Solving the (2+1)-Dimensional Derivative Toda Equation

Abstract

The main objectives of this work are to deduce the Lax pair for the Modified Toda lattice hierarchy and to find whether there exists a transformation between (2+1)-dimensional derivative Toda lattice equation and Modified Toda equation. By a new spectral problem, we obtain the Modified Toda lattice hierarchy, furthermore we give the Hirota’s form solution for the Modified Toda lattice equation, thus the Hirota’s solution for the (2+1)-dimensional derivative Toda equation can also be attained correspondingly.

1. Introduction

Toda lattice is, together with the Kortewegde Vries (KdV) equation, one of the most classical and important completely integrable systems. Several methods have been developed to reveal its profound mathematical structure [1,2]. Bogoyavlensky discovered that the classical Toda lattice (TL) is connected with the simple Lie algebras [3]. Then, Leznov and Saveliev [4] showed that the periodic TL corresponds to contragradient Lie algebras (Kac–Moody algebras). This connection enables investigation of the systems by means of the Inverse Scattering Method (ISM) [5]. On the other hand, the Sinh–Gordon equation and the Liouville equation have supersymmetric extensions, while the integration by the ISM is as applicable as before [6,7,8]. Searching exact solutions for nonlinear evolution equations is always of physical and mathematical importance and interest. There have been many well-developed methods, such as the Inverse Scattering Transform [9,10], Darboux transformation [11,12], Lie group and symmetry method [13,14], Bäcklund transformation [15], and Bilinear method [16].

In this paper, we first start the spectral problem and obtain the Modified Toda lattice hierarchy, and also give the explicit expression of the soliton solution of the Modified Toda lattice equation.

The paper is organized as follows. In Section 2, we introduce a spectral problem and attain the Modified Toda lattice hierarchy. In Section 3, we derive the soliton solution in Hirota’s form of the Modified Toda lattice equation. In Section 4, the soliton solution for the (2+1)-dimensional derivative Toda equation is given accordingly. Finally, some discussions and conclusions are given.

2. Modified Toda Lattice Equation Related to New Spectral Problem

In this section, we derive the Modified Toda lattice equation and its counterpart spectral problem.

Let us define the shift operator E, which has been used for discussing the spectral problem

$$E{f}_n=f_{n+1},E^{-1}{f}_n=f_{n-1}$$

(1)

Now, let begin the following spectral problem

$${\varphi }_{n+1}=-u_n{\varphi }_{n-1}-(\frac{1}{v_n}\lambda +\frac{1}{\lambda }){\varphi }_n,$$

(2a)

with the time evolution

$${\varphi }_{n,t}=A_n{\varphi }_n+B_n{\varphi }_{n-1},$$

(2b)

Based on the compatibility condition ${\varphi }_{n+1,t}=E{\varphi }_{n,t}$, we can obtain

$$\begin{array}{cc}\hfill & (u_n{A}_{n+1}-u_n{A}_{n-1}-u_{n,t}){\varphi }_{n-1}+(\frac{u_n}{u_{n-1}}{B}_{n-1}-B_{n+1}){\varphi }_n\hfill \\ \hfill & =[-(\frac{1}{v_n}\lambda +\frac{1}{\lambda })A_{n+1}-\frac{v_{n,t}}{v_n^2}\lambda +(\frac{1}{v_n}\lambda +\frac{1}{\lambda })A_n]{\varphi }_n\hfill \\ \hfill & +[-\frac{u_n}{u_{n-1}}(\frac{1}{v_{n-1}}\lambda +\frac{1}{\lambda })B_{n-1}+(\frac{1}{v_n}\lambda +\frac{1}{\lambda })B_n]{\varphi }_{n-1},\hfill \end{array}$$

By comparing the coefficients of ${\varphi }_n$ and ${\varphi }_{n-1}$, we can derive

$$u_n{A}_{n+1}-u_n{A}_{n-1}-u_{n,t}=(\frac{1}{v_n}\lambda +\frac{1}{\lambda })B_n-\frac{u_n}{u_{n-1}}{B}_{n-1}(\frac{1}{v_{n-1}}\lambda +\frac{1}{\lambda }),$$

