On long-time asymptotics to the nonlocal Lakshmanan -Porsezian-Daniel equation with step-like initial data

TL;DR

This paper employs the nonlinear steepest descent method to analyze the long-time behavior of solutions to the nonlocal Lakshmanan-Porsezian-Daniel equation with step-like initial data, revealing detailed asymptotic structures.

Contribution

It develops a matrix Riemann-Hilbert problem framework and introduces the Blaschke-Potapov factor to handle singularities, enabling asymptotic analysis of the nonlocal LPD equation.

Findings

Derived long-time asymptotics for the nonlocal LPD equation

Constructed a regular Riemann-Hilbert problem using BP factor

Analyzed asymptotics for different spatial regions based on phase points

Abstract

In this work, the nonlinear steepest descent method is employed to study the long-time asymptotics of the integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation with a step-like initial data: $q_{0}(x)\rightarrow0$ as $x\rightarrow-\infty$ and $q_{0}(x)\rightarrow A$ as $x\rightarrow+\infty$, where $A$ is an arbitrary positive constant. Firstly, we develop a matrix Riemann-Hilbert (RH) problem to represent the Cauchy problem of LPD equation. To remove the influence of singularities in this RH problem, we introduce the Blaschke-Potapov (BP) factor, then the original RH problem can be transformed into a regular RH problem which can be solved by the parabolic cylinder functions. Besides, under the nonlocal condition with symmetries $x\rightarrow-x$ and $t\rightarrow t$, we give the asymptotic analyses at $x>0$ and $x<0$, respectively. Finally, we derive the long-time asymptotics of the solution $q(x,t)$ corresponding to the complex case of three stationary phase points generated by phase function.