Article Contents
Article Contents

# Asymptotics for the modified witham equation

• * Corresponding author
• We consider the modified Witham equation

${{\partial }_{t}}v+{{\partial }_{x}}\sqrt{{{a}^{2}}-\partial _{x}^{2}v}={{\partial }_{x}}\left( {{v}^{3}} \right),\ \ \left( t,x \right)\in \mathbb{R}\times \mathbb{R},$

where $\sqrt{a^{2}-\partial _{x}^{2}}$ means the dispersion relation which correspond to nonlinear Kelvin and continental-shelf waves. We develop the factorization technique to study the large time asymptotics of solutions.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.

 Citation:

•  [1] M. V. Fedoryuk, Asymptotic Methods in Analysis, in Analysis. I. Integral representations and asymptotic methods, Encyclopaedia of Mathematical Sciences, 13. Springer-Verlag, Berlin, 1989. vi+238 pp. [2] N. Hayashi and E. Kaikina, Asymptotics for the third-order nonlinear Schrödinger equation in the critical case, to appear in MMAS. [3] N. Hayashi, J. Mendez-Navarro and P. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, submitted to JDE, 2016. [4] N. Hayashi and P. Naumkin, Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377. [5] N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Analysis, 116 (2015), 112-131. [6] N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions for the modified KdV equation with a fifth order dispersive term, submitted to ARMA, 2015. [7] R. S. Smith, Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 52 (1972), 379-391. [8] E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011. ⅹⅷ+423 pp. [9] G. B. Whitham, Variational methods and applications to water waves, Hyperbolic Equations and Waves (Rencontres, Battelle Res. Inst., Seattle, WA, 1968), Springer, Berlin, 1970, pp. 153-172. [10] G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974.