(3a)

$$\frac{u_n}{u_{n-1}}{B}_{n-1}-B_{n+1}=-\frac{v_{n,t}}{v_n^2}\lambda +(A_n-A_{n+1})(\frac{1}{v_n}\lambda +\frac{1}{\lambda }),$$

(3b)

Thus, by substituting the expansion

$$A_n=\sum _{j=0}^k{a}_{n,j}{\lambda }^{2(k-j)+2},B_n=\sum _{j=0}^k{b}_{n,j}{\lambda }^{2(k-j)+1},(k=1,2,\cdots )$$

(4)

into (3a) and comparing the coefficients of the same powers of $\lambda$, we have

$$\begin{array}{ccc}\hfill -u_{n,t}& =& b_{n,k}-\frac{u_n}{u_{n-1}}{b}_{n-1,k},\hfill \end{array}$$

(5a)

$$\begin{array}{ccc}\hfill u_n{a}_{n+1,j+1}-u_n{a}_{n-1,j+1}& =& \frac{1}{v_n}{b}_{n,j+1}+b_{n,j}-\frac{u_n}{u_{n-1}{v}_{n-1}}{b}_{n-1,j+1}-\frac{u_n}{u_{n-1}}{b}_{n-1,j},\hfill \end{array}$$

(5b)

$$\begin{array}{ccc}& & (j=0,1,2,\cdots ,k-1)\hfill \end{array}$$

(5c)

$$\begin{array}{ccc}\hfill u_n{a}_{n+1,0}-u_n{a}_{n-1,0}& =& \frac{1}{v_n}{b}_{n,0}-\frac{u_n}{u_{n-1}{v}_{n-1}}{b}_{n-1,0},\hfill \end{array}$$

(5d)

Further, we have the following forms:

$$\begin{array}{ccc}\hfill u_{n,t}& =& (\frac{u_n}{u_{n-1}}{E}_n^{-1}-1)b_{n,k},\hfill \\ \hfill u_n(E_n-E_n^{-1})a_{n,j+1}& =& (\frac{1}{v_n}-\frac{u_n}{u_{n-1}{v}_{n-1}}{E}_n^{-1})b_{n,j+1}+(1-\frac{u_n}{u_{n-1}}{E}_n^{-1})b_{n,j},\hfill \end{array}$$

(6a)

$$\begin{array}{ccc}& & (j=0,1,2,\cdots ,k-1)\hfill \end{array}$$

(6b)

$$\begin{array}{ccc}\hfill u_n(E_n-E_n^{-1})a_{n,0}& =& (\frac{1}{v_n}-\frac{u_n}{u_{n-1}{v}_{n-1}}{E}_n^{-1})b_{n,0},\hfill \end{array}$$

(6c)

By substituting (4) into (3b) and taking the same steps, we have

$$\begin{array}{ccc}& & v_{n,t}=v_n^2(E_n-\frac{u_n}{u_{n-1}}{E}_n^{-1})b_{n,k}+v_n^2(1-E_n)a_{n,k},\hfill \\ & & v_n^2(1-E_n)a_{n,j}+v_n^2(E_n-\frac{u_n}{u_{n-1}}{E}_n^{-1})b_{n,j}=v_n(E_n-1)a_{n,j+1},\hfill \end{array}$$

(7a)

$$\begin{array}{ccc}& & (j=0,1,2,\cdots ,k-1)\hfill \end{array}$$

(7b)

$$\begin{array}{ccc}& & (1-E_n)a_{n,0}=0,\hfill \end{array}$$

(7c)

By the (6) and (7), we can obtain

$$\begin{array}{ccc}\hfill {\left(\begin{array}{c}{u}_n\\ v_n\end{array}\right)}_t& =& L_1\left(\begin{array}{c}{a}_{n,k}\\ b_{n,k}\end{array}\right),\hfill \end{array}$$

