Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters

2013 ◽  
Vol 88 (4) ◽  
Author(s):  
Pierre Gaillard
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
Peregrine Breather

scholarly journals Two-dimensional modulation instability of wind waves

2019 ◽  
Vol 5 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulation Instability ◽  
Wind Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger ◽  
Peregrine Breather

Abstract It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients

2015 ◽  
Vol 17 (04) ◽  
pp. 1450031 ◽  
Author(s):  
Xavier Carvajal ◽  
Mahendra Panthee ◽  
Marcia Scialom
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scaling Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Dependent ◽  
Well Posedness ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
The Third

We consider the Cauchy problem associated to the third-order nonlinear Schrödinger equation with time-dependent coefficients. Depending on the nature of the coefficients, we prove local as well as global well-posedness results for given data in L2-based Sobolev spaces. We also address the scaling limit to fast dispersion management and prove that it converges in H1to the solution of the averaged equation.

Sign in / Sign up

Export Citation Format

Share Document