Mihai Mariş

On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation

in R2, where ; ? 0 and (− )1=2 is the operator de2ned by F((− )1=2u)( )=| |u( ). F or ˆ represent the Fourier transform. Eq. (1) describes the dynamics of three-dimensional, slightly nonlinear disturbances in boundary-layer shear ;ows (without the assumption of a di<erence in their scales along and across the ;ow), see [1,6]. The solitary waves of (1) are solutions of the form A(x; y; t) = v(x − ct; y) where c is the speed of the solitary wave. It seems that solitary waves play an important role in the evolution of (1). Such a solution must satisfy the equation

On the Symmetry of Minimizers

For a large class of variational problems we prove that minimizers are symmetric whenever they are C. AMS subject classifications. 35A15, 35B05, 35B65, 35H30, 35J20, 35J45, 35J50, 35J60.

Existence of nonstationary bubbles in higher dimensions

Abstract We are interested in the existence of travelling-waves for the nonlinear Schrodinger equation in R N with “ψ3−ψ5”-type nonlinearity. First, we prove an abstract result in critical point theory (a local variant of the classical saddle-point theorem). Using this result, we get the existence of travelling-waves moving with sufficiently small velocity in space dimension N⩾4.

NONEXISTENCE OF SUPERSONIC TRAVELING WAVES FOR NONLINEAR SCHRODINGER EQUATIONS WITH NONZERO CONDITIONS AT INFINITY

We prove that the nonexistence of supersonic finite-energy traveling waves for nonlinear Schrodinger equations with nonzero conditions at infinity is a general phenomenon which holds for a large class of equations. The same is true for sonic traveling waves in two dimensions. In higher dimensions we prove that sonic traveling waves, if they exist, must approach their limit at infinity in a very rigid way. In particular, we infer that there are no sonic traveling waves with finite energy and finite momentum.

Stationary solutions to a nonlinear Schrödinger equation with potential in one dimension

We study the one-dimensional Gross-Pitaevskii-Schrodinger equation with a potential U moving at velocity v . For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big.

Global Branches of Travelling-Waves to a Gross--Pitaevskii--Schrödinger System in One Dimension

We are interested in the existence of travelling-wave solutions to a system which modelizes the motion of an uncharged impurity in a Bose condensate. We prove that in space dimension one, there exist travelling-waves moving with velocity c if and only if c is less than the sound velocity at infinity. In this case we investigate the structure of the set of travelling-waves and we show that it contains global subcontinua in appropriate Sobolev spaces.

Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity

We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross–Pitaevskii and the so-called “cubic-quintic” equations) in space dimension

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity

{ N \geq 2}