(8a)

$$\begin{array}{ccc}\hfill L_1\left(\begin{array}{c}{a}_{n,j}\\ b_{n,j}\end{array}\right)& =& L_2\left(\begin{array}{c}{a}_{n,j+1}\\ b_{n,j+1}\end{array}\right),(j=0,1,2,\cdots ,k-1)\hfill \end{array}$$

(8b)

$$\begin{array}{ccc}\hfill L_2\left(\begin{array}{c}{a}_{n,0}\\ b_{n,0}\end{array}\right)& =& \left(\begin{array}{c}0\\ 0\end{array}\right),\hfill \end{array}$$

(8c)

where $L_1,L_2$ are operators defined by respectively,

$$\begin{array}{ccc}\hfill L_1& =& \left(\begin{array}{cc}0& \frac{u_n}{u_{n-1}}{E}_n^{-1}-1\\ v_n^2(1-E_n)& v_n^2(E_n-\frac{u_n}{u_{n-1}}{E}_n^{-1})\end{array}\right),\hfill \end{array}$$

(9a)

$$\begin{array}{ccc}\hfill L_2& =& \left(\begin{array}{cc}-u_n(E_n-E_n^{-1})& \frac{1}{v_n}-\frac{u_n}{u_{n-1}{v}_{n-1}}{E}_n^{-1}\\ v_n(E_n-1)& 0\end{array}\right),\hfill \end{array}$$

(9b)

Then we obtain the hierarchy

$$\begin{array}{ccc}\hfill {\left(\begin{array}{c}{u}_n\\ v_n\end{array}\right)}_t& =& {\left(L_1{L}_2^{-1}\right)}^k{L}_1\left(\begin{array}{c}{a}_{n,0}\\ b_{n,0}\end{array}\right).\hfill \end{array}$$

(10)

when $k=0$, we obtain the equation

$$\begin{array}{ccc}\hfill {\left(\begin{array}{c}{u}_n\\ v_n\end{array}\right)}_t& =& \left(\begin{array}{c}{u}_n(v_n-v_{n-1})\\ v_n^2(u_n{v}_{n-1}-u_{n+1}{v}_{n+1})\end{array}\right),\hfill \end{array}$$

(11)

where we assume $a_{n,0}=0,b_{n,0}=-u_n{v}_n$.

Besides, the Lax pair of the the above equation is provided by

$$\begin{array}{ccc}\hfill {\varphi }_{n+1}& =& -u_n{\varphi }_{n-1}-(\frac{1}{v_n}\lambda +\frac{1}{\lambda }){\varphi }_n,\hfill \end{array}$$

(12a)

$$\begin{array}{ccc}\hfill {\varphi }_{n,t}& =& -u_n{v}_n\lambda {\varphi }_{n-1},\hfill \end{array}$$

(12b)

if we take the following expansion:

$$A_n=\sum _{j=0}^k{a}_{n,j}{\lambda }^{2(j-k)},B_n=\sum _{j=0}^k{b}_{n,j}{\lambda }^{2(j-k)-1},(k=1,2,\cdots )$$

(13)

in this case, we take the same steps as the above, then we can obtain another equation:

$$\begin{array}{ccc}\hfill {\left(\begin{array}{c}{u}_n\\ v_n\end{array}\right)}_y& =& \left(\begin{array}{c}{u}_n[(u_{n+1}-u_{n-1})+(\frac{1}{v_{n-1}}-\frac{1}{v_n})]\\ -v_n(u_{n+1}-u_n)\end{array}\right).\hfill \end{array}$$

(14)

and the responding Lax pair of the the above equation is provided by

$$\begin{array}{ccc}\hfill {\varphi }_{n+1}& =& -u_n{\varphi }_{n-1}-(\frac{1}{v_n}\lambda +\frac{1}{\lambda }){\varphi }_n,\hfill \end{array}$$

(15a)

$$\begin{array}{ccc}\hfill {\varphi }_{n,y}& =& u_n({\varphi }_n+\frac{1}{\lambda }{\varphi }_{n-1}),\hfill \end{array}$$

(15b)

where the (12b) and the (15b) are compatible.

