Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation

Guillermo Terrones, David W. McLaughlin, Edward A. Overman, Arne J. Pearlstein

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)791-818
Number of pages28
JournalSIAM Journal on Applied Mathematics
Volume50
Issue number3
StatePublished - Jun 1990

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NLS Equation
Hopf bifurcation
Damped
Hopf Bifurcation
Quasiperiodicity
Bifurcation
Driver
Excitation
sine-Gordon equation
Time-periodic Solutions
Linear stability analysis
Regain
Sine-Gordon Equation
Dependent
Linear Stability Analysis
Chaotic Attractor
Spatial Pattern
Spatial Structure
Bifurcation Analysis
Periodic Solution

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation. / Terrones, Guillermo; McLaughlin, David W.; Overman, Edward A.; Pearlstein, Arne J.

In: SIAM Journal on Applied Mathematics, Vol. 50, No. 3, 06.1990, p. 791-818.

Research output: Contribution to journalArticle

Terrones, Guillermo ; McLaughlin, David W. ; Overman, Edward A. ; Pearlstein, Arne J. / Stability and bifurcation of spatially coherent solutions of the damped-driven NLS equation. In: SIAM Journal on Applied Mathematics. 1990 ; Vol. 50, No. 3. pp. 791-818.
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