scholarly journals Modulational instability for nonlinear Schrödinger equations with a periodic potential

2005 ◽  
Vol 2 (4) ◽  
pp. 335-355 ◽  
Author(s):  
Jared C. Bronski ◽  
Zoi Rapti
Keyword(s):  
Periodic Potential ◽  
Modulational Instability ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations

1998 ◽  
Vol 128 (6) ◽  
pp. 1249-1260 ◽  
Author(s):  
Silvia Cingolani ◽  
Margherita Nolasco
Keyword(s):  
Periodic Boundary ◽  
Periodic Potential ◽  
Periodic Boundary Conditions ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Stationary States ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Semiclassical States

For a class of nonlinear Schrodinger equations, we prove the existence of semiclassical stationary states with possibly infinitely many concentration points. As h → 0, these states concentrate near critical points of the potential. Furthermore, for periodic potential, these states can be constructed to satisfy periodic boundary conditions.

Coupled nonlinear Schrödinger equations for Langmuir and elecromagnetic waves and extension of their modulational instability domain

1981 ◽  
Vol 25 (3) ◽  
pp. 499-507 ◽  
Author(s):  
M. R. Gupta ◽  
B. K. Som ◽  
Brahmananda Dasgupta
Keyword(s):  
Electromagnetic Waves ◽  
Modulational Instability ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupling Coefficients ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Coupled Nonlinear Schrodinger Equations

A pair of coupled nonlinear Schrödinger equations for transverse and longitudinal waves has been derived. The coupling resulting from equalization of group velocities drives the Langmuir waves modulationally unstable for wavelengths shorter than (mi/me)½ λD and also extends the domain of modulational instability of electromagnetic waves when relativistic effects are taken into account. Instability is found to occur also for the perturbation wavenumber domain in which both the uncoupled Langmuir and electromagnetic waves are modulationally stable. This is shown to be caused by resonant four-wave interaction l + t → l′ + t′. The growth rate of the instability is, in general, of the order of but increases to the extent of a factor (c/vth)2 near the resonance Solitary wave solutions are given. Depending on the relative values of the self-modulation and coupling coefficients, the Langmuir or the transverse or both the waves may be localized in space.

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