Due to the compatibility of the (11) and (14), we can obtain the Modified Toda lattice equation

$$\begin{array}{ccc}\hfill u_{n,t}& =& u_n(v_n-v_{n-1}),\hfill \end{array}$$

(16a)

$$\begin{array}{ccc}\hfill v_{n,y}& =& -v_n(u_{n+1}-u_n).\hfill \end{array}$$

(16b)

3. $\mathit{N}$-Soliton Solutions for the Modified Toda Lattice Equation

In this section, we construct N-soliton solutions for the Modified Toda lattice equation by the Hirota bilinear method. According to the definition of the Hirota’s bilinear derivative operator D [16], we have

$$D_x^m{D}_t^{n}a·b={(\partial x-\partial x^{\prime })}^m{(\partial t-\partial t^{\prime })}^{n}a(x,t)b(x^{\prime },t^{\prime }){|}_{x^{\prime }=x,t^{\prime }=t}.$$

(17)

In order to bilinearize the Modified Toda lattice Equation (16), we replace $u_n,v_n$ by

$$u_n={(\ln \frac{g_n}{f_n})}_y,v_n={(\ln \frac{g_{n+1}}{f_n})}_t,$$

(18)

Equation (16) can be transformed into the bilinear form

$$\begin{array}{c}\hfill D_y{g}_n·f_n=f_{n-1}{g}_{n+1},\end{array}$$

(19a)

$$\begin{array}{c}\hfill D_t{g}_{n+1}·f_n=f_{n+1}{g}_n,\end{array}$$

(19b)

According to the Hirota bilinear method, we expand $f_n$ and $g_n$ into the following form

$$\begin{array}{ccc}\hfill f_n& =& 1+f_n^{\left(1\right)}\epsilon +f_n^{\left(2\right)}{\epsilon }^2+f_n^{\left(3\right)}{\epsilon }^3+\cdots ,\hfill \\ \hfill g_n& =& g_n^{\left(0\right)}+g_n^{\left(1\right)}\epsilon +g_n^{\left(2\right)}{\epsilon }^2+\cdots ,\hfill \end{array}$$

substituting them into (19) and comparing each power of $\epsilon$, we obtain

$$\begin{array}{c}\hfill D_y{g}_n^{\left(0\right)}·1=g_{n+1}^{\left(0\right)},\end{array}$$

(20a)

$$\begin{array}{c}\hfill D_y{g}_n^{\left(1\right)}·1-g_{n+1}^{\left(1\right)}=g_{n+1}^{\left(0\right)}{f}_{n-1}^{\left(1\right)}-D_y{g}_n^{\left(0\right)}·f_n^{\left(1\right)},\end{array}$$

(20b)

$$\begin{array}{c}\hfill D_y{g}_n^{\left(2\right)}·1-g_{n+1}^{\left(2\right)}=g_{n+1}^{\left(1\right)}{f}_{n-1}^{\left(1\right)}+g_{n+1}^{\left(0\right)}{f}_{n-1}^{\left(2\right)}-D_y{g}_n^{\left(1\right)}·f_n^{\left(1\right)}-D_y{g}_n^{\left(0\right)}·f_n^{\left(2\right)},\\ \hfill \cdots \cdots ;& \hfill & \hfill \end{array}$$

(20c)

$$\begin{array}{c}\hfill D_t{g}_{n+1}^{\left(0\right)}·1=g_n^{\left(0\right)},\end{array}$$

(21a)

$$\begin{array}{c}\hfill D_t{g}_{n+1}^{\left(1\right)}·1-g_n^{\left(1\right)}=g_n^{\left(0\right)}{f}_{n+1}^{\left(1\right)}-D_t{g}_{n+1}^{\left(0\right)}·f_n^{\left(1\right)},\end{array}$$

