The variational principle for nonlinear waves in dissipative systems

  title={The variational principle for nonlinear waves in dissipative systems},
  author={D. J. Kaup and Boris A. Malomed},
  journal={Physica D: Nonlinear Phenomena},
Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks.
The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the
Numerical Simulations of Snake Dissipative Solitons in Complex Cubic-Quintic Ginzburg-Landau Equation
Numerical simulations of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of
Perturbed dissipative solitons: A variational approach
We adopt a variational technique to study the dynamics of perturbed dissipative solitons, whose evolution is governed by a Ginzburg--Landau equation (GLE). As a specific example of such solitons, we
Solving Soliton Perturbation Problems by Introducing Rayleigh's Dissipation Function
We solve soliton perturbation problem in nonlinear optical system by introducing Rayleigh's dissipation function in the framework of variational approach. The adopted process facilitates variational
Soliton solutions of nonlinear partial differential equations using variational approximations and inverse scattering techniques
Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their
Stability of the fixed points of the complex Swift-Hohenberg equation
We performed an investigation of the stability of fixed points in the complex Swift- Hohenberg equation using a variational formulation. The analysis is based on fixed points Euler-Lagrange equations
Analysis of non-paraxial solitons using a collective variable approach
Based on the nonlinear Helmholtz equation, we consider a model of the propagation of non-paraxial optical solitons, namely the non-paraxial nonlinear Schr?dinger equation. Using this model, we
Pulses and snakes in Ginzburg–Landau equation
Using a variational formulation for partial differential equations combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of


Dynamics of Solitons in Nearly Integrable Systems
A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that
The Inverse scattering transform fourier analysis for nonlinear problems
A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution
Internal dynamics of a vector soliton in a nonlinear optical fiber.
  • Kaup, Malomed, Tasgal
  • Physics, Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1993
A system of ordinary differential equations for the evolution of the ansatz parameters is derived and a continuous family of stationary solutions to these equations is found which can be interpreted as vector solitons with an arbitrary polarization.
Variational approach to nonlinear pulse propagation in optical fibers
The problem of nonlinear pulse propagation in optical fibers, as governed by the nonlinear Schr\"odinger equation, is reformulated as a variational problem. By means of Gaussian trial functions and a
A Perturbation Expansion for the Zakharov–Shabat Inverse Scattering Transform
For any nonlinear evolution equation which is reasonably close to a nonlinear evolution equation that can be exactly solved by the Zakharov–Shabat inverse scattering transform, the total time
Polarization dynamics and interactions of solitons in a birefringent optical fiber.
  • Malomed
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1991
Dynamics of vector solitons are studied within the framework of the general model of coupled nonlinear Schr\"odinger equations, and it is demonstrated that a stable bound state is possible.
Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system
An analytical investigation is made of the asymptotic propagation properties of pulses evolving from nonsoliton initial conditions in an optical-fiber communication system. Explicit analytical