Riemann-Hilbert approach and N-soliton solution for the Chen-Lee-Liu equation

Abstract

In this paper, we consider the Riemann-Hilbert method for the Chen-Lee-Liu equation and obtain a new formula of the explicit N-soliton solution (see equation (44) in Sect. 4), which is expressed by a ratio of two $(N+1)\times (N+1)$ determinants. Meanwhile, we have provided a way to deal with an integral factor in the sectional analytic functions $P_{\pm }$. Furthermore, by applying asymptotic analysis and using the property of the Cauchy determinant, the simple elastic interactions of N-soliton are concluded.

Appendix A. The proof of Theorem 4.3

Proof

Since $\mathrm{Im}\lambda _k^2>0(k=1,2,\cdots ,N)$, the asymptotic behavior of $\exp (-\theta _k)$ is decided by $x+4(\mathrm{Re}\lambda _k^2)t$. Denoting the vicinity of $x=-4(\mathrm{Re}\lambda _k^2)t$ as $\Omega _k$, in the limit of $t\rightarrow +\infty$, then we may obtain

$$\begin{aligned} x+4(\mathrm{Re}\lambda _j^2)t=-4(\mathrm{Re}\lambda _k^2-\mathrm{Re}\lambda _j^2)t\rightarrow +\infty \quad j<k,\\ x+4(\mathrm{Re}\lambda _j^2)t=-4(\mathrm{Re}\lambda _k^2-\mathrm{Re}\lambda _j^2)t\rightarrow -\infty \quad j>k. \end{aligned}$$

Thus,

$$\begin{aligned} \exp (-\theta _j)\rightarrow 0\quad j<k; \quad \exp (\theta _j)\rightarrow 0\quad j>k. \end{aligned}$$

Then, in the vicinity of $\Omega _k$

$$\begin{aligned} \mathrm{det}M_1\sim&\left| \begin{array}{cccccccc} \frac{-2\lambda _1^{*}}{\lambda _1^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _1^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}}{\lambda _1^2-\lambda _k^{*2}}&{}0&{}\cdots &{}0&{}1\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots \\ \frac{-2\lambda _1^{*}}{\lambda _{k-1}^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _{k-1}^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}}{\lambda _{k-1}^2-\lambda _k^{*2}}&{}0&{}\cdots &{}0&{}1\\ \frac{-2\lambda _1^{*}}{\lambda _{k}^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _{k}^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}-2\lambda _ke^{-\theta _k-2\theta _k^*}}{\lambda _{k}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _N^{*2}}&{}1\\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{k+1}e^{-2\theta _k^*}}{\lambda _{k+1}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{N}^{*2}}&{}0\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots \\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{N}e^{-2\theta _k^*}}{\lambda _{N}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{N}^{*2}}&{}0\\ 0&{}\cdots &{}0&{}e^{-2\theta _k^*}&{}1&{}\cdots &{}1&{}0\\ \end{array}\right| \\&\times \exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\ =&(-1)^{N-k+1}\left| \begin{array}{ccccc} \frac{-2\lambda _1^*}{\lambda _1^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^*}{\lambda _1^2-\lambda _{k-1}^{*2}}&{}1\\ \vdots &{} &{}\vdots &{}\vdots \\ \frac{-2\lambda _1^*}{\lambda _k^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^*}{\lambda _k^2-\lambda _{k-1}^{*2}}&{}1\\ \end{array}\right| \left| \begin{array}{ccccc} \frac{-2\lambda _{k+1}^*e^{-2\theta _k^*}}{\lambda _{k+1}^2-\lambda _{k}^{*2}} &{}\frac{-2\lambda _{k+1}^*}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}} &{}\cdots &{}\frac{-2\lambda _{k+1}^*}{\lambda _{k+1}^2-\lambda _{N}^{*2}} \\ \vdots &{}\vdots &{}&{}\vdots \\ \frac{-2\lambda _{N}^*e^{-2\theta _k^*}}{\lambda _{N}^2-\lambda _{k}^{*2}} &{}\frac{-2\lambda _{N}^*}{\lambda _{N}^2-\lambda _{k+1}^{*2}} &{}\cdots &{}\frac{-2\lambda _{N}^*}{\lambda _{N}^2-\lambda _{N}^{*2}}\\ e^{-2\theta _k^*}&{}1&{}\cdots &{}1\\ \end{array}\right| \\&\times \exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\ =&-\prod \limits _{j=1}^{k-1}(-2\lambda _j^*)\prod \limits _{j=1}^{k-1}\frac{\lambda _k^2-\lambda _j^2}{\lambda _k^2-\lambda _j^{*2}}\prod \limits _{j=k+1}^{N}(-2\lambda _j)\prod \limits _{j=k+1}^{N}\frac{\lambda _j^{*2}-\lambda _k^{*2}}{\lambda _j^2-\lambda _k^{*2}}\\&\exp \left[ \sum _{j=1}^{k}(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\&\times \exp (-2\theta _k^*) C(\lambda _1^2,\cdots ,\lambda _{k-1}^2)C(\lambda _{k+1}^2,\cdots ,\lambda _N^2), \end{aligned}$$

