Stability and Bifurcation of Spatially Coherent Solutions of the Damped-Driven NLS Equation

An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitati...

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Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 50; no. 3; pp. 791 - 818
Main Authors Terrones, Guillermo, McLaughlin, David W., Overman, Edward A., Pearlstein, Arne J.
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.06.1990
Subjects
Applied mathematics
Approximation
Boundary conditions
Chaos theory
Eigenfunctions
Eigenvalues
Exact sciences and technology
Function theory, analysis
Mathematical manifolds
Mathematical methods in physics
Metastable atoms
Overman
Physics
Solitons
Waveforms
Damping
Non linear equation
Stability
Analytical method
Chaotic attractor
Bifurcation
Schroedinger equation
Hopf bifurcation
Numerical calculation
Periodic solution
Sine Gordon equation
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Summary:An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about x = 0 or L/2. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude Γ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a Γ-window of finite width before undergoing a third Hopf bifurcation to quasi-periodicity. The analytical results compare favorably with numerical solutions and provide the requisite ingredients for construction of chaotic attractors for this system.
ISSN:0036-1399
1095-712X
DOI:10.1137/0150046