigma }_{xx}^r-2p_x{\sigma }^r-2q_x{\sigma }^s-2{\sigma }_x^{p}r-2{\sigma }_x^{q}s\hfill \\ \hfill & -2p{\sigma }_x^r-2q{\sigma }_x^s-2{\sigma }^p{r}_x-2{\sigma }^q{s}_x=0,\hfill \\ \hfill & {\sigma }_t^s-\beta {\sigma }_{xxx}^q+\frac{1}{2}{\beta }^2{\sigma }_{xxx}^q-\frac{1}{2}{\sigma }_{xx}^s+\frac{1}{2}\beta {\sigma }_{xx}^s-2p_x{\sigma }^s-2q_x{\sigma }^r-2{\sigma }_x^{p}s-2{\sigma }_x^{q}r\hfill \\ \hfill & -2p{\sigma }_x^s-2q{\sigma }_x^r-2{\sigma }^p{s}_x-2{\sigma }^q{r}_x=0,\hfill \end{array}$$

(16b)

$$\begin{array}{c}\hfill {\sigma }^p=\frac{{\psi }_{i,x}({\sigma }_{xx}^{{\psi }_i}+2{\sigma }_t^{{\psi }_i})-{\varphi }_{i,x}({\sigma }_{xx}^{{\varphi }_i}+2{\sigma }_t^{{\varphi }_i})}{4({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}-\frac{A_0({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2B_0{\psi }_{i,x}{\varphi }_{i,x}}{4{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2},\\ \hfill {\sigma }^q=\frac{{\psi }_{i,x}({\sigma }_{xx}^{{\varphi }_i}+2{\sigma }_t^{{\varphi }_i})-{\varphi }_{i,x}({\sigma }_{xx}^{{\psi }_i}+2{\sigma }_t^{{\psi }_i})}{4({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}-\frac{B_0({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2A_0{\psi }_{i,x}{\varphi }_{i,x}}{4{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2},\end{array}$$

(16c)

$$\begin{array}{cc}\hfill {\sigma }^r=& -\frac{\beta }{2}\frac{A_1({\psi }_{i,x}^3+3{\varphi }_{i,x}^2{\psi }_{i,x})-B_1({\varphi }_{i,x}^3+3{\psi }_{i,x}^2{\varphi }_{i,x})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^3}+\frac{\beta }{2}\frac{A_2({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2B_2{\psi }_{i,x}{\varphi }_{i,x})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}\hfill \\ \hfill & +\frac{\beta }{4}\frac{A_3({\psi }_{i,x}^2-{\varphi }_{i,x}^2)-2B_3{\psi }_{i,x}{\varphi }_{i,x}}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\beta \frac{A_5({\psi }_i^3+3{\varphi }_i^2{\psi }_i)-B_5({\varphi }_i^3+3{\psi }_i^2{\varphi }_i)}{{({\psi }_i^2-{\varphi }_i^2)}^3}\hfill \\ \hfill & -\frac{\beta }{4}\frac{A_4{\psi }_{i,x}-B_4{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2},\hfill \\ \hfill {\sigma }^s=& -\frac{\beta }{2}\frac{B_1({\psi }_{i,x}^3+3{\varphi }_{i,x}^2{\psi }_{i,x})-A_1({\varphi }_{i,x}^3+3{\psi }_{i,x}^2{\varphi }_{i,x})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^3}+\frac{\beta }{2}\frac{B_2({\psi }_{i,x}^2+{\varphi }_{i,x}^2)-2A_2{\psi }_{i,x}{\varphi }_{i,x})}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}\hfill \\ \hfill & +\frac{\beta }{4}\frac{B_3({\psi }_{i,x}^2-{\varphi }_{i,x}^2)-2A_3{\psi }_{i,x}{\varphi }_{i,x}}{{({\psi }_{i,x}^2-{\varphi }_{i,x}^2)}^2}-\beta \frac{B_5({\psi }_i^3+3{\varphi }_i^2{\psi }_i)-A_5({\varphi }_i^3+3{\psi }_i^2{\varphi }_i)}{{({\psi }_i^2-{\varphi }_i^2)}^3}\hfill \\ \hfill & -\frac{\beta }{4}\frac{B_4{\psi }_{i,x}-A_4{\varphi }_{i,x}}{{\psi }_{i,x}^2-{\varphi }_{i,x}^2},\hfill \end{array}$$