where $C(\lambda _1^2,\cdots ,\lambda _{k}^2)$ denotes the determinant of Cauchy matrix $(\frac{1}{\lambda _j^2-\lambda _l^{*2}})_{k\times k}$, $j,l=1,2,\cdots ,k$. Similarly, in the vicinity of $\Omega _k$, we have

$$\begin{aligned} {{\text{ d }et}}M_2\sim&\left| \begin{array}{cccccccc} \frac{-2\lambda _1^{*}}{\lambda _1^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _1^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}}{\lambda _1^2-\lambda _k^{*2}}&{}0&{}\cdots &{}0&{}1\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots \\ \frac{-2\lambda _1^{*}}{\lambda _{k-1}^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _{k-1}^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}}{\lambda _{k-1}^2-\lambda _k^{*2}}&{}0&{}\cdots &{}0&{}1\\ \frac{-2\lambda _1^{*}}{\lambda _{k}^2-\lambda _1^{*2}} &{}\cdots &{}\frac{-2\lambda _{k-1}^{*}}{\lambda _{k}^2-\lambda _{k-1}^{*2}}&{}\frac{-2\lambda _k^{*}-2\lambda _ke^{-\theta _k-2\theta _k^*}}{\lambda _{k}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _N^{*2}}&{}1\\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{k+1}e^{-2\theta _k^*}}{\lambda _{k+1}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{N}^{*2}}&{}0\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots \\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{N}e^{-2\theta _k^*}}{\lambda _{N}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{N}^{*2}}&{}0\\ \frac{2}{\lambda _1^*}&{}\cdots &{}\frac{2}{\lambda _{k-1}^*}&{}\frac{2}{\lambda _k^*}&{}0&{}\cdots &{}0&{}1\\ \end{array}\right| \\&\times \exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\ \\ =&\left| \begin{array}{cccccccc} \frac{-2\lambda _1^{2}}{(\lambda _1^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k-1}^{2}}{(\lambda _1^2-\lambda _{k-1}^{*2})\lambda _{k-1}^*}&{}\frac{-2\lambda _k^{2}}{(\lambda _1^2-\lambda _k^{*2})\lambda _k^*}&{}0&{}\cdots &{}0\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots \\ \frac{-2\lambda _{1}^{2}}{(\lambda _{k-1}^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k-1}^{2}}{(\lambda _{k-1}^2-\lambda _{k-1}^{*2})\lambda _{k-1}^*}&{}\frac{-2\lambda _{k}^{2}}{(\lambda _{k-1}^2-\lambda _k^{*2})\lambda _k^*}&{}0&{}\cdots &{}0\\ \frac{-2\lambda _1^{2}}{(\lambda _{k}^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k-1}^{2}}{(\lambda _{k}^2-\lambda _{k-1}^{*2})\lambda _{k-1}^*}&{}\frac{-2\lambda _{k}^{2}}{(\lambda _{k}^2-\lambda _{k}^{*2})\lambda _{k}^*}+\frac{-2\lambda _ke^{-2\theta _k-2\theta _k^*}}{\lambda _{k}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _N^{*2}}\\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{k+1}e^{-2\theta _k^*}}{\lambda _{k+1}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{N}^{*2}}\\ \vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots \\ 0&{}\cdots &{}0&{}\frac{-2\lambda _{N}e^{-2\theta _k^*}}{\lambda _{N}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{N}^{*2}}\\ \end{array}\right| \\&\times \exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\ =&\left( \left| \begin{array}{cccccccc} \frac{-2\lambda _1^{2}}{(\lambda _1^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k-1}^{2}}{(\lambda _1^2-\lambda _{k-1}^{*2})\lambda _{k-1}^*}\\ \vdots &{}&{}\vdots \\ \frac{-2\lambda _{1}^{2}}{(\lambda _{k-1}^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k-1}^{2}}{(\lambda _{k-1}^2-\lambda _{k-1}^{*2})\lambda _{k-1}^*}\\ \end{array}\right| \left| \begin{array}{cccccccc} \frac{-2\lambda _ke^{-2\theta _k-2\theta _k^*}}{\lambda _{k}^2-\lambda _k^{*2}}&{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _ke^{-2\theta _k}}{\lambda _k^2-\lambda _N^{*2}}\\ \frac{-2\lambda _{k+1}e^{-2\theta _k^*}}{\lambda _{k+1}^2-\lambda _{k}^{*2}}&{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{N}^{*2}}\\ \vdots &{}\vdots &{}&{}\vdots \\ \frac{-2\lambda _{N}e^{-2\theta _k^*}}{\lambda _{N}^2-\lambda _{k}^{*2}}&{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{N}^{*2}}\\ \end{array}\right| +\right. \\&\left. \left| \begin{array}{cccccccc} \frac{-2\lambda _1^{2}}{(\lambda _1^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k}^{2}}{(\lambda _1^2-\lambda _k^{*2})\lambda _k^*}\\ \vdots &{}&{}\vdots \\ \frac{-2\lambda _1^{2}}{(\lambda _{k}^2-\lambda _1^{*2})\lambda _1^*} &{}\cdots &{}\frac{-2\lambda _{k}^{2}}{(\lambda _{k}^2-\lambda _{k}^{*2})\lambda _{k}^*}\\ \end{array}\right| \left| \begin{array}{cccccccc} \frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{k+1}}{\lambda _{k+1}^2-\lambda _{N}^{*2}}\\ \vdots &{}&{}\vdots \\ \frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{k+1}^{*2}}&{}\cdots &{}\frac{-2\lambda _{N}}{\lambda _{N}^2-\lambda _{N}^{*2}}\\ \end{array}\right| \right) \times \\&\exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] \\ =&\left( \prod \limits _{j=1}^{k-1}\frac{-2\lambda _j^2}{\lambda _j^*}C(\lambda _1^2,\cdots ,\lambda _{k-1}^2)\prod \limits _{j=k}^{N}(-2\lambda _j)C(\lambda _k^2,\cdots ,\lambda _N^2)e^{-2\theta _k-2\theta _k^*} \right. \\&\left. + \prod \limits _{j=1}^{k}\frac{-2\lambda _j^2}{\lambda _j^*}C(\lambda _1^2,\cdots , \lambda _{k}^2)\prod \limits _{j=k+1}^{N}(-2\lambda _j)C(\lambda _{k+1}^2,\cdots , \lambda _N^2)\right) \\&\times \exp \left[ \sum _{j=1}^k(\theta _j+\theta _j^*)-\sum _{j=k+1}^N(\theta _j+\theta _j^*)\right] . \end{aligned}$$

Due to the recursive property of Cauchy determinant, it gives rise to the following two identities

$$\begin{aligned}&\frac{C(\lambda _k^2,\cdots ,\lambda _N^2)}{C(\lambda _{k+1}^2,\cdots ,\lambda _N^2)}=\prod \limits _{j=k+1}^{N}\frac{\lambda _k^2-\lambda _j^2}{\lambda _k^2-\lambda _j^{*2}}\prod \limits _{j=k+1}^{N}\frac{\lambda _j^{*2}-\lambda _k^{*2}}{\lambda _j^2-\lambda _k^2}\frac{1}{\lambda _k^2-\lambda _k^{*2}}, \\&\quad \frac{C(\lambda _1^2,\cdots ,\lambda _k^2)}{C(\lambda _{1}^2,\cdots ,\lambda _{k-1}^2)}=\prod \limits _{j=1}^{k-1}\frac{\lambda _j^{*2}-\lambda _k^{*2}}{\lambda _j^2-\lambda _k^{*2}}\prod \limits _{j=1}^{k-1}\frac{\lambda _k^{2}-\lambda _j^{2}}{\lambda _k^2-\lambda _j^{*2}}\frac{1}{\lambda _k^2-\lambda _k^{*2}}. \end{aligned}$$

