Abstract
This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbedp-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbedp-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.
Highlights
The main concerns of the paper are the travelling waves for the generalized nonlinear Schrodinger NLS equation with the free initial condition in the following form see 1 for generalized NLS : ivt − div |∇v|p−2∇v ν|v|q−2v, p > 1, 1.1 v|∂Q 0, where v : v t, x1, x2, . . . , xn 1 and ν is a parameter
For the general case ν / 0 we study the existence of positive solutions and variational principles in some special cases
We study separately two cases: ν 0 and ν / 0
Summary
This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrodinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbed p-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbed p-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. A “duality principle” between the Robin and the Steklov problems is presented
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