(16d)

$${\sigma }_x^{{\psi }_i}={\sigma }^{f_i},{\sigma }_x^{{\varphi }_i}={\sigma }^{g_i},$$

(16e)

$${\sigma }_x^{f_i}={\sigma }^{h_i},{\sigma }_x^{g_i}={\sigma }^{k_i},i=1,2,\dots ,n,$$

(16f)

where

$$\begin{array}{cc}\hfill & A_0={\psi }_{i,xx}{\sigma }_x^{{\psi }_i}+{\varphi }_{i,xx}{\sigma }_x^{{\varphi }_i}+2{\psi }_{i,t}{\sigma }_x^{{\psi }_i}+2{\varphi }_{i,t}{\sigma }_x^{{\varphi }_i},\hfill \\ \hfill & B_0={\psi }_{i,xx}{\sigma }_x^{{\varphi }_i}+{\varphi }_{i,xx}{\sigma }_x^{{\psi }_i}+2{\psi }_{i,t}{\sigma }_x^{{\varphi }_i}+2{\varphi }_{i,t}{\sigma }_x^{{\psi }_i},\hfill \\ \hfill & A_1=({\psi }_{i,xx}^2+{\varphi }_{i,xx}^2){\sigma }_x^{{\psi }_i}+2{\psi }_{i,xx}{\varphi }_{i,xx}{\sigma }_x^{{\varphi }_i},\hfill \\ \hfill & B_1=({\psi }_{i,xx}^2+{\varphi }_{i,xx}^2){\sigma }_x^{{\varphi }_i}+2{\psi }_{i,xx}{\varphi }_{i,xx}{\sigma }_x^{{\psi }_i},\hfill \\ \hfill & A_2={\psi }_{i,xx}{\sigma }_{xx}^{{\psi }_i}+{\varphi }_{i,xx}{\sigma }_{xx}^{{\varphi }_i}+{\psi }_{i,xx}{\sigma }_t^{{\psi }_i}+{\varphi }_{i,xx}{\sigma }_t^{{\varphi }_i}+{\psi }_{i,t}{\sigma }_{xx}^{{\psi }_i}+{\varphi }_{i,t}{\sigma }_{xx}^{{\varphi }_i}+{\psi }_{i,xt}{\sigma }_x^{{\psi }_i}+{\varphi }_{i,xt}{\sigma }_x^{{\varphi }_i},\hfill \\ \hfill & B_2={\psi }_{i,xx}{\sigma }_{xx}^{{\varphi }_i}+{\varphi }_{i,xx}{\sigma }_{xx}^{{\psi }_i}+{\psi }_{i,xx}{\sigma }_t^{{\varphi }_i}+{\varphi }_{i,xx}{\sigma }_t^{{\psi }_i}+{\psi }_{i,t}{\sigma }_{xx}^{{\varphi }_i}+{\varphi }_{i,t}{\sigma }_{xx}^{{\psi }_i}+{\psi }_{i,xt}{\sigma }_x^{{\varphi }_i}+{\varphi }_{i,xt}{\sigma }_x^{{\psi }_i},\hfill \\ \hfill & A_3={\psi }_{i,xxx}{\sigma }_x^{{\psi }_i}+{\varphi }_{i,xxx}{\sigma }_x^{{\varphi }_i},B_3={\varphi }_{i,xxx}{\sigma }_x^{{\psi }_i}+{\psi }_{i,xxx}{\sigma }_x^{{\varphi }_i},\hfill \\ \hfill & A_4={\sigma }_{xxx}^{{\psi }_i}+2{\sigma }_{xt}^{{\psi }_i},B_4={\sigma }_{xxx}^{{\varphi }_i}+2{\sigma }_{xt}^{{\varphi }_i},\hfill \\ \hfill & A_5=({\psi }_{i,xx}{\psi }_{i,t}+{\varphi }_{i,xx}{\varphi }_{i,t}){\sigma }_x^{{\psi }_i}+({\psi }_{i,xx}{\varphi }_{i,t}+{\varphi }_{i,xx}{\psi }_{i,t}){\sigma }_x^{{\varphi }_i},\hfill \\ \hfill & B_5=({\psi }_{i,xx}{\varphi }_{i,t}+{\varphi }_{i,xx}{\psi }_{i,t}){\sigma }_x^{{\psi }_i}+({\psi }_{i,xx}{\psi }_{i,t}+{\varphi }_{i,xx}{\varphi }_{i,t}){\sigma }_x^{{\varphi }_i}.\hfill \end{array}$$

