Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation

Author(s): Yue Liu
Journal: Trans. Amer. Math. Soc. 353 (2001), 191-208.
MSC (2000): Primary 35Q53, 35B60, 76B25
Posted: June 8, 2000
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form \begin{equation*}( u_{t} + u_{xxx} + u^{p} u_{x} )_{x} = u_{yy} \quad (x, y) \in R^{2}, t \ge 0. \end{equation*} It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity $ p \ge 4/3 $ with $ p $ the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if $ 2 < p < 4$; that is, the solutions blow up in finite time provided they start near an unstable solitary wave.

References:

[AlPeSa]
Alexander, J.C., Pego, R. L., and Sachs, R. L., On the transverse instability of solitary eaves in the Kadomtsev-Petviashvili equation, Phys. Lett. A 226 (1997), 187-192. MR 97j:35132

[AbSe]
Ablowitz, M., and Segur, H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. MR 84a:35251

[BeBoMa]
Benjamin, T., Bona, J., and Mahony, J., Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. London A272 (1972), 47-78. MR 55:898

[BeIlNi]
Besov, O. V., Il$'$in, V. P., and Nikolski{\u{\i}}\kern.15em, S. M., Integral Representations of Functions and Imbedding Theorems, Vol. I, J. Wiley, New York, 1978. MR 80f:46030a

[Bo]
Bona, J., On the stability of solitary waves, Proc. Roy. Soc. London A344 (1975), 363-374. MR 52:7292

[BoDoKa]
Bona, J., Dougalis, V. A. and Karakashian, O. A., Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. Appl. Ser. A 12 (1986), 859-884. MR 87j:65108

[Bou]
Bourgain, J., On the Cauchy problem for the Kadomtsev-Petviashvili equations, Geom. $ \qquad \qquad $ Funct. Anal. 3(4) (1993), 315-341. MR 94d:35142

[BoLiu]
Bona, J. and Liu, Yue, Instability of solitary waves of the Kadomtsev-Petviashvili equation in three dimensions, preprint.

[BoSoSt]
Bona, J., Souganidis, P. and Strauss, W., Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A 411 (1987), 395-412. MR 88m:35128

[BoSa1]
de Bouard A. and Saut, J.C., Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 211-236. MR 98a:35115

[BoSa2]
de Bouard A. and Saut, J. C., Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal. 28 (1997), 1064-1085. MR 99c:35208

[BoSa3]
de Bouard A. and Saut, J. C., Remarks on the stability of generalized KP solitary waves, Contemp. Math., 200, Amer. Math. Soc., Providence, R.I., 1996, 75-84. MR 97a:35157

[BrLi1]
Brezis, H., and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. MR 84e:28003

[BrLi2]
Brezis, H., and Lieb, E., Minimum action solutions of some vector field equations, Comm. Math. Phys. 96 (1984), 97-113. MR 86d:35045

[FoSu]
Fokas, A. S. and Sung, L. Y., On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations, Inverse Problems 8 (1992), 673-708. MR 93h:35177

[IsMeSt]
Isaza, P., Mejía, J., and Stallbohm, V., Local solution for the Kadomtsev-Petviashvili equation in $ R^{2},$ J. Math. Anal. Appl. 196 (1995), 566-587. MR 96j:35217

[KaPe]
Kadomtsev, B. B. and Petviashvili, V. I., On the stability of solitary waves in weakly dispersive media, Soviet Phys. Dokl. 15(6) (1970), 539-541.

[KePoVe]
Kenig, C. E., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 94h:35229

[LiuWa]
Liu, Yue and Wang, Xiao-Ping, Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Comm. Math. Phys. 183 (1997), 253-266. MR 99b:35184

[Liu]
Liu, Yue, Blow up and instability of solitary waves to a generalized Kadomtsev-Petviashvili equation in three dimensions, in preparation.

[PeYa]
Petviashvili, V. and Yan'kov, V., Solitons and turbulence, in B. B. Kadomtsev (ed.) Rev. Plasma Phys. XIV, (1989), 1-62.

[Sa1]
Saut, J., Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Math. J. 42(3) (1993), 1011-1026. MR 95j:35200

[Sa2]
Saut, J., Recent results on the generalized Kadomtsev-Petviashvili equations, Acta Appl. Math. 39 (1995), 477-487. MR 96a:35178

[SoSt]
Souganidis, P. and Strauss, W., Instability of nonlinear of a class of dispersive solitary waves, Proc. Roy. Soc. Edinburgh, 114A, (1990) 195-212. MR 92a:35143

[To]
Tom, M. M., On the Generalized Kadomtsev-Petviashvili equation, Contemp. Math., Amer. Math. Soc., 1996, 193-210. MR 97e:35165

[TuFa]
Turitsyn, S. and Falkovitch, G., Stability of magneto elastic solitons and self-focusing of sound in antiferromagnet, Soviet Phys. JETP 62 (1985), 146-152.

[Uk]
Ukai, S., Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 36 (1989), 193-209. MR 90i:35270

[WaAbSe]
Wang, X-P., Ablowitz, M., and Segur, H., Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Physica D 78 (1994), 241-265. MR 95g:35182

[We]
Weinstein, M., On the solitary traveling wave of the generalized Korteweg-de Vries equation, Amer. Math. Soc., Lectures in Appl. Math. 23 (1986). MR 88a:35218

[Wi]
Wickerhauser, M. V., Inverse scattering for the heat operators and evolutions in $2 + 1$ variables, Comm. Math. Phys. 108 (1987), 67-89. MR 87m:35224

[Za]
Zakharov, V., On the stochastization of one dimensional chains of nonlinear oscillators, Sov. Phys. JETP 38 (1974), 108-110.

[Zh]
Zhou, X., Inverse scattering transform for the time dependent Schödinger equation with applications to the KPI equation, Comm. Math. Phys. 128 (1990), 551-564. MR 91j:34138

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35Q53, 35B60, 76B25

Retrieve articles in all Journals with MSC (2000): 35Q53, 35B60, 76B25

Additional Information:

Yue Liu
Affiliation: Department of Mathematics, The University of Texas at Arlington, Arlington, Texas 76019
Email: liu@math.uta.edu

DOI: 10.1090/S0002-9947-00-02465-X
PII: S 0002-9947(00)02465-X
Received by editor(s): April 6, 1998
Received by editor(s) in revised form: September 2, 1998
Posted: June 8, 2000
Copyright of article: Copyright 2000, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google