Global dynamics of small solutions to the modified fractional Korteweg-de Vries and fractional cubic nonlinear Schrödinger equations
Jean-Claude Saut, Yuexun Wang
Abstract
This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and fractional cubic nonlinear Schrödinger (fNLS) equations, with the dispersions |D|α∂x and |D|α+1, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a logarithmic phase correction. Our results cover the full range −1<α<1, α≠0 for both the modified fKdV and fNLS equations.
Topics & Concepts
MathematicsNonlinear systemCover (algebra)Mathematical analysisLogarithmFractional calculusInitial value problemCauchy distributionRange (aeronautics)Cauchy problemPhase (matter)Applied mathematicsDynamics (music)Type (biology)Advanced Mathematical Physics ProblemsNonlinear Waves and SolitonsFractional Differential Equations Solutions