Let us first consider the special case: for any fixed $m,c_m\ne 0,$ while $c_j=0$, $j\ne m$ in Equation (13). In this case, the localized symmetry for $\{p,q,r,s,f_m,g_m,h_m,k_m,{\psi }_m,{\varphi }_m\}$ can be obtained from Equation (9):

$$\begin{array}{c}\hfill {\sigma }^p=-\frac{1}{2}{c}_m{f}_m,{\sigma }^q=-\frac{1}{2}{c}_m{g}_m,\end{array}$$

(17a)

$$\begin{array}{c}\hfill {\sigma }^r=\frac{\beta }{2}{c}_m{h}_m,{\sigma }^s=\frac{\beta }{2}{c}_m{k}_m,\end{array}$$

(17b)

$$\begin{array}{c}\hfill {\sigma }^{{\psi }_m}=-c_m({\psi }_m^2+{\varphi }_m^2),{\sigma }^{{\varphi }_m}=-2c_m{\psi }_m{\varphi }_m,\end{array}$$

(17c)

$$\begin{array}{c}\hfill {\sigma }^{f_m}=-2c_m(f_m{\psi }_m+g_m{\varphi }_m),{\sigma }^{g_m}=-2c_m(g_m{\psi }_m+f_m{\varphi }_m),\end{array}$$

(17d)

$$\begin{array}{c}\hfill {\sigma }^{h_m}=-2c_m(f_m^2+g_m^2)-2c_m(h_m{\psi }_m+k_m{\varphi }_m),{\sigma }^{k_m}=-4c_m{f}_m{g}_m-2c_m(k_m{\psi }_m+h_m{\varphi }_m).\end{array}$$

(17e)

In Equation (15c), p and q can be eliminated by taking $i=m$ and $i=j$, respectively, and then we obtain

$$\begin{array}{cc}\hfill & {\psi }_{j,xx}=\frac{A_6{\psi }_{m,x}-B_6{\varphi }_{m,x}}{{\psi }_{m,x}^2-{\varphi }_{m,x}^2},\hfill \\ \hfill & {\varphi }_{j,xx}=\frac{B_6{\psi }_{m,x}-A_6{\varphi }_{m,x}}{{\psi }_{m,x}^2-{\varphi }_{m,x}^2},\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill & A_6={\psi }_{j,x}{\psi }_{m,xx}+{\varphi }_{j,x}{\varphi }_{m,xx}+2{\psi }_{j,x}{\psi }_{m,t}+2{\varphi }_{j,x}{\varphi }_{m,t}-2{\psi }_{m,x}{\psi }_{j,t}-2{\varphi }_{m,x}{\varphi }_{j,t},\hfill \\ \hfill & B_6={\psi }_{j,x}{\varphi }_{m,xx}+{\varphi }_{j,x}{\psi }_{m,xx}+2{\psi }_{j,x}{\varphi }_{m,t}+2{\varphi }_{j,x}{\psi }_{m,t}-2{\psi }_{m,x}{\varphi }_{j,t}-2{\varphi }_{m,x}{\psi }_{j,t}.\hfill \end{array}$$

