Related papers: Shock problem for MKdV equation: Long time Dynamic…

We consider the rigorous derivation of asymptotic formulas for initial-boundary value problems using the nonlinear steepest descent method. We give detailed derivations of the asymptotics in the similarity and self-similar sectors for the…

In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates…

This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection…

In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions \begin{align*} m_t+2\kappa q_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+\kappa,\\…

We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider rapidly decreasing initial data admitting only a finite number of moments. For the so-called…

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering…

We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion…

In 1978, A. C. Newell [SIAM J. Appl. Math. 35(4) (1978) 650-664] proposed an exactly solvable model called Newell equation, which simulates the investigation of significant interaction mechanism between long and short waves. Nearly fifty…

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…

This paper is concerned with the initial value problem for semilinear wave equation with structural damping $u_{tt}+(-\Delta)^{\sigma}u_t -\Delta u =f(u)$, where $\sigma \in (0,\frac{1}{2})$ and $f(u) \sim |u|^p$ or $u |u|^{p-1}$ with $p> 1…

We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow…

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory…

The three-wave resonant interaction (three-wave) equation not only possesses $3\times 3$ matrix spectral problem, but also being absence of stationary phase points, which give rise to difficulty on the asymptotic analysis with stationary…

We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We…

We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function''…

This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation $$ u_t(x,t)+6u(x,t)u(-x,-t)u_x(x,t)+u_{xxx}(x,t)=0, $$ with the oscillating step-like boundary conditions: $u(x,t)\to 0$ as $x\to-\infty$ and…

In these lecture notes, we address the problem of large-time asymptotic behaviour of the solutions to scalar convection-diffusion equations set in ${R}^N$. The large-time asymptotic behaviour of the solutions to many convection-diffusion…

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2…

We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the…

We consider the focusing integrable nonlocal nonlinear Schr\"odinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as…