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

Wen-yu zhou, Shou-Fu Tian, Xiao-fan Zhang

Published 2023-07-19Version 1

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.