The semiclassical limit of a weakly coupled nonlinear focusing Schrodinger system in presence of a nonconstant potential is studied. The initial data is of the form $(u_{1},u_{2})$ with $u_{i}=r_{i}\big( \frac{x-\tilde x}{\vep}\big)e^{\frac{{\rm i}}{\vep} x\cdot \tilde{\xi}}$, where $(r_{1},r_{2}) $ is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For $\vep$ sufficiently small, the solution $(\phi_{1},\phi_{2})$ will been shown to have, locally in time, the form $(r_{1}\big(\frac{x- x(t)}{\vep}\big) e^{\frac{{\rm i}}{\vep} x\cdot {\xi}(t)},r_{2}\big(\frac{x- x(t)}{\vep}\big) e^{\frac{{\rm i}}{\vep} x\cdot {\xi}(t)})$, where $(x(t),\xi(t))$ is the solution of the Hamiltonian system $\dot x(t)=\xi(t)$, $\dot \xi(t)=-\nabla V(x(t))$ with $x(0)=\tilde{x}$ and $\xi(0)=\tilde{\xi}$.
Soliton dynamics for CNLS systems with potentials
PELLACCI, Benedetta;
2010-01-01
Abstract
The semiclassical limit of a weakly coupled nonlinear focusing Schrodinger system in presence of a nonconstant potential is studied. The initial data is of the form $(u_{1},u_{2})$ with $u_{i}=r_{i}\big( \frac{x-\tilde x}{\vep}\big)e^{\frac{{\rm i}}{\vep} x\cdot \tilde{\xi}}$, where $(r_{1},r_{2}) $ is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For $\vep$ sufficiently small, the solution $(\phi_{1},\phi_{2})$ will been shown to have, locally in time, the form $(r_{1}\big(\frac{x- x(t)}{\vep}\big) e^{\frac{{\rm i}}{\vep} x\cdot {\xi}(t)},r_{2}\big(\frac{x- x(t)}{\vep}\big) e^{\frac{{\rm i}}{\vep} x\cdot {\xi}(t)})$, where $(x(t),\xi(t))$ is the solution of the Hamiltonian system $\dot x(t)=\xi(t)$, $\dot \xi(t)=-\nabla V(x(t))$ with $x(0)=\tilde{x}$ and $\xi(0)=\tilde{\xi}$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
