# American Institute of Mathematical Sciences

July 2018, 17(4): 1407-1448. doi: 10.3934/cpaa.2018069

## Asymptotics for the modified witham equation

 1 Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan 2 Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

* Corresponding author

Received  August 2016 Revised  May 2017 Published  April 2018

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.
Citation: Nakao Hayashi, Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the modified witham equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1407-1448. doi: 10.3934/cpaa.2018069
##### References:
 [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.

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##### References:
 [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.
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