(21b)

$$\begin{array}{c}\hfill D_t{g}_{n+1}^{\left(2\right)}·1-g_n^{\left(2\right)}=g_n^{\left(1\right)}{f}_{n+1}^{\left(1\right)}+g_n^{\left(0\right)}{f}_{n+1}^{\left(2\right)}-D_t{g}_{n+1}^{\left(1\right)}·f_n^{\left(1\right)}-D_t{g}_{n+1}^{\left(0\right)}·f_n^{\left(2\right)},\\ \hfill \cdots \cdots .& & \hfill \end{array}$$

(21c)

To obtain N-soliton-like solutions, one can take

$$g_n^{\left(0\right)}=e^{{\xi }_1},f_n^{\left(1\right)}=\sum _{j=1}^N{e}^{{\eta }_j},g_n^{\left(1\right)}=\sum _{j=1}^N{\alpha }_j{e}^{{\xi }_1+{\eta }_j},$$

(22)

where

$${\xi }_1={\omega }_{1}t+k_{1}y+l_{1}n+{\xi }_1^{\left(0\right)},{\omega }_1=e^{-l_1},k_1=e^{l_1},{\eta }_j=a_{j}t+b_{j}y+c_{j}n+{\eta }_j^{\left(0\right)},$$

(23a)

$$a_j=e^{\frac{c_j}{2}}-e^{-\frac{c_j}{2}},b_j=e^{-\frac{c_j}{2}}-e^{\frac{c_j}{2}},{\alpha }_j=\frac{1+e^{l_1-\frac{c_j}{2}}}{1+e^{l_1+\frac{c_j}{2}}}(j=1,2,\cdots ,N).$$

(23b)

then from (20) and (21), $\left\{f_n^{\left(j\right)}\right\}$ and $\left\{g_n^{\left(j\right)}\right\}$ can be derived step by step.

When $N=1$, we can have the following truncated $f_n$ and $g_n$ as solutions

$$f_n=1+e^{{\eta }_1},$$

(24)

$$g_n=e^{{\xi }_1}+{\alpha }_1{e}^{{\xi }_1+{\eta }_1},$$

(25)

where we have taken $\epsilon =1$.

We depict the shapes of

$$\begin{array}{ccc}\hfill u_n& =& {\left(\ln \frac{e^{{\xi }_1}(1+{\alpha }_1{e}^{{\eta }_1})}{1+e^{{\eta }_1}}\right)}_y,\hfill \end{array}$$

(26a)

$$\begin{array}{ccc}\hfill v_n& =& {\left(\ln \frac{e^{{\xi }_1+l_1}(1+{\alpha }_1{e}^{{\eta }_1+c_1})}{1+e^{{\eta }_1}}\right)}_t,\hfill \end{array}$$

(26b)

in Figure 1.

In the case of N = 2, we have

$$f_n=1+\epsilon (e^{{\eta }_1}+e^{{\eta }_2})+{\epsilon }^2{e}^{{\eta }_1+{\eta }_2+A_{12}},$$

(27)

$$g_n=e^{{\xi }_1}+\epsilon ({\alpha }_1{e}^{{\xi }_1+{\eta }_1}+{\alpha }_2{e}^{{\xi }_1+{\eta }_2})+{\epsilon }^2{\alpha }_1{\alpha }_2{e}^{{\xi }_1+{\eta }_1+{\eta }_2+A_{12}},$$

(28)

and the corresponding 2-soliton-like solution is ($\epsilon =1$)

$$\begin{array}{ccc}\hfill u_n& =& {\left(\ln \frac{e^{{\xi }_1}[1+{\alpha }_1{e}^{{\eta }_1}+{\alpha }_2{e}^{{\eta }_2}+{\alpha }_1{\alpha }_2{e}^{{\eta }_1+{\eta }_2+A_{12}}]}{1+e^{{\eta }_1}+e^{{\eta }_2}+e^{{\eta }_1+{\eta }_2+A_{12}}}\right)}_y,\hfill \end{array}$$

