Antidark solitons and soliton molecules in a (3 + 1)-dimensional nonlinear evolution equation

Abstract

We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg–de Vries equation. On the basis of the decomposition approach, the N-antidark soliton solution on a finite background is constructed by using the Darboux transformation together with the limit technique. The asymptotic analysis for the N-antidark soliton solution is performed, and the collision between multiple antidark solitons is proved to be elastic. Under the velocity resonant mechanism, the antidark soliton molecules on the (x, t), (y, t), (y, z) and (t, z) planes are found instead of the (x, y) and (x, z) planes. Based on the three- and the four-antidark soliton solutions, the elastic collision between a soliton molecule and a common soliton and the elastic collision between two soliton molecules are analytically demonstrated, respectively. These results may be useful for the study of soliton molecules in hydrodynamics and nonlinear optics.

Appendix: The derivation of Eq. (22)

Proof

Considering the following $N\times N$ determinant with respect to $\zeta$

$$\begin{aligned} g(\zeta )=\det \left( \dfrac{\beta _{j}}{\chi _l-\chi _{j}^{*}}\delta _{jl}\zeta +\dfrac{1}{\chi _l-\chi _j^{*}} \mathrm{e}^{\mathrm{i}(\theta _l-\theta _{j}^{*})} \right) _{1\le j,l\le N}, \end{aligned}$$

we have the expansion

$$\begin{aligned} g(\zeta )=\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) (\zeta ^N+a_1\zeta ^{N-1}+\cdots +a_N). \end{aligned}$$

Then, one can compute that

$$\begin{aligned} a_N= & {} g(0)/\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) \\= & {} \det \left( \dfrac{1}{\chi _l-\chi _j^{*}} \mathrm{e}^{\mathrm{i}(\theta _l-\theta _{j}^{*})} \right) /\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) . \end{aligned}$$

In terms of the determinant of the following Cauchy matrix

$$\begin{aligned} \det \left( \dfrac{1}{\chi _l-\chi _j^{*}}\right) =\prod _{i=1}^{N}\dfrac{1}{\chi _i-\chi _i^{*}}\prod _{1\le j<l}^{N}\dfrac{|\chi _j-\chi _l|^2}{|\chi _j-\chi _l^{*}|^2}, \end{aligned}$$

one obtains

$$\begin{aligned} a_N= & {} \mathrm{e}^{\mathrm{i}(\theta _1-\theta _1^{*})+\mathrm{i}(\theta _2-\theta _2^{*})+\cdots +\mathrm{i}(\theta _N-\theta _N^{*})}\dfrac{1}{\beta _1\beta _2\cdots \beta _N}\\&\prod _{1\le j<l}^{N} \dfrac{|\chi _j-\chi _l|^2}{|\chi _j-\chi _l^{*}|^2} \\= & {} \mathrm{e}^{-2(\sum _{j=1}^{N}K_j+\sum _{1\le j<l}^{N}\mu _j\mu _lA_{jl}) }. \end{aligned}$$

Furthermore, we can calculate that

$$\begin{aligned} \begin{array}{l} a_{N-1}= g'(0)/\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) \\ =\displaystyle \mathrm{e}^{\mathrm{i}(\theta _2-\theta _2^{*})+\mathrm{i}(\theta _3-\theta _3^{*})+\cdots +\mathrm{i}(\theta _N-\theta _N^{*})}\dfrac{1}{\beta _2\beta _3\cdots \beta _N}\\ \prod _{2\le j<l}^{N}\dfrac{|\chi _j-\chi _l|^2}{|\chi _j-\chi _l^{*}|^2}\\ \displaystyle \qquad \qquad +\mathrm{e}^{\mathrm{i}(\theta _1-\theta _1^{*})+\mathrm{i}(\theta _3-\theta _3^{*})+\cdots +\mathrm{i}(\theta _N-\theta _N^{*})}\dfrac{1}{\beta _1\beta _3\cdots \beta _N}\\ \qquad \prod _{ {{\begin{array}{c} 1\le j<l,\\ j\ne 2\\ \end{array}}}}^{N}\dfrac{|\chi _j-\chi _l|^2}{|\chi _j-\chi _l^{*}|^2}+\cdots \\ \qquad \qquad \displaystyle +\mathrm{e}^{\mathrm{i}(\theta _1-\theta _1^{*})+\mathrm{i}(\theta _2-\theta _2^{*})+\cdots +\mathrm{i}(\theta _{N-1}-\theta _{N-1}^{*})}\\ \dfrac{1}{\beta _1\beta _2\cdots \beta _{N-1}}\prod _{1\le j<l}^{N-1}\dfrac{|\chi _j-\chi _l|^2}{|\chi _j-\chi _l^{*}|^2}. \end{array} \end{aligned}$$

Continuing the above process by following

$$\begin{aligned} a_{N-2}= & {} \dfrac{1}{2!}g''(0)/\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) ,\cdots ,\\ a_{1}= & {} \dfrac{1}{(N-1)!}g^{N-1}(0)/\left( \prod _{i=1}^{N}\dfrac{\beta _i}{\chi _i-\chi _i^{*}}\right) , \end{aligned}$$

one can infer that

$$\begin{aligned}&\widehat{M}=g(1)=\det \left( \dfrac{\beta _{j}}{\chi _l-\chi _{j}^{*}}\delta _{jl}+\dfrac{1}{\chi _l-\chi _j^{*}} \mathrm{e}^{\mathrm{i}(\theta _l-\theta _{j}^{*})} \right) \\&\quad = \sum _{\mu =0,1}\mathrm{e}^{-2(\sum _{j=1}^{n}\mu _jK_j+\sum _{1\le j<l}^{n}\mu _j\mu _lA_{jl})}. \end{aligned}$$

$\square$