A normal form for Hamiltonian-Hopf bifurcations in nonlinear Schrödinger equations with general external potentials (Q2794983)

This paper concerns stability of solitons of a nonlinear Schrödinger equation with a potential NEWLINE\[NEWLINEiU_t+U_{xx}-V(x)U+\sigma|U|^2U=0. \eqno{(1)} NEWLINE\]NEWLINE A soliton has the form NEWLINE\[NEWLINEU(x,t)=e^{i\mu t}u(x). NEWLINE\]NEWLINE Assuming that near a Hamiltonian-Hopf bifurcation point \(\mu=\mu_0\) the solution of (1) can be expanded in the powers of a small parameter \(\epsilon\), and by matching the first a few terms in the expansion, the author derives a normal form, which is a second order nonlinear ODE: NEWLINE\[NEWLINEB_{TT}-\hat\beta B+\hat\gamma |B|^2B=0. \eqno{(2)} NEWLINE\]NEWLINE It is shown that if \(\hat\gamma\) is complex then the solution of (2) will blow up in finite time, which suggests instability of the soliton; and if \(\hat\gamma\) is real then (2) can have periodic solutions, which suggests stability of the soliton. This prediction is verified by numerical examples.