TY - JOUR

T1 - Drift of spectrally stable shifted states on star graphs

AU - Kairzhan, Adilbek

AU - Pelinovsky, Dmitry E.

AU - Goodman, Roy H.

N1 - Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated by an arbitrary distance along the edges and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable due to an irreversible drift along the incoming edge toward the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

AB - When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated by an arbitrary distance along the edges and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable due to an irreversible drift along the incoming edge toward the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

KW - Conserved quantities

KW - Modulation equations

KW - Nonlinear Schrödinger equation

KW - Orbital stability of nonlinear waves

KW - Reflectionless transmission

KW - Symmetries

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U2 - 10.1137/19M1246146

DO - 10.1137/19M1246146

M3 - Article

AN - SCOPUS:85077759517

SN - 1536-0040

VL - 18

SP - 1723

EP - 1755

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 4

ER -