Persistence of kinks in the NLS equation with competing nonlinearities
Abstract
External potentials break the translational invariance of nonlinear Schrödinger (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 s0 of the pinned kinks is selected near roots of an effective potential Veff(s). We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of Veff'(s0) 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.
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BibTeX
@misc{kevrekidis2026persistence,
title = {{Persistence of kinks in the NLS equation with competing nonlinearities}},
author = {P. G. Kevrekidis and D. E. Pelinovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02541},
eprint = {2610.02541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02541},
}
RIS
TY - UNPB
TI - Persistence of kinks in the NLS equation with competing nonlinearities
AU - P. G. Kevrekidis
AU - D. E. Pelinovsky
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02541
UR - https://arxiv.org/abs/2610.02541
AB - External potentials break the translational invariance of nonlinear Schrödinger (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 s_0 of the pinned kinks is selected near roots of an effective potential V_eff(s). We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of V_eff'(s_0) 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.
ER -