arXiv subjects

Wei-Qi Peng

Publications and source records attributed to Wei-Qi Peng.

8 recordsLinked to original sources

arXiv2026

Rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation

This paper numerically investigates the statistical properties of rogue waves and their generation mechanisms in integrable turbulence, taking the Gerdjikov-Ivanov (GI) equation as the research object. The eigenvalue spectra of the analytical solutions and the chaotic wave field are calculated using the Fourier collocation method. Subsequently, taking a plane wave with random noise as the initial condition, the evolution of chaotic wave fields is simulated using the split-step Fourier (SSF) method. Numerical results show that the larger the initial disturbance intensity, the faster the wave field converges to a chaotic state, and the higher the peak amplitude after convergence, the higher the tail of the probability density function, and the significantly higher probability of rogue wave occurrence. Moreover, as the initial disturbance intensity increases, the turbulence type transitions from breather turbulence to soliton turbulence. In addition, the evolution of the wave-action spectrum is studied. The research has found that the wave-action spectrum of the GI equation shows an asymmetric distribution during the time evolution process, and this asymmetry persists even after the system reaches a steady state.

nlin.PS↗

arXiv2024

$PT$ Symmetric PINN for integrable nonlocal equations: Forward and inverse problems

Since the $PT$-symmetric nonlocal equations contain the physical information of the $PT$-symmetric, it is very appropriate to embed the physical information of the $PT$-symmetric into the loss function of PINN, named PTS-PINN. For general $PT$-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms, due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component, so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the $PT$-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

physics.comp-ph↗

arXiv2023

Long time and Painlevé-type asymptotics for the defocusing Hirota equation with finite density initial data

In this work, we consider the Cauchy problem for the defocusing Hirota equation with a nonzero background \begin{align} \begin{cases} iq_{t}+α\left[q_{xx}-2\left(\left\vert q\right\vert^{2}-1\right)q\right]+iβ\left(q_{xxx}-6\left\vert q\right\vert^{2}q_{x}\right)=0,\quad (x,t)\in \mathbb{R}\times(0,+\infty),\\ q(x,0)=q_{0}(x),\qquad \underset{x\rightarrow\pm\infty 1}{\lim} q_{0}(x)=\pm 1, \qquad q_{0}\mp 1\in H^{4,4}(\mathbb{R}). \end{cases} \nonumber \end{align} According to the Riemann-Hilbert problem representation of the Cauchy problem and the $\bar{\partial}$ generalization of the nonlinear steepest descent method, we find different long time asymptotics types for the defocusing Hirota equation in oscillating region and transition region, respectively. For the oscillating region $ξ<-8$, four phase points appear on the jump contour $\mathbb{R}$, which arrives at an asymptotic expansion,given by \begin{align} q(x,t)=-1+t^{-1/2}h+O(t^{-3/4}).\nonumber \end{align} It consists of three terms. The first term $-1$ is leading term representing a nonzero background, the second term $t^{-1/2}h$ originates from the continuous spectrum and the third term $O(t^{-3/4})$ is the error term due to pure $\bar{\partial}$-RH problem. For the transition region $\vertξ+8\vert t^{2/3}<C$, three phase points raise on the jump contour $\mathbb{R}$. Painlevé asymptotics expansion is obtained \begin{align} q(x,t)=-1-(\frac{15}{4}t)^{-1/3}\varrho+O(t^{-1/2}),\nonumber \end{align} in which the leading term is a solution to the Painlevé II equation, the last term is a residual error being from pure $\bar{\partial}$-RH problem and parabolic cylinder model.

nlin.SI↗

arXiv2023

Long-time asymptotics for the Elastic Beam equation in the solitonless region via $\bar{\partial}$ methods

