Discretization of the modified Korteweg–de Vries–sine Gordon equation

Abstract

We provide an integrable discretization of the modified Korteweg–de Vries–sine Gordon equation. The discrete form is a coupled system and is derived via the Cauchy matrix approach by introducing suitable discrete plane wave factors. Solutions and a Lax pair are constructed in this approach. The dynamics of some solutions are illustrated. The modified Korteweg–de Vries–sine Gordon equation is recovered in the continuum limit.

Appendix: Solution of Eq. (2.1) and (2.3)

We list solutions $(\boldsymbol r,\boldsymbol M)$ of the coupled system (2.1) and (2.3) when $\boldsymbol K$ is in the canonical form and $\boldsymbol s\in\mathbb{C}_N$. We can also refer to Appendices A and B in [16] or Appendix A in [21] for similar formulas. We first introduce some notation (cf.[15], [16]).

We consider the following matrices: an $N\times N$ diagonal matrix

$$ \Gamma_{\mathrm d}^{[N]}(\{k_{j}\}_1^{N})=\operatorname{diag}(k_1,k_2,\ldots,k_{N}),$$

(A.1)

an $N\times N$ Jordan matrix

$$ \Gamma_{\mathrm J}^{[N]}(k_1)=\begin{pmatrix} k_1 & 0 & 0 &\ldots & 0 \\ 1 & k_1 & 0 &\ddots &\vdots \\ 0 & 1 & k_1 &\ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \ldots & 0& 1 & k_1 \\ \end{pmatrix}_{\!{}^{\scriptstyle N\times N}},$$

(A.2)

a lower triangular Toeplitz matrix

$$ T^{[N]}(\{a_{j}\}_1^{N})=\begin{pmatrix} a_1 & 0 & 0 &\ldots & 0 \\ a_2 & a_1 & 0 &\ddots & \vdots \\ a_3 & a_2 & a_1 &\ddots & 0 \\ \vdots & \ddots & \ddots &\ddots & 0 \\ a_N & \ldots & a_3 & a_2 & a_1 \end{pmatrix}_{\!{}^{\scriptstyle N\times N}},$$

(A.3)

and a skew triangular Toeplitz matrix

$$ H^{[N]}(\{b_{j}\}_1^{N})=\begin{pmatrix} b_1 & \ldots & b_{N-2} & b_{N-1} & b_N\\ \vdots & \cdot^{\displaystyle{\displaystyle\cdot}^{\displaystyle\cdot}} & b_{N-1} & b_{N}& 0 \\ b_{N-2} & \cdot^{\displaystyle{\displaystyle\cdot}^{\displaystyle\cdot}} & b_N & 0 & 0\\ b_{N-1} & \cdot^{\displaystyle{\displaystyle\cdot}^{\displaystyle\cdot}} & \cdot^{\displaystyle{\displaystyle\cdot}^{\displaystyle\cdot}} & \cdot^{\displaystyle{\displaystyle\cdot}^{\displaystyle\cdot}} & \vdots \\ b_N & 0 & 0 & \ldots & 0 \end{pmatrix}_{\!{}^{\scriptstyle N\times N}}.$$

(A.4)

We introduce discrete PWFs

$$ \rho_i=\biggl(\frac{p+k_i}{p-k_i}\biggr)^{\!n}\biggl(\frac{q+k_i}{q-k_i}\biggr)^{\!m}\biggl(\frac{q^3k_i+1}{q^3k_i-1}\biggr)^{\!m}\rho^{(0)}_i$$

(A.5)

where $\rho^{0}_i$ are constants. In addition, we introduce $N$th-order vectors

$$ \boldsymbol r_{\mathrm d}^{N}(\{k_{j}\}_1^{N})=(\rho_1,\ldots,\rho_{N})^{\mathrm T},\qquad \boldsymbol r_{\mathrm J}^{N}(k_1)=(r_1,\ldots,r_{N})^{\mathrm T},$$

(A.6)

where

$$r_1=\rho_1,\qquad r_i=\frac{\partial_{k_1}^{i-1}\rho_1}{(i-1)!}\quad\text{for}\quad i=1,\ldots,N.$$

We also introduce the matrices

$$ \begin{alignedat}{3} &G_{\mathrm d}^{[N]}(\{k_{j}\}_1^{N})=(g_{i,j})_{N\times N},&\qquad &g_{i,j}=\frac{1}{k_i+k_{j}}, \\ &G_{\mathrm{dJ}}^{[N_1,N_2]}(\{k_i\}_1^{N_1};a)=(g_{i,j})_{N_1\times N_2},&\qquad &g_{i,j}=-\biggl(\frac{-1}{k_i+a}\biggr)^{\!j}, \\ &G_{\mathrm{JJ}}^{[N_1,N_2]}(a;b)=(g_{i,j})_{N_1\times N_2},&\qquad &g_{i,j}=\mathrm{C}_{i+j-2}^{i-1}\frac{(-1)^{i+j}}{(a+b)^{i+j-1}}, \\ &G_{\mathrm J}^{[N]}(a)=G_{\mathrm{JJ}}^{[N,N]}(a;a)=(g_{i,j})_{N\times N},&\qquad &g_{i,j}=\mathrm{C}_{i+j}^{i-1}\frac{(-1)^{i+j}}{(2a)^{i+j-1}}, \end{alignedat}$$

(A.7)

where $\mathrm{C}_{j}^{i}=\frac{j!}{i!(j-i)!}$ for $j\ge i$.

