Hostname: page-component-75d7c8f48-9kl9f Total loading time: 0 Render date: 2026-03-14T07:35:40.557Z Has data issue: false hasContentIssue false
  • English
  • Français

On The Uniqueness of Solutions for Kawahara Type Equations

Part of: Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  10 April 2018

Marcos A. de Farias ,
Cezar I. Kondo  and
José Ruidival dos Santos Filho
Marcos A. de Farias
Affiliation:
Universidade Federal de São Carlos PO Box 676, 13565-905, São Carlos-SP, Brazil (dcik@dm.ufscar.br; marcspbo@gmail.com; santos@dm.ufscar.br)
Cezar I. Kondo*
Affiliation:
Universidade Federal de São Carlos PO Box 676, 13565-905, São Carlos-SP, Brazil (dcik@dm.ufscar.br; marcspbo@gmail.com; santos@dm.ufscar.br)
José Ruidival dos Santos Filho
Affiliation:
Universidade Federal de São Carlos PO Box 676, 13565-905, São Carlos-SP, Brazil (dcik@dm.ufscar.br; marcspbo@gmail.com; santos@dm.ufscar.br)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we extend to Kawahara type equations a uniqueness result obtained by C. E. Kenig, G. Ponce, and L. Vega for KdV type equations. We prove that, under certain decay's conditions, the null solution is the unique solution.

Keywords

Kawahara type equationsKdV type equationsuniqueness problems

MSC classification

Primary: 35A02: Uniqueness problems
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) 35Q35: PDEs in connection with fluid mechanics

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 623 - 646
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Benney, D. J., A general theory for interactions between short and long waves, Stud. Appl. Math. 56 (1977), 81–94.Google Scholar
2.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., On uniqueness properties of solutions of Schrödinger equations, Comm. PDE 31 (2006), 1811–1823.CrossRefGoogle Scholar
3.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal. 244 (2007), 504–535.CrossRefGoogle Scholar
4.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., Convexity of free solutions of Schrödinger equations with Gaussian decay, Math. Res. Lett. 15 (2008), 957–972.CrossRefGoogle Scholar
5.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., Hardy's uncertainly principle, convexity and Schrödinger equations, J. Eur. Math. Soc. 10 (2008), 882–907.Google Scholar
6.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., The sharp hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 1 (2010), 163–187.Google Scholar
7.Escauriaza, L., Kenig, C. E., Ponce, G. and Vega, L., Unique continuation for Schrodinger evolutions, with applications to profiles of concentration and traveling waves, Commun. Math. Phys. 305 (2011), 487–512.Google Scholar
8.Iguchi, T., A long wave approximation for capillary-gravity waves and the Kawahara equations, Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007), 179–220.Google Scholar
9.Jeffrey, A. and Mohamad, M. N. B., Traveling wave solutions to a higher order KdV equation, Chaos Solitons Fractals 1 (1991), 187–194.Google Scholar
10.Kakutani, T. and Ono, H., Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan 5 (1969), 1305–1318.CrossRefGoogle Scholar
11.Kato, T., On the Cauchy problem for the (generalized) Korteweg–de Vries equation, In Studies in applied mathematics, Advances in Mathematics Supplementary Studies, Volume 8, pp. 93–128 (Academic Press, 1983).Google Scholar
12.Kawahara, T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan 33 (1972), 260–264.Google Scholar
13.Kenig, C. E., Ponce, G. and Vega, L., Lower bounds for non-trivial travelling wave solutions of equations of KdV type, Nonlinearity 25 (2012), 1235–1245.CrossRefGoogle Scholar
14.Schneider, G. and Wayne, C. E., The rigorous approximation of long-wavelength capillary-gravity waves, Arch. Rational Mech. Anal. 162 (2002), 247–285.CrossRefGoogle Scholar
15.Yamamoto, Y., On gravity-surface tension waves in liquids, J. Phys. Soc. Japan 55 (1986), 1523–1527.Google Scholar