Articles you may be interested inGround states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator

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In this paper we present a unified picture concerning general splitting methods for solving a large class of semilinear problems: nonlinear Schrödinger, Schrödinger-Poisson, Gross-Pitaevskii equations, etc. This picture includes as particular instances known schemes such as Lie-Trotter, Strang, and Ruth-Yoshida. The convergence result is presented in suitable Hilbert spaces related to the time regularity of the solution and is based on Lipschitz estimates for the nonlinearity. In addition, with extra requirements both on the regularity of the initial datum and on the nonlinearity, we show the linear convergence of these methods. We finally mention that in some special cases in which the linear convergence result is known, the assumptions we made are less restrictive.

We investigate one particular prototypical system of evolution equations in 1+1 dimensions, which models the (nonlocal) interaction of two waves in quadratic nonlinear media. These equations are integrable as they possess a Lax pair. We consider the spectral method of solving the associated initial value problem and, as recently pointed out, we show that generically the associated initial value problem cannot be linearized. We further elaborate on the integrability issue for this model by displaying the corresponding conservation laws and the motion of a particular soliton.

We study a 2D system that couples a Schrödinger evolution equation to a nonlinear elliptic equation and models the propagation of a laser beam in a nematic liquid crystal. The nonlinear elliptic equation describes the response of the director angle to the laser beam electric field. We obtain results on wellposedness and solitary wave solutions of this system, generalizing results for a well-studied simpler system with a linear elliptic equation for the director field. The analysis of the nonlinear elliptic problem shows the existence of an isolated global branch of solutions with director angles that remain bounded for arbitrary electric field. The results on the director equation are also used to show local and global existence, as well as decay for initial conditions with sufficiently small L 2 −norm. For sufficiently large L 2 −norm we show the existence of energy minimizing optical solitons with radial, positive and monotone profiles.