可积非局部Lakshmanan-Porsezian-Daniel方程的逆散射变换
Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation
摘要：
本文引入了一个广义非局部Lakshmanan-Porsezian-Daniel（LPD）方程，并建立了它作为一个无穷维Hamilton动力系统的可积性.受Ablowitz和Musslimani（2016 Nonlinearity 29 915）思想的启发，我们成功地推导出了非局域LPD方程的逆散射变换（IST）。首先构造了该方程的直接散射问题，讨论了本征函数和散射数据的一些重要对称性。利用一种新的左右Riemann-Hilbert（RH）问题，分析了逆散射问题，并恢复了势函数。通过引入无反射情形的特殊条件，成功地导出了该方程的时间周期孤子解公式。以$J=\overline{J}=1,2,3$和$4$为例，我们得到了一些有趣的现象，如呼吸子型孤子、弧型孤子、三孤子和四孤子。通过图解分析，进一步考虑了参数$δ$对这些解的影响.最后，在几种特殊的初始条件下，研究了本征值和守恒量。
英文摘要：
In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. Motivated by the ideas of Ablowitz and Musslimani (2016 Nonlinearity 29 915), we successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take $J=\overline{J}=1,2,3$ and $4$ for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter $δ$ on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.
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