Substituting Equation (17a) into Equation (16c) with $i=j$ and vanishing ${\psi }_{j,xx}$ and ${\varphi }_{j,xx}$ through Equation (18), we have

$$\begin{array}{cc}\hfill & {\sigma }^{{\psi }_j}=-c_m({\psi }_m{\psi }_j+{\varphi }_m{\varphi }_j),\hfill \\ \hfill & {\sigma }^{{\varphi }_j}=-c_m({\psi }_m{\varphi }_j+{\varphi }_m{\psi }_j).\hfill \end{array}$$

(19)

The symmetries for $f_j$, $g_j$, $h_j$, and $k_j$ can be obtained from Equation (19) as follows:

$$\begin{array}{c}\hfill {\sigma }^{f_j}=-c_m({\psi }_m{f}_j+{\varphi }_m{g}_j+f_m{\psi }_j+g_m{\varphi }_j),\\ \hfill {\sigma }^{g_j}=-c_m({\psi }_m{g}_j+{\varphi }_m{f}_j+f_m{\varphi }_j+g_m{\psi }_j),\end{array}$$

(20)

$$\begin{array}{c}\hfill {\sigma }^{h_j}=-c_m(h_j{\psi }_m+k_j{\varphi }_m+2f_m{f}_j+2g_m{g}_j+h_m{\psi }_j+k_m{\varphi }_j),\\ \hfill {\sigma }^{k_j}=-c_m(h_j{\varphi }_m+k_j{\psi }_m+2f_m{g}_j+2g_m{f}_j+k_m{\psi }_j+h_m{\varphi }_j).\end{array}$$

(21)

Further, Equation (14) can be obtained by taking a linear combination of the above results for $m=1,2,\dots ,n$. Therefore, we have completed the proof of Theorem 2. □

According to Lie’s first theorem, the initial value problem of the Lie point symmetry Equation (14) has the following form:

$$\begin{array}{c}\hfill \frac{dP\left(ϵ\right)}{dϵ}=-\frac{1}{2}\sum _{j=1}^n{c}_j{F}_j\left(ϵ\right),\frac{dQ\left(ϵ\right)}{dϵ}=-\frac{1}{2}\sum _{j=1}^n{c}_j{G}_j\left(ϵ\right),\end{array}$$

(22a)

$$\begin{array}{c}\hfill \frac{dR\left(ϵ\right)}{dϵ}=\frac{1}{2}\beta \sum _{j=1}^n{c}_j{H}_j\left(ϵ\right),\frac{dS\left(ϵ\right)}{dϵ}=\frac{1}{2}\beta \sum _{j=1}^n{c}_j{k}_j\left(ϵ\right),\end{array}$$

(22b)

$$\begin{array}{cc}\hfill & \frac{d\mathsf{\Psi }\left(ϵ\right)}{dϵ}=-c_i({\mathsf{\Psi }}_i^2\left(ϵ\right)+{\mathsf{\Phi }}_i^2\left(ϵ\right))-\sum _{j\ne i}^n{c}_j({\mathsf{\Psi }}_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)+{\mathsf{\Phi }}_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)),\hfill \\ \hfill & \frac{d\mathsf{\Phi }\left(ϵ\right)}{dϵ}=-2c_i{\mathsf{\Psi }}_i\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)-\sum _{j\ne i}^n{c}_j({\mathsf{\Psi }}_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+{\mathsf{\Phi }}_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)),\hfill \end{array}$$