Based on above results, in the vicinity of $\Omega _k$, we may obtain

$$\begin{aligned} q&=4\mathrm{i}\frac{\mathrm{det}M_1}{\mathrm{det}M_2}\\&\sim \frac{2\mathrm{i}\lambda _k^*(\lambda _k^2-\lambda _k^{*2})}{\prod \limits _{j=1}^{k-1}\frac{\lambda _j^{2}}{\lambda _j^{*2}}\left( \prod \limits _{j=1}^{k-1}\frac{\lambda _k^{2}-\lambda _j^{*2}}{\lambda _k^{2}-\lambda _j^{2}}\prod \limits _{j=k+1}^{N}\frac{\lambda _k^2-\lambda _j^2}{\lambda _k^2-\lambda _j^{*2}}\lambda _k\lambda _k^*\exp (-2\theta _k)+\prod \limits _{j=1}^{k-1}\frac{\lambda _j^{*2}-\lambda _k^{*2}}{\lambda _j^2-\lambda _k^{*2}}\prod \limits _{j=k+1}^{N}\frac{\lambda _j^{2}-\lambda _k^{*2}}{\lambda _j^{*2}-\lambda _k^{*2}}\lambda _k^2\exp (2\theta _k^*)\right) }. \end{aligned}$$

Let

$$\begin{aligned} \alpha _k=\prod \limits _{j=1}^{k-1}\frac{\lambda _k^2-\lambda _j^{*2}}{\lambda _k^2-\lambda _j^{2}},\quad \beta _k=\prod \limits _{j=k+1}^{N}\frac{\lambda _k^2-\lambda _j^{2}}{\lambda _k^2-\lambda _j^{*2}}. \end{aligned}$$

(A.1)

Then, we obtain a simple form of the asymptotic behavior for q as $t\rightarrow +\infty$

$$\begin{aligned} q\sim \frac{2\mathrm{i}\lambda _k^*(\lambda _k^2-\lambda _k^{*2})\exp (2\mathrm{i}\theta _{k,\mathrm{I}}-\mathrm{i}~\mathrm{arg}\alpha _k-\mathrm{i}~\mathrm{arg}\beta _k)}{\prod \limits _{j=1}^{k-1}\frac{\lambda _j^2}{\lambda _j^{*2}}\left( \lambda _k^2\exp (2\theta _{k,\mathrm{R}}-\ln |\alpha _k\beta _k|)+\lambda _k\lambda _k^*\exp (-2\theta _{k,\mathrm{R}}+\ln |\alpha _k\beta _k|)\right) }. \end{aligned}$$

(A.2)

Similarly, setting $t\rightarrow -\infty$ and taking the similar procedure as above, we have

$$\begin{aligned} q\sim \frac{2\mathrm{i}\lambda _k^*(\lambda _k^2-\lambda _k^{*2})\exp (2\mathrm{i}\theta _{k,\mathrm{I}}+\mathrm{i}~\mathrm{arg}\alpha _k+\mathrm{i}~\mathrm{arg}\beta _k)}{\prod \limits _{j=k+1}^{N}\frac{\lambda _j^2}{\lambda _j^{*2}}\left( \lambda _k^2\exp (2\theta _{k,\mathrm{R}}+\ln |\alpha _k\beta _k|)+\lambda _k\lambda _k^*\exp (-2\theta _{k,\mathrm{R}}-\ln |\alpha _k\beta _k|)\right) }. \end{aligned}$$

(A.3)

It is remarked that from (A.1), the above discussion corresponding to the asymptotic behavior of the k-th soliton requires $1<k<N$. In order to obtain the asymptotic behaviors of the 1st and N-th solitons, it can be also similarly proved: if $k=1$, we just need to define $\alpha _1=1,\prod \limits _{j=1}^{k-1}\frac{\lambda _j^2}{\lambda _j^{*2}}=1$ , and define $\beta _N=1,\prod \limits _{j=k+1}^{N}\frac{\lambda _j^2}{\lambda _j^{*2}}=1$ for the case $k=N$. Thus, on the whole plane, we may conclude that q has the asymptotic behavior as (49). $\square$