(29a)

$$\begin{array}{ccc}\hfill v_n& =& {\left(\ln \frac{e^{{\xi }_1+l_1}[1+{\alpha }_1{e}^{{\eta }_1+c_1}+{\alpha }_2{e}^{{\eta }_2+c_2}+{\alpha }_1{\alpha }_2{e}^{{\eta }_1+{\eta }_2+c_1+c_2+A_{12}}]}{1+e^{{\eta }_1}+e^{{\eta }_2}+e^{{\eta }_1+{\eta }_2+A_{12}}}\right)}_t,\hfill \end{array}$$

(29b)

This procedure can be continued to the three-soliton, four-soliton solutions and so on. Generally, we have

$$f_n=\sum _{\mu =0,1}\exp \left[\sum _{j=1}^N{\mu }_j{\eta }_j+\sum _{1\le j<l}^N{\mu }_j{\mu }_l{A}_{jl}\right],$$

(30)

$$g_n=e^{{\xi }_1}\sum _{\mu =0,1}\left(\sum _{j=1}^N{\alpha }_j{e}^{{\mu }_j}\right)\exp [\sum _{j=1}^N{\mu }_j{\eta }_j+\sum _{1\le j<l}^N{\mu }_j{\mu }_l{A}_{jl}],$$

(31)

where ${\xi }_j$ and ${\eta }_j$ are given by (23)

$$e^{A_{jl}}=\frac{sin{h}^2\frac{c_j-c_l}{4}}{sin{h}^2\frac{c_j+c_l}{4}},$$

(32)

and the sum is taken over all possible combinations of ${\mu }_j=0,1$ for $j=1,2,\cdots ,N$.

4. (2+1)-Dimensional Derivative Toda Equation

A (2+1)-dimensional derivative Toda equation is derived from the Lax triad composed of the Kaup–Newell, the negative Kaup–Newell and a discrete eigenvalue problem [17,18,19].

$$\frac{{\partial }^2{\phi }_n}{\partial x\partial y}+(e^{{\phi }_{n+1}-{\phi }_n}-e^{{\phi }_n-{\phi }_{n-1}})\frac{\partial {\phi }_n}{\partial x}=0,$$

(33)

and the Modified Toda lattice is proposed by Hirota [16],

$$\begin{array}{ccc}& & u_{n,t}=u_n(v_n-v_{n-1}),\hfill \end{array}$$

(34a)

$$\begin{array}{ccc}& & v_{n,y}=-v_n(u_{n+1}-u_n),\hfill \end{array}$$

(34b)

if we take the following transformation

$$u_n=e^{{\phi }_n-{\phi }_{n-1}},v_n={\phi }_{n,x},$$

(35)

Then, the Equation (34) can be transformed to the Equation (33). Thus, solving the (2+1)-dimensional derivative Toda equation can reduced to solve the Modified Toda lattice equation.

5. Discussions

The Hirota method can be used not only to study the integrability of ordinary nonlinear equations, but also to study the integrability of supersymmetry systems [20]. The supersymmetry extension of nonlinear evolution equation is a coupling system of the Bose field and Fermion field. The corresponding results may have potential applications in nonlinear optics, plasma physics and fluid dynamics [21]. The Lax pairs of the (2+1)-dimensional derivative Toda equation play an important role in the nonlinear integral system. From the Lax pairs, we can derive solutions of evolution equation hierarchies, and meanwhile, many integrable properties can be researched such as conservation laws, Hamiltonian structure, $\tau$-symmmetry and so on. Conservation laws play an important role, the existence of which for the integrable lattice equations may further confirm their integrability. Finally the symmetries and infinitely many conservation laws for this system are further considered.

6. Conclusions

In this research, we have considered the (2+1)-dimensional derivative Toda equation and derived the transformation between (2+1)-dimensional derivative Toda equation and Modified Toda lattice equation. We derived the spectral problem related to the Modified Toda lattice equation and give the Hirota’s form soliton solution. It seems that solitary wave solutions are merely special cases in one family.