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12
RELATIVE PERIODIC SOLUTIONS OF THE COMPLEX GINZBURGLANDAU EQUATION
, 2005
"... A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the onedimensional complex GinzburgLandau equation (CGLE) with periodic boundary conditions. A relative periodic ..."
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A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the onedimensional complex GinzburgLandau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation’s symmetry group. With the method used, relative periodic solutions are represented by a spacetime Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories.
A Perturbative Analysis of Modulated Amplitude Waves in BoseEinstein Condensates
, 2008
"... We apply Lindstedt’s method and multiple scale perturbation theory to analyze spatiotemporal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasionedimensional BoseEinstein condensates (BECs) with meanfield interactions. We determine the dependence of the intens ..."
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We apply Lindstedt’s method and multiple scale perturbation theory to analyze spatiotemporal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasionedimensional BoseEinstein condensates (BECs) with meanfield interactions. We determine the dependence of the intensity of modulated amplitude waves (MAWs) on their wave number. We also explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.
Stationary modulatedamplitude waves in the 1D complex GinzburgLandau equation
 PHYSICA D
, 2002
"... We reformulate the onedimensional complex GinzburgLandau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulatedamplitude wave solutions. Approximate anal ..."
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We reformulate the onedimensional complex GinzburgLandau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulatedamplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.
Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations
 ADV COMPUT MATH
, 2012
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BoseEinstein condensates in superlattices
 SIAM Journal of Applied Dynamical Systems
, 2005
"... We consider the GrossPitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigarshaped BoseEinstein condensates (BECs) in such lattice potentials. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability numerical ..."
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Cited by 4 (2 self)
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We consider the GrossPitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigarshaped BoseEinstein condensates (BECs) in such lattice potentials. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability numerically. We find that the superlattice can operate as an efficient (symmetric and asymmetric) matterwave splitter. We then focus on spatially extended wavefunctions in the form of modulated amplitude waves. With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply secondorder multiple scale perturbation theory to study harmonic resonances with respect to a single lattice wavenumber as well as the two types of mixed resonances that result from interactions of both wavenumbers of the superlattice. In each case, we determine the resulting system’s equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions by direct simulations of the GP equation, identifying them as typically stable solutions of the model. PACS: 05.45.a, 03.75.Lm, 05.30.Jp, 05.45.Ac 1
Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I: General presentation and periodic solutions
, 2002
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Modulated amplitude waves with nonzero phases
 in BoseEinstein condensates, arXiv:1103.5277v1
, 2011
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Patterns of Sources and Sinks in the Complex Ginzburg–Landau Equation with Zero Linear Dispersion
, 2010
"... The complex Ginzburg–Landau equation with zero linear dispersion occurs in a wide variety of contexts as the modulation equation near the supercritical onset of a homogeneous oscillation. The analysis of its coherent structures is therefore of great interest. Its fundamental spatiotemporal pattern ..."
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The complex Ginzburg–Landau equation with zero linear dispersion occurs in a wide variety of contexts as the modulation equation near the supercritical onset of a homogeneous oscillation. The analysis of its coherent structures is therefore of great interest. Its fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves and plane waves). In the past decade interfaces separating regions with different wavetrains have been studied in detail, as they occur both in simulations and in real experiments. The basic interface types are sources and sinks, distinguished by the signs of the opposing group velocities of the adjacent wavetrains. In this paper we study existence conditions for propagating patterns composed of sources and sinks. Our analysis is based on a formal asymptotic expansion in the limit of large sourcesink separation and small speed of propagation. The main results concern the possible relative locations of sources and sinks in such a pattern. We show that sources and sinks are to leading order coupled only to their nearest neighbors, and that the separations of a source from its neighboring sinks, L+ and L − say, satisfy a phase locking condition, whose leading order form is derived explicitly. Significantly this leading order phase locking condition is independent of the propagation speed. The solutions of the condition form a discrete infinite sequence of curves in the L+–L − plane.