Abstract
We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.
Highlights
This paper is concerned with the study of solutions to a nonlinear Schrodinger (NLS) type equation which, in a specific non-relativistic limit proper to nuclear physics, describes the behavior of a particle inside the atomic nucleus
In [7], Lewin and Rota Nodari proved the uniqueness, modulo translations and multiplication by a phase factor, and the non-degeneracy of the positive solution to (1.6). The proof of this result is based on the remark that equation (1.6) can be written in terms of u = arcsin(φ) as simpler nonlinear Schrodinger equation
We outline the numerical approach for the time evolution of initial data according to (1.1). This code is applied to perturbations of the ground states for various values of the nonlinearity parameter α and for initial data from the Schwartz class of rapidly decreasing functions
Summary
This paper is concerned with the study of solutions to a nonlinear Schrodinger (NLS) type equation which, in a specific non-relativistic limit proper to nuclear physics, describes the behavior of a particle inside the atomic nucleus. Note that solutions to (1.6) do not have a simple scaling property in the parameter b as ground states for the standard NLS equation This makes it necessary to study several values of b in this context. In [7], Lewin and Rota Nodari proved the uniqueness, modulo translations and multiplication by a phase factor, and the non-degeneracy of the positive solution to (1.6) The proof of this result is based on the remark that equation (1.6) can be written in terms of u = arcsin(φ) as simpler nonlinear Schrodinger equation. This code is applied to perturbations of the ground states for various values of the nonlinearity parameter α and for initial data from the Schwartz class of rapidly decreasing functions. A formal derivation of the equation (1.1) is presented in Appendix A
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