Related papers: Large time asymptotics for a cubic nonlinear Schr\…
This paper studies a space-inhomogeneous Boltzmann-Nordheim equation with pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem in a setting with large bounded L1 initial data. The main results are existence,…
The Cauchy problem for a quasi-linear parabolic equation with a small parameter at a higher derivative is considered. The initial step-like function contains another small parameter. Formal asymptotic solutions of the problem in small…
In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schr\"{o}dinger equations. Our aim of this research is to specify the asymptotic profile of the solution in $L^{\infty}$ as $t \to \infty$. It is…
In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these…
The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth…
We study the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions. We show global existence of the solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result…
The problem on the asymptotics for the solution of multidimensional nonlinear Boussinesq equation with respect to a small parameter $\ve$ is considered. The asymptotic expansion of the solution of this problem with respect to $\ve\to0$ for…
We present doubly-periodic solutions of the infinitely extended nonlinear Schrodinger equation with an arbitrary number of higher-order terms and corresponding free real parameters. Solutions have one additional free variable parameter that…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq…
We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions, and give an asymptotic description of…
The results from J. Stat. Phys. 159:668-712 & 163:1350-1393, on a quadratic kinetic equation in the analysis of the long time asymptotics of weak turbulence theory for the nonlinear Schr\"odinger equation, are summarized and placed in…
We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove…
We consider the $1d$ cubic nonlinear Schr\"odinger equation with an external potential $V$ that is non-generic. Without making any parity assumption on the data, but assuming that the zero energy resonance of the associated Schr\"odinger…
Massive and massless Dirac equations with Lorentz-covariant cubic nonlinearities are considered in spatial dimension $d=2,3$. Global well-posedness of the Cauchy problem for small initial data in scale-invariant Sobolev spaces and…
We discuss a one-dimensional version of the Landau-Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear…
We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…
We consider the cubic nonlinear Schr\"{o}dinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result…
We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global…
We present some new persistence results for the non-periodic two-component Camassa-Holm (2CH) system in weighted $L_p$ spaces. Working with moderate weight functions that are commonly used in time-frequency analysis, the paper generalizes…
We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.