With this notation, we list solutions $(\boldsymbol r,\boldsymbol M)$ of coupled system (2.1) and (2.3).

Case 1.

If

$$ \boldsymbol K=\Gamma_{\mathrm d}^{[N]}(\{k_{j}\}_1^{N}),$$

(A.8)

then

$$ \boldsymbol r=\boldsymbol r_{\mathrm d}^{N}(\{k_{j}\}_1^{N}),\qquad \boldsymbol M=\biggl(\frac{\rho_i s_{j}}{k_i+k_{j}}\biggr)_{{}^{\scriptstyle N\times N}}.$$

(A.9)

Case 2.

If

$$ \boldsymbol K=\Gamma_{\mathrm J}^{[N]}(k_1),$$

(A.10)

then

$$ \boldsymbol r=\boldsymbol r_{\mathrm J}^{N}(k_1),\qquad \boldsymbol M=\boldsymbol F\boldsymbol G\boldsymbol H,$$

(A.11)

where

$$\boldsymbol F=T^{[N]}(\{\rho_{j}\}_1^{N}),\qquad\boldsymbol G=\boldsymbol G_{\mathrm J}^{[N]}(k_1),\qquad \boldsymbol H=H^{[N]}(\{s_{j}\}_1^{N}).$$

Case 3.

If

$$ \boldsymbol K=\operatorname{diag} \bigl(\Gamma_{\mathrm d}^{[N_1]}(\{k_{j}\}_1^{N_1}),\Gamma_{\mathrm J}^{[N_2]}(k_{N_1+1}),\Gamma_{\mathrm J}^{[N_3]}(k_{N_1+2}), \ldots,\Gamma_{\mathrm J}^{[N_{s}]}(k_{N_1+(s-1)})\bigr),$$

(A.12)

where $\sum_{j=1}^{s}N_{j}=N$, then

$$\boldsymbol r=\begin{pmatrix} \boldsymbol r_{\mathrm d}^{N_1}(\{k_{j}\}_1^{N_1}) \\ \boldsymbol r_{\mathrm J}^{N_2}(k_{N_1+1}) \\ \boldsymbol r_{\mathrm J}^{N_3}(k_{N_1+2}) \\ \vdots\\ \boldsymbol r_{\mathrm J}^{N_{s}}(k_{N_1+(s-1)}) \end{pmatrix}, \quad \boldsymbol M=\boldsymbol F\boldsymbol G\boldsymbol H,$$

where

$$\begin{aligned} \, &\boldsymbol F=\operatorname{diag}\bigl(\Gamma_{\mathrm d}^{[N_1]}(\{\rho_{j}\}_1^{N_1}), T^{[N_2]}(\{r_{j}\}_{N_1+1}^{N_1+N_2}), \\ &\kern 100pt T^{[N_3]}(\{r_{j}\}_{N_1+N_2+1}^{N_1+N_2+N_3}),\ldots, T^{[N_s]}(\{r_{j}\}_{1+\sum_{j=1}^{s-1}N_{j}}^{N})\bigr), \\ &\boldsymbol H=\operatorname{diag}\bigl(\Gamma_{\mathrm d}^{[N_1]}(\{\sigma_{j}\}_1^{N_1}), H^{[N_2]}(\{s_{j}\}_{N_1+1}^{N_1+N_2}), \\ &\kern 100pt H^{[N_3]}(\{s_{j}\}_{N_1+N_2+1}^{N_1+N_2+N_3}),\ldots, H^{[N_s]}(\{s_{j}\}_{1+\sum_{j=1}^{s-1}N_{j}}^{N})\bigr), \end{aligned}$$

and $\boldsymbol G$ is a symmetric matrix with the block structure

$$\boldsymbol G=\boldsymbol G^{\mathrm T}=(\boldsymbol G_{i,j})_{1\le i,j\le s}$$

with

$$\begin{aligned} \, &\boldsymbol G_{1,1}=\boldsymbol G_{\mathrm d}^{[N_1]}(\{k_{j}\}_1^{N_1}), && \\ &\begin{aligned} \, &\boldsymbol G_{1,j}=\boldsymbol G^{\mathrm T}_{j,1}=G_{DJ}^{[N_1,N_{j}]}(\{k_{j}\}_1^{N_1};k_{N_{j-1}+1}), \\ &\boldsymbol G_{i,j}=\boldsymbol G^{\mathrm T}_{j,i}=G_{JJ}^{[N_i,N_{j}]}(k_{N_{i-1}+1};k_{N_{j-1}+1}), \end{aligned}\qquad 1<i\le j\le s. \end{aligned}$$