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Register a new userA survey on asymptotic stability of ground states of nonlinear Schrodinger equations II
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We give short survey on the question of asymptotic stability of ground states of nonlinear Schrodinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
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We investigate the stability of ground states to a nonlinear focusing Schrodinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground states in the spirit of one due to Weinstein [{it SIAM J. Math. Anal.}, 16(1985),472-491].
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t to infty$ for a general class of initial data whose difference from the non-vanishing background possesss a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).
We consider the Schrodinger--Poisson--Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electron field is described by the $N$-particle Schrodinger equation with antisymmetric wave function. Our main results are i) the global dynamics with moving ions, and ii) the orbital stability of periodic ground state under a novel Jellium and Wiener-type conditions on the ion charge density. Under Jellium condition both ionic and electronic charge densities of the ground state are uniform.
We study the existence of ground states for the coupled Schrodinger system begin{equation} left{begin{array}{lll} displaystyle -Delta u_i+lambda_i u_i= mu_i |u_i|^{2q-2}u_i+sum_{j eq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i u_iin H^1(mathbb{R}^n), quad i=1,ldots, d, end{array}right. end{equation} $ngeq 1$, for $lambda_i,mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called symmetric attractive case) and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrodinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to CNLS with a cubic nonlinearity and give numerical evidences for the theoretical results.
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