Research on Soliton Solutions of Variable Coefficient Coupled Nonlinear Schrödinger Equations by Different Methods
摘要
This paper studies the soliton solutions of a class of variable-coefficient 2-coupled Schrödinger equations. Firstly, the integrability of the equation is tested by the Painlevé analysis method, and the specific constraint conditions for the equation under Painlevé integrability are derived. Secondly, the Riemann-Hilbert method and the Hirota bilinear method are respectively employed for analysis, and the N-soliton solution is derived via these two distinct approaches. For the Riemann-Hilbert method, the variable-coefficient model is transformed into a constant-coefficient form via variable replacement. The Riemann-Hilbert problem is then constructed based on the Lax pair, and the N-soliton solution expression is directly obtained by discussing the regular and non-regular cases. For the Hirota bilinear method, the bilinear equation in a functional form is directly obtained via rational transformation, and the N-soliton solution of the equation is derived through mathematical induction. Furthermore, to explore the dynamic behavior and evolution of these solutions, we present the time-dependent evolution diagrams of solitons via numerical simulation methods. On this basis, we conduct a more comprehensive analysis of the influence of several variable-coefficient functions on soliton propagation. The influence law of the function