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Pfaffian solution for dark-dark soliton to the coupled complex modified Korteweg-de Vries equation

Chenxi Li, Xiaochuan Liu, Bao-Feng Feng

TL;DR

This work addresses constructing multi-dark soliton solutions for the $ccmKdV$ equation under nonzero boundary conditions. The authors develop a pfaffian representation of dark-dark solitons by bilinearizing the system, linking it to the discrete BKP hierarchy via a Miwa transformation, and applying a complex-conjugate reduction. They derive explicit one- and two-soliton solutions and analyze their dynamics, revealing phase shifts during collisions while preserving the dark soliton profiles. This advances pfaffian-based soliton theory in multi-component, nonzero-boundary regimes and suggests directions for semi-discrete extensions.

Abstract

In this paper, we study coupled complex modified Korteweg-de Vries (ccmKdV) equation by combining the Hirota's method and the Kadomtsev-Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under nonzero boundary condition is linked to the discrete BKP hierarchy through Miwa transformation. Based on this finding, we construct the dark-dark soliton solution in the pfaffian form. The dynamical behaviors for one- and two-soliton are analyzed and illustrated.

Pfaffian solution for dark-dark soliton to the coupled complex modified Korteweg-de Vries equation

TL;DR

This work addresses constructing multi-dark soliton solutions for the equation under nonzero boundary conditions. The authors develop a pfaffian representation of dark-dark solitons by bilinearizing the system, linking it to the discrete BKP hierarchy via a Miwa transformation, and applying a complex-conjugate reduction. They derive explicit one- and two-soliton solutions and analyze their dynamics, revealing phase shifts during collisions while preserving the dark soliton profiles. This advances pfaffian-based soliton theory in multi-component, nonzero-boundary regimes and suggests directions for semi-discrete extensions.

Abstract

In this paper, we study coupled complex modified Korteweg-de Vries (ccmKdV) equation by combining the Hirota's method and the Kadomtsev-Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under nonzero boundary condition is linked to the discrete BKP hierarchy through Miwa transformation. Based on this finding, we construct the dark-dark soliton solution in the pfaffian form. The dynamical behaviors for one- and two-soliton are analyzed and illustrated.

Paper Structure

This paper contains 6 sections, 4 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bilinearization and dark soliton solutions under nonzero boundary condition
  3. Derivation of the dark soliton solution
  4. Dynamics of one- and two-soliton solutions
  5. Conclusion
  6. Appendix: Proof of the lemma 3.2

Key Result

Theorem 2.1

The dark-dark soliton solution of the ccmKdV equation (ccmkdv1)--(ccmkdv2) satisfies where The pfaffian $\tau_{k_1,k_2}$ given by with the parameters $p_i$ and $\xi_{i 0}$ are complex constants and satisfying the reduction condition where $p_{2N+1-i}=p^*_i,$ and $\xi_{2N+1-i} = \xi^*_{i} + \mathrm{i} \pi/2$ for $1 \leq i \leq N.$

Figures (2)

  • Figure 1: One dark soliton solutions to the ccmKdV equation (\ref{['ccmkdv1']})--(\ref{['ccmkdv2']}) with parameters $p_1$ = 0.88 + i, $\rho1 = 1,\,\alpha1 = 2$, $\rho2 = 1,\, \alpha2 = 1.$ (a) and (b) are profiles of $|u_1|$ and $|u_2|$, respectively.
  • Figure 2: Two dark soliton solutions to the ccmKdV equation (\ref{['ccmkdv1']})--(\ref{['ccmkdv2']}) with parameters $p_1$ = 1.53 + i, $p_2$ = 1.49 + 2i, $\rho_1 = 2,\alpha_1 = 2.3$, $\rho_2 = 1, \alpha_2 = 1.5$, (a) the profile of $|u_1|$, (b) the profile of $|u_2|$, (c) and (d) are counter plots of $|u_1|$ and $|u_2|$ respectively.

Theorems & Definitions (7)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof