Related papers: Non-relativistic limit for the cubic nonlinear Kle…
In the paper we consider the nonexistence of global solutions of the Cauchy problem for coupled Klein-Gordon equations of the form \begin{eqnarray*} \left\{\begin{array}{l} u_{tt}-\Delta u+m_1^2 u+K_1(x)u=a_1|v|^{q+1}|u|^{p-1}u…
We briefly review results on Colombeau type generalized solutions to the Cauchy problem for linear Schr\"odinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main…
We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for…
We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of…
We consider the initial value problem for a system of cubic nonlinear Schr\"odinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude…
The long-wavelength, weak-dispersion limit of the discrete nonlinear Schr\"odinger equation with long-range dispersion is analytically considered. This continuum approximation is carried out irrespective of the dispersion range and hence…
We study the nonrelativistic limit of the motion of a classical particle in a model of deformed special relativity and of the corresponding generalized Klein-Gordon and Dirac equations, and show that they reproduce nonrelativistic classical…
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of…
We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular…
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…
We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a…
We prove a sharp bilinear inequality for the Klein-Gordon equation on $\sr^{d+1}$, for any $d \geq 2$. This extends work of Ozawa-Rogers and Quilodr\'an for the Klein-Gordon equation and generalises work of Bez-Rogers for the wave equation.…
In this article, we consider the kinetic derivative nonlinear Schr\"odinger equation (KDNLS), which is a one-dimensional nonlinear Schr\"odinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. For the…
We consider the initial value problem for the Dirac-Klein-Gordon equations in two space dimensions. Global regularity for $C^\infty$ data was proved by Gr\"unrock and Pecher. Here we consider analytic data, proving that if the initial…
We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting…
This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state…
In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously…
The Cauchy problem is studied for systems of quasi-linear wave equations with multiple speeds in two space dimensions. Using the method of Klainerman and Sideris together with the localized energy estimate, we give an alternative proof of a…
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…
The s-wave Klein-Gordon equation for the bound states is separated in two parts to see clearly the relativistic contributions to the solution in the non-relativistic limit. The reliability of the model is discussed with the specifically…