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Stability of solitary waves in dispersive media described by a fifth- order evolution equation. (English) Zbl 0755.76022
Summary: The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well- known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability is established. The shape of solitary waves under analysis is determined by a numerical solution of the boundary- value problem followed by a correction using the Picard method of 4-12 orders of accuracy.

76B25 Solitary waves for incompressible inviscid fluids
35Q53 KdV equations (Korteweg-de Vries equations)
35B35 Stability in context of PDEs
76B45 Capillarity (surface tension) for incompressible inviscid fluids
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[1] Abdelouhab, L., Bona, J.L., Felland, M., and Saut, J.-C. (1989). Non-local models for nonlinear, dispersive waves. Phys. D, 40, 360-392. · Zbl 0699.35227
[2] Albert, J.P., and Bona, J.L. (1989). Total positivity and the stability of internal waves in stratified fluids of finite depth. Report No. AML 47, Penn State University Report Series. · Zbl 0723.76106
[3] Albert, J.P., Bona, J.L., and Henry D.B. (1987). Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Phys. D, 24, 343-366. · Zbl 0634.35079
[4] Amann, H. (1976). Fixed-point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev., 18, 620-709. · Zbl 0345.47044
[5] Amick, C.J., and Kirchgässner, K. (1989). A theory of solitary water waves in the presence of surface tension. Arch. Rational Mech. Anal., 105, 1-49. · Zbl 0666.76046
[6] Amick, C.J., and Toland, J.F. (1981a). On solitary water of finite amplitude. Arch. Rational Mech. Anal., 76, 9-95. · Zbl 0468.76025
[7] Amick, C.J., and Toland, J.F. (1981b). On periodic water waves in the long-wave limit. Philos. Trans. Roy. Soc. London Ser. A, 303, 633-673. · Zbl 0482.76029
[8] Amick, C.J., and Turner, R.E.L. (1986). A global theory of internal solitary waves in two-fluid systems. Trans. Amer. Math. Soc., 298, 431-484. · Zbl 0631.35029
[9] Amick, C.J., and Turner, R.E.L. (1989). Small internal waves in two-fluid systems. Arch. Rational Mech. Anal., 108, 111-139. · Zbl 0681.76103
[10] Bardos, C. (1971). A regularity theorem for parabolic equations. J. Funct. Anal., 7, 311-322. · Zbl 0214.12302
[11] Beale, J.T. (1977). The existence of solitary water waves. Comm. Pure Appl. Math., 30, 373-389. · Zbl 0379.35055
[12] Benjamin, T.B. (1966). Internal waves of finite amplitude and permanent form. J. Fluid Mech., 25, 241-270. · Zbl 0145.23602
[13] Benjamin, T.B. (1972). The stability of solitary waves. Proc. Roy. Soc. London Ser. A, 328, 153-183.
[14] Benjamin, T.B., Bona, J.L., and Bose, D.K. (1988). Solitary-wave solutions of nonlinear problems. Report No. AML 30, Penn State University Report Series. · Zbl 0707.35131
[15] Bennet, D.P., Brown, R.W., Stansfield, S.E., Stroughair, J.D., and Bona, J.L. (1983). The stability of internal solitary waves. Math. Proc. Cambridge Philos. Soc., 94, 351-379. · Zbl 0574.76028
[16] Bona, J.L. (1972). Fixed-Point Theorems for Fréchet Spaces. Lecture Notes in Mathematics, Vol. 280. Springer-Verlag, Berlin, pp. 219-222.
[17] Bona, J.L. (1975). On the stability of solitary waves. Proc. Roy. Soc. London Ser. A, 344, 363-374. · Zbl 0328.76016
[18] Bona, J.L., and Sachs, R.L. (1988). Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. Phys., 118, 15-29. · Zbl 0654.35018
[19] Bona, J.L., and Sachs, R.L. (1989). The existence of internal solitary waves in a two-fluid system near the KdV limit. Geophys. Astrophys. Fluid Dynamics, 47, 25-51. · Zbl 0708.76136
[20] Bona, J.L., Bose, D.K., and Benjamin, T.B. (1976). Solitary-Wave Solutions for Some Model Equations for Waves in Nonlinear Dispersive Media. Lecture Notes in Mathematics, Vol. 503. Springer-Verlag, Berlin, pp. 207-218. · Zbl 0346.76012
[21] Bona, J.L., Bose, D.K., and Turner, R.E.L. (1983). Finite amplitude steady waves in stratified fluids. J. Math. Pures Appl., 62, 389-440. · Zbl 0491.35049
[22] Bona, J.L., Souganidis, P.E., and Strauss, W.A. (1987). Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London Ser. A, 411, 395-412. · Zbl 0648.76005
[23] Boyd, J.P. (1991). Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg-de Vries equation. Phys. D, 48, 129-146. · Zbl 0728.35100
[24] Courant, R., and Hilbert, D. (1953). Methods of Mathematical Physics, Vol. 1. Interscience, New York. · Zbl 0051.28802
[25] Dugundji, J. (1951). An extension of Tietze’s theorem. Pacific J. Math., 1, 353-367. · Zbl 0043.38105
[26] Friedrichs, K.O., and Hyers, D.H. (1954). The existence of solitary waves. Comm. Pure Appl. Math., 7, 517-550. · Zbl 0057.42204
[27] Granas, A. (1972). The Leray-Shauder index and the fixed-point theory for arbitrary ANRs. Bull. Math. Soc. France, 100, 209-228. · Zbl 0236.55004
[28] Grillakis, M., Shatah, J., and Strauss, W. (1987). Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal., 74, 160-197. · Zbl 0656.35122
[29] Hunter, J., and Sheurle, J. (1988). Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D, 32, 253-268. · Zbl 0694.35204
[30] Il’ichev, A.T., and Semenov, A.Yu. (1991). Stability of subcritical solitary wave solutions to fifth-order evolution equation. Preprint No. 28, General Physics Institute, U.S.S.R. Academy of Science, Moscow.