In this work, we study the Cauchy problem of the Elastic Beam equation with initial value in weighted Sobolev space $H^{1,1}(\mathbb{R})$ via the $\bar{\partial}$-steepset descent method. Begin with the Lax pair of the Elastic Beam equation, we successfully derive the basic Riemann-Hilbert problem, which can be used to represent the solutions of the Elastic Beam equation. Then, considering the solitonless region and using the $\bar{\partial}$-steepset descent method, we analyse the long-time asymptotic behaviors of the solutions for the Elastic Beam equation.

math.AP↗

arXiv2023

Long-time asymptotics for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation with decaying initial value problem

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basis Riemann-Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived. For the integrable nonlocal Lakshmanan-Porsezian-Daniel equation, the asymptotic behavior is different from the local model, due to they have different symmetry for the scattering matrix. Besides, since the increase of real stationary phase points also makes the asymptotic behavior have more complex error term which has nine possibilities in our analysis.

nlin.SI↗

arXiv2022

Long-time asymptotics for the reverse space-time nonlocal Hirota equation with decaying initial value problem: Without solitons

In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the $\vartheta(λ_{i})(i=0, 1)$ would like to be imaginary, which results in the $δ_{λ_{i}}^{0}$ contains an increasing $t^{\frac{\pm Im\vartheta(λ_{i})}{2}}$, and then the asymptotic behavior for nonlocal Hirota equation becomes differently.

math.AP↗

arXiv2021

$N$-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm

We systematically investigate the nonlocal Hirota equation with nonzero boundary conditions via Riemann-Hilbert method and multi-layer physics-informed neural networks algorithm. Starting from the Lax pair of nonzero nonlocal Hirota equation, we first give out the Jost function, scattering matrix, their symmetry and asymptotic behavior. Then, the Riemann-Hilbert problem with nonzero boundary conditions are constructed and the precise formulaes of $N$-double poles solutions and $N$-simple poles solutions are written by determinants. Different from the local Hirota equation, the symmetry of scattering data for nonlocal Hirota equation is completely different, which results in disparate discrete spectral distribution. In particular, it could be more complicated and difficult to obtain the symmetry of scattering data under the circumstance of double poles. Besides, we also analyse the asymptotic state of one-double poles solution as $t\rightarrow \infty$. Whereafter, the PINN algorithm is applied to research the data-driven soliton solutions of the nonzero nonlocal Hirota equation by using the training data obtained from the Riemann-Hilbert method. Most strikingly, the integrable nonlocal equation is firstly solved via PINN algorithm. As we all know, the nonlocal equations contain the $\mathcal{PT}$ symmetry $\mathcal{P}:x\rightarrow -x,$ or $\mathcal{T}:t\rightarrow -t,$ which are different with local ones. Adding the nonlocal term into the NN, we can successfully solve the integrable nonlocal Hirota equation by PINN algorithm. The numerical results indicate the algorithm can well recover the data-driven soliton solutions of the integrable nonlocal equation. Noteworthily, the inverse problems of the integrable nonlocal equation are discussed for the first time through applying the PINN algorithm to discover the parameters of the equation in terms of its soliton solution.

nlin.SI↗

arXiv2019

Long-time asymptotics in the modified Landau-Lifshitz equation with nonzero boundary conditions

In this work, we consider the long-time asymptotics of the modified Landau-Lifshitz equation with nonzero boundary conditions (NZBCs) at infinity. The critical technique is the deformations of the corresponding matrix Riemann-Hilbert problem via the nonlinear steepest descent method, as well as we employ the $g$-function mechanism to eliminate the exponential growths of the jump matrices. The results indicate that the solution of the modified Landau-Lifshitz equation with nonzero boundary conditions admits two different asymptotic behavior corresponding to two types of regions in the $xt$-plane. They are called the plane wave region with $x<(β-4\sqrt{2}q_{0})t, x>(β+4\sqrt{2}q_{0})t$, and the modulated elliptic wave region with $(β-4\sqrt{2}q_{0})t<x< (β+4\sqrt{2}q_{0})t$, respectively.

nlin.SI↗