P. G. Kevrekidis and D. E. Pelinovsky
Persistence of kinks in the NLS equation with competing noninearities
Abstract:
External potentials break the translational invariance of nonlinear Schrodinger (NLS) equations and
generally convert a continuous family of standing wave solutions into
isolated pinned states. We study this mechanism for the prototypical case of a kink in the NLS
equation with competing focusing cubic and defocusing quintic nonlinearities. The Lyapunov-Schmidt
reduction method is used to prove that the location of the pinned kinks is selected
near simple roots of an effective potential. We then determine how the translational zero
eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of
the derivative of the effective potential at the simple root distinguishes
eigendirections of positive and negative energy. We further show that if
the Jacobian operator admits a negative eigenvalue, then the stability problem contains a real
unstable eigenvalue. The derivation of the asymptotic expansion for the unstable eigenvalue
is nonstandard because the zero eigenvalue is embedded in the continuous spectrum of the
linearized operator. Finally, we show that the persistence of kinks lacks parity symmetry even
if the external potential is odd. Numerical continuation, spectral computations with domains
adapted to the slowly decaying eigenfunctions, and direct dynamical simulations of the time
evolution corroborate the analytical predictions for different representative examples.
Keywords:
nonlinear Schrodinger equation; competiting nonlinearity; kink; spectral stability; persistence; external potentials.