(22c)

$$\begin{array}{cc}\hfill \frac{dF\left(ϵ\right)}{dϵ}=& -2c_i(F_i\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)+G_i\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right))\hfill \\ \hfill & -\sum _{j\ne i}^n{c}_j(F_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)+G_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+F_i\left(ϵ\right){\mathsf{\Psi }}_j\left(ϵ\right)+G_i\left(ϵ\right){\mathsf{\Phi }}_j\left(ϵ\right)),\hfill \\ \hfill \frac{dG\left(ϵ\right)}{dϵ}=& -2c_i(F_i\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+G_i\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right))\hfill \\ \hfill & -\sum _{j\ne i}^n{c}_j(F_i\left(ϵ\right){\mathsf{\Phi }}_j\left(ϵ\right)+G_i\left(ϵ\right){\mathsf{\Psi }}_j\left(ϵ\right)+F_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+G_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)),\hfill \end{array}$$

(22d)

$$\begin{array}{cc}\frac{dH\left(ϵ\right)}{dϵ}=\hfill & -2c_i(F_i^2\left(ϵ\right)+G_i^2\left(ϵ\right))-2c_i(H_i\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)+K_i\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right))\hfill \\ & -\sum _{j\ne i}^n{c}_j(H_i\left(ϵ\right){\mathsf{\Psi }}_j\left(ϵ\right)+K_i\left(ϵ\right){\mathsf{\Phi }}_j\left(ϵ\right)+2F_j\left(ϵ\right)F_i\left(ϵ\right)+2G_j\left(ϵ\right)G_i\left(ϵ\right)+H_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)+K_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)),\hfill \\ \frac{dK\left(ϵ\right)}{dϵ}=\hfill & -4c_i{F}_i\left(ϵ\right)G_i\left(ϵ\right)-2c_i(H_i\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+K_i\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right))\hfill \\ & -\sum _{j\ne i}^n{c}_j(H_i\left(ϵ\right){\mathsf{\Phi }}_j\left(ϵ\right)+K_i\left(ϵ\right){\mathsf{\Psi }}_j\left(ϵ\right)+2F_j\left(ϵ\right)G_i\left(ϵ\right)+2G_j\left(ϵ\right)F_i\left(ϵ\right)+H_j\left(ϵ\right){\mathsf{\Phi }}_i\left(ϵ\right)+K_j\left(ϵ\right){\mathsf{\Psi }}_i\left(ϵ\right)),\hfill \end{array}$$

(22e)

$$\begin{array}{cc}\hfill & P\left(0\right)=p,Q\left(0\right)=q,R\left(0\right)=r,S\left(0\right)=s,{\mathsf{\Psi }}_i\left(0\right)={\psi }_i,{\mathsf{\Phi }}_i\left(0\right)={\varphi }_i,\hfill \\ \hfill & F_i\left(0\right)=f_i,G_i\left(0\right)=g_i,H_i\left(0\right)=h_i,K_i\left(0\right)=k_i,\hfill \end{array}$$

(22f)

$i=1,2,\dots ,n.$

Then, one can get the following N-th Bäcklund transformation for the extended system Equation (15) by solving Equation (22).

Theorem 3.

If $\{p,q,r,s,f_i,g_i,h_i,k_i,{\psi }_i,{\varphi }_i\}$ is a solution of the coupled system Equation (15), then so is$\{P\left(ϵ\right),Q\left(ϵ\right),R\left(ϵ\right),S\left(ϵ\right),F_i\left(ϵ\right),G_i\left(ϵ\right),H_i\left(ϵ\right),K_i\left(ϵ\right),{\mathsf{\Psi }}_i\left(ϵ\right),{\mathsf{\Phi }}_i\left(ϵ\right)\}$, $(i=1,2,\dots ,n)$, where

$$\begin{array}{c}\hfill P\left(ϵ\right)=p-\frac{1}{4}{(ln(M+N)+ln(M-N))}_x,Q\left(ϵ\right)=q-\frac{1}{4}{(ln(M+N)-ln(M-N))}_x,\end{array}$$

