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Asymptotics for the modified witham equation

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  • 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.


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    [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.
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    [10] G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974.
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