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Multistable pulselike solutions in a parametrically driven Ginzburg-Landau equation
I V Barashenkov1, S Cross, Boris A Malomed
1Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa. igor@cenerentola.mth.uct.ac.za
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
Summary
Parametric drivers can stabilize unstable pulselike solutions in the cubic complex Ginzburg-Landau equation. This study reveals a method generating stable, multi-humped solutions, converging to a bound state of fronts.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Pulselike solutions of the cubic complex Ginzburg-Landau equation are typically unstable.
- Quintic terms are known to stabilize these solutions.
Purpose of the Study:
- To explore an alternative stabilization mechanism for pulselike solutions using a parametric driver.
- To investigate the generation and properties of stable solutions in the complex Ginzburg-Landau equation.
Main Methods:
- Numerical continuation of solutions was employed, varying the diffusion coefficient (c).
- Analysis started from the nonlinear Schrödinger limit (c=0).
Main Results:
- A sequence of coexisting stable solutions with an increasing number of humps was generated recursively.
- The solution sequence "converged" to a long pulse, interpretable as a bound state of two fronts with opposite polarities.
Conclusions:
- Parametric driving offers an alternative to quintic terms for stabilizing pulselike solutions.
- The study demonstrates the existence of complex, stable, multi-humped solutions and their convergence to a specific bound state.