Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg–de Vries equations☆
By Jaime Angulo Pava. 2007
121 citations recorded by the source corpus.
Abstract This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrodinger equation i u t + u x x + | u | 2 u = 0 posed in R , and the modified Korteweg–de Vries equation u t + 3 u 2 u x + u x x x = 0 posed in R . Our principal goal in this paper is the study of positive periodic wave solutions of the equation ϕ ω ″ + ϕ ω 3 − ω ϕ ω = 0 , called dnoidal waves. A proof of the existence of a smooth curve of solutions with a fixed fundamental period L, ω ∈ ( 2 π 2 / L 2 , + ∞ ) → ϕ ω ∈ H per ∞ ( [ 0 , L ] ) , is given. It is also shown that these solutions are nonlinearly stable in the energy space H per 1 ( [ 0 , L ] ) and unstable by perturbations with period 2L in the case of the Schrodinger equation.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/daae6670ea00edaa74743b75de3a6381a3a4e2c5
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In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- The Stability Analysis of the Periodic Traveling Wave Solutions of the mKdV Equation (2011, background): “The orbital stability of the dn solutions was first studied in [5], where they were proven to be orbitally stable with respect to periodic perturbations of the same period.”