(23a)

$$\begin{array}{c}\hfill R\left(ϵ\right)=r-\frac{\beta }{4}{(ln(M+N)+ln(M-N))}_{xx},S\left(ϵ\right)=s-\frac{\beta }{4}{(ln(M+N)-ln(M-N))}_{xx},\end{array}$$

(23b)

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_i\left(ϵ\right)=\frac{N_{i}N-M_{i}M}{M^2-N^2},{\mathsf{\Phi }}_i\left(ϵ\right)=\frac{M_{i}N-N_{i}M}{M^2-N^2},\end{array}$$

(23c)

$$\begin{array}{c}\hfill F_i\left(ϵ\right)={\mathsf{\Psi }}_{i,x}\left(ϵ\right),G_i\left(ϵ\right)={\mathsf{\Phi }}_{i,x}\left(ϵ\right),\end{array}$$

(23d)

$$\begin{array}{c}\hfill H_i\left(ϵ\right)={\mathsf{\Psi }}_{i,xx}\left(ϵ\right),K_i\left(ϵ\right)={\mathsf{\Phi }}_{i,xx}\left(ϵ\right),\end{array}$$

(23e)

where

$$M=\left|\begin{array}{cccccc}{c}_1ϵ{\psi }_1+1& c_1ϵA_{1,2}& \cdots & c_1ϵA_{1,j}& \cdots & c_1ϵA_{1,n}\\ c_2ϵA_{1,2}& c_2ϵ{\psi }_2+1& \cdots & c_2ϵA_{2,j}& \cdots & c_2ϵA_{2,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_jϵA_{1,j}& c_jϵA_{2,j}& \cdots & c_jϵ{\psi }_j+1& \cdots & c_jϵA_{j,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_nϵA_{1,n}& c_nϵA_{2,n}& \cdots & c_nϵA_{j,n}& \cdots & c_nϵ{\psi }_n+1\end{array}\right|,$$

$$N=\left|\begin{array}{cccccc}{c}_1ϵ{\varphi }_1+1& c_1ϵB_{1,2}& \cdots & c_1ϵB_{1,j}& \cdots & c_1ϵB_{1,n}\\ c_2ϵB_{1,2}& c_2ϵ{\varphi }_2+1& \cdots & c_2ϵB_{2,j}& \cdots & c_2ϵB_{2,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_jϵB_{1,j}& c_jϵB_{2,j}& \cdots & c_jϵ{\varphi }_j+1& \cdots & c_jϵB_{j,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_nϵB_{1,n}& c_nϵB_{2,n}& \cdots & c_nϵB_{j,n}& \cdots & c_nϵ{\varphi }_n+1\end{array}\right|,$$

$$M_i=\left|\begin{array}{cccccccc}{c}_1ϵ{\psi }_1+1& c_1ϵA_{1,2}& \cdots & c_1ϵA_{1,i-1}& c_1ϵA_{1,i}& c_1ϵA_{1,i+1}& \cdots & c_1ϵA_{1,n}\\ c_2ϵA_{1,2}& c_2ϵ{\psi }_2+1& \cdots & c_2ϵA_{2,i-1}& c_2ϵA_{2,i}& c_2ϵA_{2,i+1}& \cdots & c_2ϵA_{2,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_{i-1}ϵA_{1,i-1}& c_{i-1}ϵA_{2,i-1}& \cdots & c_{i-1}ϵ{\psi }_{i-1}+1& c_{i-1}ϵA_{i-1,i}& c_{i-1}ϵA_{i-1,i+1}& \cdots & c_{i-1}ϵA_{i-1,n}\\ A_{1,i}& A_{2,i}& \cdots & A_{i,i-1}& {\psi }_i& A_{i,i+1}& \cdots & A_{i,n}\\ c_{i+1}ϵA_{1,i+1}& c_{i+1}ϵA_{2,i+1}& \cdots & c_{i+1}ϵA_{i-1,i+1}& c_{i+1}ϵA_{i,i+1}& c_{i+1}ϵ{\psi }_{i+1}+1& \cdots & c_{i+1}ϵA_{i+1,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_nϵA_{1,n}& c_nϵA_{2,n}& \cdots & c_nϵA_{i-1,n}& c_nϵA_{i,n}& c_nϵA_{i+1,n}& \cdots & c_nϵ{\psi }_n+1\end{array}\right|,$$

