Solvable nonlinear evolution PDEs in multidimensional space.
Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods
Some relatively new techniques for nonlinear problems.
Spatial patterns for the three species Gross-Pitaevskii system in the plane.
Spectral invariants for coupled spin-oscillators
This text deals with inverse spectral theory in a semiclassical setting. Given a quantum system, the haunting question is “What interesting quantities can be discovered on the spectrum that can help to characterize the system ?” The general framework will be semiclassical analysis, and the issue is to recover the classical dynamics from the quantum spectrum. The coupling of a spin and an oscillator is a fundamental example in physics where some nontrivial explicit calculations can be done.
Spectrum of the linearized operator for the Ginzburg-Landau equation.
Spikes in two coupled nonlinear Schrödinger equations
Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation
We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.
Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction
Stability of solitary waves for derivative nonlinear Schrödinger equation
Stability of solitary waves for nonlinear Schrödinger equation
Stability of standing waves for nonlinear Schrödinger equations with potentials
Stability of the NLS equation with viscosity effect.
Stabilization of Schrödinger equation in exterior domains
We prove uniform local energy estimates of solutions to the damped Schrödinger equation in exterior domains under the hypothesis of the Exterior Geometric Control. These estimates are derived from the resolvent properties.
Standing wave solutions of Schrödinger systems with discontinuous nonlinearity in anisotropic media.
Standing waves for nonlinear Schrödinger equations with singular potentials
Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions
Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.
Stochastic nonlinear Schrödinger equations driven by a fractional noise, well-posedness, large deviations and support.
Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains
We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
Structured matrix numerical solution of the nonlinear Schrödinger equation by the inverse scattering transform.