Numerical Study of Blowup in the Davey-Stewartson System

@article{Klein2011NumericalSO,
  title={Numerical Study of Blowup in the Davey-Stewartson System},
  author={Christian Klein and Benson K. Muite and K. Roidot},
  journal={arXiv: Analysis of PDEs},
  year={2011}
}
Nonlinear dispersive partial differential equations such as the nonlinear Schr\"odinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa exact solutions. It is shown in this way that the lump is unstable to both blowup and dispersion, and that blowup in the Ozawa solution is generic. 

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References

SHOWING 1-10 OF 36 REFERENCES

Numerical study of the Davey-Stewartson system

We deal with numerical analysis and simulations of the Davey-Stewartson equations which model, for example, the evolution of water surface waves. This time dependent PDE system is particularly

On the initial value problem for the Davey-Stewartson systems

In the theory of water waves, the 2D generalisation of the usual cubic 1D Schrodinger equation turns out to be a family of systems: the Davey-Stewartson systems. For special values of the parameters

Exact blow-up solutions to the Cauchy problem for the Davey–Stewartson systems

  • T. Ozawa
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1992
We present exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. It is shown that for any prescribed blow-up time there is an exact solution whose mass density converges to

Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations

TLDR
Numerical simulations are performed to analyze various qualitative properties of the Kadomtsev–Petviashvili type equations: blow-up versus long time behavior, stability and instability of solitary waves.

Fourth Order Time-Stepping for Kadomtsev-Petviashvili and Davey-Stewartson Equations

Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schr\"odinger equation and higher dimensional generalizations thereof can have solutions which

Numerical Computation of Solutions of the Critical Nonlinear Schrödinger Equation after the Singularity

  • P. Stinis
  • Mathematics
    Multiscale Model. Simul.
  • 2012
TLDR
A reduced model is constructed which allows us to follow the solution after the formation of the singularity of the one-dimensional critical nonlinear Schrodinger with periodic boundary conditions and initial data that give rise to a finite time singularity.

A new type of soliton behavior of the Davey-Stewartson equations in a plasma system

The Davey-Stewartson equations are derived in a plasma system by the reductive perturbation method. Modulational instability of a plane wave is discussed including a finite ion temperature effect.

Long-time decay of the solutions of the Davey—Stewartson II equations

SummaryUsing the method of inverse scattering, the sup-norms of the solutions of the Davey—Stewartson II equations are shown to decay in the order of 1/¦t¦ as ¦t¦ goes to infinity. In the focusing

A New-Type of Soliton Behavior in a Two Dimensional Plasma System

The Davey-Stewartson equations are derived in a plasma system by the reductive perturbation method. Certain particular solutions are obtained by means of a linearization technique. One of them shows

Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation

Purely dispersive equations, such as the Korteweg-de Vries and the nonlinear Schrequations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a