Related papers: Soliton resolution for the complex short pulse equ…
Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution…
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…
We revisit the asymptotic analysis of the KdV shock problem in the soliton region. Our approach is based on the analysis of the associated Riemann-Hilbert problem and we extend the domain of validity of the asymptotic formulas while at the…
In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive…
In this paper, we prove the asymptotic stability of solitary waves to 1D nonlinear Schr\"odinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by…
The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff…
We consider the Cauchy problem for the defocusing complex mKdV equation with finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be…
In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\_t -\Delta u= u^{3 },\ u(0,x)=u\_0$. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the…
We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and…
We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order…
We characterize the soliton solutions of the nonlinear Schroedinger equation on the half line with linearizable boundary conditions. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering…
We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $\epsilon \le \sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\log h|/h$ and errors of size…
We consider the Benjamin-Ono equation with a slowly varying potential $u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is…
We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable,…
Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schr\"odinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper…
In this paper we construct a weakly-nonlinear d'Alembert-type solution of the Cauchy problem for a Boussinesq-Klein-Gordon equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the…
We find for the first time the asymptotic representation of the solution to the space dependent variable order fractional diffusion and Fokker-Planck equations. We identify a new advection term that causes ultra-slow spatial aggregation of…
We investigate the multi-soliton solutions to the generalized discrete KdV equation. In some cases a soliton with smaller amplitude moves faster than that with larger amplitude unlike the soliton solutions of the KdV equation. This…
We study the semilinear Poisson equation \begin{equation} \label{pro} \Delta u = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation…
We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…