$$N_i=\left|\begin{array}{cccccccc}{c}_1ϵ{\varphi }_1+1& c_1ϵB_{1,2}& \cdots & c_1ϵB_{1,i-1}& c_1ϵB_{1,i}& c_1ϵB_{1,i+1}& \cdots & c_1ϵB_{1,n}\\ c_2ϵB_{1,2}& c_2ϵ{\varphi }_2+1& \cdots & c_2ϵB_{2,i-1}& c_2ϵB_{2,i}& c_2ϵB_{2,i+1}& \cdots & c_2ϵB_{2,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_{i-1}ϵB_{1,i-1}& c_{i-1}ϵB_{2,i-1}& \cdots & B_{i-1}ϵ{\varphi }_{i-1}+1& c_{i-1}ϵB_{i-1,i}& c_{i-1}ϵB_{i-1,i+1}& \cdots & c_{i-1}ϵB_{i-1,n}\\ B_{1,i}& B_{2,i}& \cdots & B_{i,i-1}& {\varphi }_i& B_{i,i+1}& \cdots & B_{i,n}\\ c_{i+1}ϵB_{1,i+1}& c_{i+1}ϵB_{2,i+1}& \cdots & c_{i+1}ϵB_{i-1,i+1}& c_{i+1}ϵB_{i,i+1}& c_{i+1}ϵ{\varphi }_{i+1}+1& \cdots & c_{i+1}ϵB_{i+1,n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_nϵB_{1,n}& c_nϵB_{2,n}& \cdots & c_nϵB_{i-1,n}& c_nϵB_{i,n}& c_nϵB_{i+1,n}& \cdots & c_nϵ{\varphi }_n+1\end{array}\right|,$$

with

$$\begin{array}{c}\hfill A_{i,j}=\frac{1}{2}{({\psi }_i{\psi }_j+{\varphi }_i{\varphi }_j+{\psi }_i{\varphi }_j+{\varphi }_i{\psi }_j)}^{\frac{1}{2}}+\frac{1}{2}{({\psi }_i{\psi }_j+{\varphi }_i{\varphi }_j-{\psi }_i{\varphi }_j-{\varphi }_i{\psi }_j)}^{\frac{1}{2}},\\ \hfill B_{i,j}=\frac{1}{2}{({\psi }_i{\psi }_j+{\varphi }_i{\varphi }_j+{\psi }_i{\varphi }_j+{\varphi }_i{\psi }_j)}^{\frac{1}{2}}-\frac{1}{2}{({\psi }_i{\psi }_j+{\varphi }_i{\varphi }_j-{\psi }_i{\varphi }_j-{\varphi }_i{\psi }_j)}^{\frac{1}{2}}.\end{array}$$

From any seed solution of the strongly coupled B–B system, one can get an infinite number of new solutions because n is an arbitrary positive integer.

4. Conclusions

To sum up, a new strongly coupled B–B system was first introduced. Then, the Schwarzian form of the strongly coupled B–B system was obtained by using the truncated Painlevé expansion, and the corresponding residual was the nonlocal symmetry. To localize the residual symmetry, we introduced a suitable enlarged system. According to Lie’s first theorem, the ralated finite transformation was found. Further, the N-th Bäcklund transformation of the strongly coupled Burgers system was obtained by localizing the linear superposition of multiple residual symmetries, and the N-th Bäcklund transformation was expressed by determinants in a compact form.