Snake solitons in the cubic-quintic Ginzburgr-Landau equation
Abbreviated Journal Title
Math. Comput. Simul.
Ginzburg-Landau equation; Dissipative solitons; Nonlinear waves; Snaking; Solitons; Nonlinera PDE; TIME-PERIODIC SOLUTIONS; DISSIPATIVE SYSTEMS; WAVES; FRONTS; PULSES; SINKS; Computer Science, Interdisciplinary Applications; Computer Science, ; Software Engineering; Mathematics, Applied
Comprehensive numerical simulations of pulse solutions of the cubic-quintic Ginzburg-Landau equation (CGLE), a canonical equation governing. the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intuguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz pulsating, creeping, snake. erupting. and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades. these dissipative solitons [C.I. Christov. M G Velarde. Dissipative solitons, Physica D 86 (1995) 3231 are not stationary in time Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this paper, we address the issues of central interest in the area. i.e.. the conditions for the occurrence of the live categories of dissipative solitons. as well the dependence of both their shape and their stability oil the various parameters of the CGLE. viz the nonlinearity. dispersion. linear and nonlinear gain. loss and spectral filtering parameters Our predictions on the variation of the soliton amplitudes. widths and periods with the CGLE parameters agree with simulation results First, we develop and discuss a variational formalism within which to explore the Various Classes of dissipative solitons Given the complex dynamics of the various dissipative Solutions. this formulation is. of necessity. significantly generalized overall earlier approaches in several crucial ways. Firstly, the starting formulation for the Lagrangian is recent and not well explored. Also. the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable(1)) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler-Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons or plain pulses. We use dynamical systems theory 10 focus Oil more complex attractors. viz periodic. quasiperiodic. and chaotic ones Periodic evolution of the trial function parameters oil stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent. In particular, Pulsating and snake dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves, their regimes of Occurrence. bifurcations the parameter dependences of the amplitudes. widths. and periods agree with simulation results [1.26] (C) 2009 IMACS Published by Elsevier B.V. All rights reserved.
Mathematics and Computers in Simulation
Article; Proceedings Paper
"Snake solitons in the cubic-quintic Ginzburgr-Landau equation" (2009). Faculty Bibliography 2000s. 1858.