scholarly journals Variational approach to the modulational instability

2004 ◽  
Vol 69 (1) ◽  
Author(s):  
Z. Rapti ◽  
P. G. Kevrekidis ◽  
A. Smerzi ◽  
A. R. Bishop
Keyword(s):  
Variational Approach ◽  
Modulational Instability

Variational approach to Langmuir waves described by the Zakharov equations

1991 ◽  
Vol 45 (2) ◽  
pp. 203-212 ◽  
Author(s):  
J. C. Bhakta
Keyword(s):  
Dispersion Relation ◽  
Variational Approach ◽  
Modulational Instability ◽  
Optimization Method ◽  
Pulse Velocity ◽  
Nonlinear Dispersion ◽  
Initial Pulse ◽  
Langmuir Waves ◽  
Nonlinear Dispersion Relation ◽  
Stokes Waves

Nonlinear modulational instability and the evolution of a pulse that is initially non-solitonic for Langmuir waves described by the Zakharov equations are considered. The average Lagrangian method has been used to derive the nonlinear dispersion relation for Stokes waves. It is found that the region of instability increases with the amplitude of the perturbed Langmuir field. The propagation of the pulse is studied with the help of the Rayleigh-Ritz optimization method. It is noted that the width of the pulse oscillates rapidly for high initial pulse velocity.

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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