[31] Kawahara, T. (1972). Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan, 33, 260-264.
[32] Keady, G., and Norbury, J. (1978). On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc., 83, 137-157. · Zbl 0393.76015
[33] Krasnoselskii, M.A. (1964). Positive Solutions of Operator Equations. Nordhoff, Amsterdam.
[34] Krasovskii, Yu.P. (1961). On the theory of steady state waves of large amplitude. U.S.S.R. Comput. Math. and Math. Phys., 1, 996-1018. · Zbl 0158.12003
[35] Lavrentiev, M.A. (1947). On the theory of long waves. Dokl. Akad. Nauk Ukrain. SSR, 8, 13-69 (in Ukranian).
[36] Leray, J., and Shauder, J. (1934). Topologie et equations fonctionelles. Ann. Sci. École Norm. Sup., 51, 45-78. · JFM 60.0322.02
[37] Lions, J.-L. (1969). Quelques méthodes de résolution des problemès aux limites non-linéaries. Dunod, Paris.
[38] Lions, J.-L., and Magenes, E. (1968). Problemès aux limites non homogènes et applications, Vol. 1. Dunod, Paris. · Zbl 0165.10801
[39] Marchenko, A.V. (1988). Long waves in a shallow liquid beneath an ice sheet. Prikl. Mat. Mekh., 52, 230-235 (in Russian).
[40] Nagumo, M. (1951). Degree of mapping in convex linear topological spaces. Amer. J. Math., 73 497-511. · Zbl 0043.17801
[41] Pomeau, Y., Ramani, A., and Grammaticos, B. (1988). Structural stability of the KdV solitons under a singular perturbation. Phys. D, 31, 127-134. · Zbl 0695.35161
[42] Rozhdestvensky, B.L., and Yanenko, N.N. (1983). Systems of Quasi-Linear Equations. American Mathematical Society Monograph 55. American Mathematical Society, Providence, RI.
[43] Saut, J.-C. (1979). Sur quelques généalizations de l’équation de Korteweg-de Vries. J. Math. Pures Appl., 58, 21-61. · Zbl 0449.35083
[44] Saut, J.-C., and Temam, R. (1976). Remarks on the Korteweg-de Vries equation. Israel J. Math., 24, 78-87. · Zbl 0334.35062
[45] Ter-Krikorov, A.M. (1960). The existence of periodic waves which degenerate into a solitary wave. J. Appl. Math. Mech., 24, 930-939. · Zbl 0101.20704
[46] Ter-Krikorov, A.M. (1961). Surface solitary wave in fluid with vortices. Vych. Mat. Mat. Fiz., 1, 1077-1088 (in Russian).
[47] Ter-Krikorov, A.M. (1963). Théorie exacte des ondes longues stationaries dans un liquide hétérogène. J. Mèc., 2, 351-376.
[48] Voliak, K.I., Semenov, A.Yu., and Shugan, I.V. (1989). Interaction between surface and internal waves. Gen. Phys. Inst. USSR Acad. Sci. Proc., 18, 3-32 (in Russian).
[49] Wenstein, M.I. (1986). Liapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math., 39, 51-68. · Zbl 0594.35005
[50] Wenstein, M.I. (1987). Existence and dynamic stability of solitary wave solutions of equations, arising in long wave propagation. Comm. Partial Differential Equations, 12, 1133-1173. · Zbl 0657.73040
[51] Yamamoto, Y., and Takizawa, E.I. (1981). On a solution of non-linear evolution equation of fifth order. J. Phys. Soc. Japan, 50, 1421-1422.
[52] Zufiria, J.A. (1987). Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth. J. Fluid. Mech., 184, 183-206. · Zbl 0634.76016
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