Blow-up in the Boussinesq equation

Sergei K. Turitsyn

doi:10.1103/PhysRevE.47.R796

Rapid Communication
Access by Xinjiang University

Blow-up in the Boussinesq equation

Sergei K. Turitsyn

Institüt für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-4000 Düsseldorf 1, Germany

PDF

Phys. Rev. E 47, R796(R) – Published 1 February, 1993

DOI: https://doi.org/10.1103/PhysRevE.47.R796

Abstract

The Boussinesq equation is known to be the fundamental nonlinear model, describing wave propagation in a weakly dispersive nonlinear medium. In this paper, we prove the existence of the collapse dynamics for the two basic forms of the Boussinesq equation in the case of the periodic boundary conditions. The sufficient criterion of the blow-up is found analytically.

References (24)

On leave from Institute of Automation and Electrometry, Russian Academy of Sciences, Novosibirsk 630090, Russia.
J. Boussinesq, J. Math. Pure Appl. 17, 55 (1872).
J. Boussinesq, Mem. Pres. Div. Sav. Acad. Sci. Inst. Fr. 23, 1 (1877).
L. Iskandar and P. C. Jain, Proc. Ind. Acad. Sci. (Math. Sci.) 89, 171 (1980).
M. P. Soerensen, P. L. Christiansen and P. S. Lomdahl, J. Acoust. Soc. Am. 76, 871 (1984).
P. Rosenau, Phys. Lett. 118A, 222 (1986).
J. Bona, W. G. Pritchard and L. R. Scott, Lect. Appl. Math. 20, 229 (1982).
N. J. Zabusky, Nonlinear Partial Differential Equations (Academic, New York, 1967).
G. B. Whitham Linear and Nonlinear Waves (Wiley Interscience, New York, 1975).
M. P. Soerensen, P. L. Christiansen, P. S. Lomdahl and O. Skovgaard, J. Acoust. Soc. Am. 81, 1718 (1987).
K. E. Longren, in Solitons in Action, edited by K. Longren and A. Scott (Academic, New York, 1978), pp. 127 152.
L. Xu, D. H. Auston and A. Hasegawa, Phys. Rev. A 45, 3184 (1991).
V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, New York, 1975).
S. K. Turitsyn and G. E. Falkovich, Zh. Eksp. Teor. Fiz. 89, 258 (1985) [Sov. Phys. JETP 62, 146 (1985)].
V. E. Zakharov, Zh. Eksp. Teor. Fiz. 65, 219 (1973) [Sov. Phys. JETP 38, 108 (1974)].
G. E. Falkovich, G. Papanicolaou, C. Sulem and P. L. Salem, Phys. Lett. A 99, 271 (1983).
J. L. Bona and R. L. Sachs, Commun. Math. Phys. 118, 15 (1988).
V. Kalantarov and O. Ladyzhenskaya, J. Sov. Math. 10, 53 (1978).
R. L. Sachs, Appl. Anal. 36, 145 (1990).
H. A. Levine, Trans. Am. Math. Soc. 192, 1 (1974).
J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
D. W. McLaughlin, G. Papanico, C. Sulem and P. L. Sulem, Phys. Rev. A 34, 1200 (1986).
S. K. Turitsyn, in Singularities in Fluids, Plasmas and Optics, edited by R. Caflisch (Kluwer Academic, Dordrecht, 1992).
H. A. Levine, SIAM Rev. 32, 283 (1990).

[c1] On leave from Institute of Automation and Electrometry, Russian Academy of Sciences, Novosibirsk 630090, Russia.

[c2] J. Boussinesq, J. Math. Pure Appl. 17, 55 (1872).

[c3] J. Boussinesq, Mem. Pres. Div. Sav. Acad. Sci. Inst. Fr. 23, 1 (1877).

[c4] L. Iskandar and P. C. Jain, Proc. Ind. Acad. Sci. (Math. Sci.) 89, 171 (1980).

[c5] M. P. Soerensen, P. L. Christiansen and P. S. Lomdahl, J. Acoust. Soc. Am. 76, 871 (1984).

[c6] P. Rosenau, Phys. Lett. 118A, 222 (1986).

[c7] J. Bona, W. G. Pritchard and L. R. Scott, Lect. Appl. Math. 20, 229 (1982).

[c8] N. J. Zabusky, Nonlinear Partial Differential Equations (Academic, New York, 1967).

[c9] G. B. Whitham Linear and Nonlinear Waves (Wiley Interscience, New York, 1975).

[c10] M. P. Soerensen, P. L. Christiansen, P. S. Lomdahl and O. Skovgaard, J. Acoust. Soc. Am. 81, 1718 (1987).

[c11] K. E. Longren, in Solitons in Action, edited by K. Longren and A. Scott (Academic, New York, 1978), pp. 127 152.

[c12] L. Xu, D. H. Auston and A. Hasegawa, Phys. Rev. A 45, 3184 (1991).

[c13] V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, New York, 1975).

[c14] S. K. Turitsyn and G. E. Falkovich, Zh. Eksp. Teor. Fiz. 89, 258 (1985) [Sov. Phys. JETP 62, 146 (1985)].

[c15] V. E. Zakharov, Zh. Eksp. Teor. Fiz. 65, 219 (1973) [Sov. Phys. JETP 38, 108 (1974)].

[c16] G. E. Falkovich, G. Papanicolaou, C. Sulem and P. L. Salem, Phys. Lett. A 99, 271 (1983).

[c17] J. L. Bona and R. L. Sachs, Commun. Math. Phys. 118, 15 (1988).

[c18] V. Kalantarov and O. Ladyzhenskaya, J. Sov. Math. 10, 53 (1978).

[c19] R. L. Sachs, Appl. Anal. 36, 145 (1990).

[c20] H. A. Levine, Trans. Am. Math. Soc. 192, 1 (1974).

[c21] J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).

[c22] D. W. McLaughlin, G. Papanico, C. Sulem and P. L. Sulem, Phys. Rev. A 34, 1200 (1986).

[c23] S. K. Turitsyn, in Singularities in Fluids, Plasmas and Optics, edited by R. Caflisch (Kluwer Academic, Dordrecht, 1992).

[c24] H. A. Levine, SIAM Rev. 32, 283 (1990).

Physical Review E

Export citation

Export citation

Blow-up in the Boussinesq equation

Abstract

References (24)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation