Riemann-Hilbert方法和一个新的四分量非线性薛定谔方程的$N$孤子解
Riemann-Hilbert approach and $N$-soliton solutions for a new four-component nonlinear Schrödinger equation
摘要：
本文首次提出了一个新的四分量非线性薛定谔方程，并利用Riemann-Hilbert方法对其进行了研究。首先，我们导出了四分量非线性薛定谔方程的$5\times5$矩阵谱问题的拉克斯对。然后基于拉克斯对，分析了谱问题和Jost函数的解析性质，由此成功地建立了方程的Riemann-Hilbert问题。通过求解无反射的Riemann-Hilbert问题，得到了该方程的$N$ -孤子解.最后，我们导出了方程解的两种特殊情况：$N=1$和$N=2$，并对单孤子解和双孤子解的局域结构和动力学行为进行了图解分析.
英文摘要：
A new four-component nonlinear Schrödinger equation is first proposed in this work and studied by Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a $5\times5$ matrix spectral problem for the four-component nonlinear Schrödinger equation. Then based on the Lax pair, we analyze the spectral problem and the analytical properties of the Jost functions, from which the Riemann-Hilbert problem of the equation is successfully established. Moreover, we obtain the $N$-soliton solutions of the equation by solving the Riemann-Hilbert problem without reflection. Finally, we derive two special cases of the solutions to the equation for $N=1$ and $N=2$, and the local structure and dynamic behavior of the one-and two-soliton solutions are analyzed graphically.
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