Related papers: Asymptotic behavior for a Schr\"odinger equation w…

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work,…

In this paper, we study the focusing nonlinear Schr\"odinger equation with exponential nonlinearities \[ i \partial_t u + \Delta u = - \left(e^{4\pi |u|^2} - 1 - 4\pi \mu |u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}…

We study the leading order behaviour of positive solutions of the equation -\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete…

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{\"o}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law''…

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and…

We consider the Cauchy problem for the focusing nonlinear Schr\"odinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using…

In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption \begin{align} \begin{cases} u_t = \mathcal{L} u-u^p\quad& \mbox{in}\quad \mathbb…

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer,…

The stochastic Landau-Lifshitz-Bloch equation describes the phase spins in a ferromagnetic material and has significant role in simulating heat-assisted magnetic recording. In this paper, we consider the deviation of the solution to the 1-D…

We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schr\"{o}dinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity…

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the…

This paper is concerned with the $L^{2}$-decay estimate of solutions to nonlinear dissipative Schr\"odinger equations with power-type nonlinearity of the order $p$. It is known that the sign of the real part of the dissipation coefficient…

In this paper, we consider one--parameter ($\lambda>0$) families of Li\'enard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of $\lambda>0$. To prove our…

\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right) u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where…

In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\…

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=\Delta u^m$ in $({\mathbb R}^n\setminus\{0\})\times(0,\infty)$ in the subcritical case $0<m<\frac{n-2}{n}$, $n\ge3$. Firstly, we prove the…

We consider the long time behavior of the solutions of the coupled Schr\"odinger-KdV systems \begin{eqnarray*} \left\{ \begin{array}{llll}i\partial_tu+\partial^2_xu=\alpha uv+\beta u|u|^2,\hskip30pt (x,t)\in \mathbb{R}\times…

In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic…

We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schr\"odinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.

We study numerically the semiclassical limit for the nonlinear Schroedinger equation thanks to a modification of the Madelung transform due to E.Grenier. This approach is naturally asymptotic preserving, and allows for the presence of…