Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

Abstract:

In this paper we study the existence and qualitative property of standing wave solutions $\psi (x,t) = e^{-\frac {iEt}{\hbar }} u(x)$ for the nonlinear Schrödinger equation $i\hbar \frac {\partial \psi }{\partial t} + \frac {\hbar ^2}{2m} \Delta \psi - W(x) \psi + |\psi |^{p-1} \psi = 0$ with $E$ being a critical frequency in the sense that $\inf \limits _{x\in \mathbb {R}^N} W(x)=E.$ We show that if the zero set of $V=W-E$ has $k$ isolated connected components $Z_i (i=1,\cdots , k)$ such that the interior of $Z_i$ is not empty and $\partial Z_i$ is smooth, $V$ has $t$ isolated zero points, $b_i$, $i=1,\cdots ,t$, and $V$ has $l$ critical points $a_i(i=1,\cdots ,l)$ such that $V(a_i)>0$, then for $\hbar > 0$ small, there exists a standing wave solution which is trapped in a neighborhood of $\bigcup _{i=1} Z_i\cup \bigl (\bigcup _{i=1}^t\{b_i\})\cup \bigl (\bigcup _{i=1}^l\{a_i\}\bigr ).$ Moreover the amplitudes of the standing wave around $\bigcup ^k_{i=1} Z_i$, $\bigcup ^t_{i=1}\{b_i\}$ and $\bigcup ^l_{i=1}\{a_i\}$ are of a different order of $\hbar$. This type of multi-scale solution has never before been obtained.