Abstract
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 are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.
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Abdelouhab, L., Bona, J.L., Felland, M., and Saut, J.-C. (1989). Non-local models for nonlinear, dispersive waves. Phys. D, 40, 360–392.
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.
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.
Amann, H. (1976). Fixed-point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev., 18, 620–709.
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.
Amick, C.J., and Toland, J.F. (1981a). On solitary water of finite amplitude. Arch. Rational Mech. Anal., 76, 9–95.
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.
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.
Amick, C.J., and Turner, R.E.L. (1989). Small internal waves in two-fluid systems. Arch. Rational Mech. Anal., 108, 111–139.
Bardos, C. (1971). A regularity theorem for parabolic equations. J. Funct. Anal., 7, 311–322.
Beale, J.T. (1977). The existence of solitary water waves. Comm. Pure Appl. Math., 30, 373–389.
Benjamin, T.B. (1966). Internal waves of finite amplitude and permanent form. J. Fluid Mech., 25, 241–270.
Benjamin, T.B. (1972). The stability of solitary waves. Proc. Roy. Soc. London Ser. A, 328, 153–183.
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.
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.
Bona, J.L. (1972). Fixed-Point Theorems for Fréchet Spaces. Lecture Notes in Mathematics, Vol. 280. Springer-Verlag, Berlin, pp. 219–222.
Bona, J.L. (1975). On the stability of solitary waves. Proc. Roy. Soc. London Ser. A, 344, 363–374.
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.
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.
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.
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.
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.
Boyd, J.P. (1991). Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg-de Vries equation. Phys. D, 48, 129–146.
Courant, R., and Hilbert, D. (1953). Methods of Mathematical Physics, Vol. 1. Interscience, New York.
Dugundji, J. (1951). An extension of Tietze's theorem. Pacific J. Math., 1, 353–367.
Friedrichs, K.O., and Hyers, D.H. (1954). The existence of solitary waves. Comm. Pure Appl. Math., 7, 517–550.
Granas, A. (1972). The Leray-Shauder index and the fixed-point theory for arbitrary ANRs. Bull. Math. Soc. France, 100, 209–228.
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.
Hunter, J., and Sheurle, J. (1988). Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D, 32, 253–268.
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.
Kawahara, T. (1972). Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan, 33, 260–264.
Keady, G., and Norbury, J. (1978). On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc., 83, 137–157.
Krasnoselskii, M.A. (1964). Positive Solutions of Operator Equations. Nordhoff, Amsterdam.
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.
Lavrentiev, M.A. (1947). On the theory of long waves. Dokl. Akad. Nauk Ukrain. SSR, 8, 13–69 (in Ukranian).
Leray, J., and Shauder, J. (1934). Topologie et equations fonctionelles. Ann. Sci. École Norm. Sup., 51, 45–78.
Lions, J.-L. (1969). Quelques méthodes de résolution des problemès aux limites non-linéaries. Dunod, Paris.
Lions, J.-L., and Magenes, E. (1968). Problemès aux limites non homogènes et applications, Vol. 1. Dunod, Paris.
Marchenko, A.V. (1988). Long waves in a shallow liquid beneath an ice sheet. Prikl. Mat. Mekh., 52, 230–235 (in Russian).
Nagumo, M. (1951). Degree of mapping in convex linear topological spaces. Amer. J. Math., 73 497–511.
Pomeau, Y., Ramani, A., and Grammaticos, B. (1988). Structural stability of the KdV solitons under a singular perturbation. Phys. D, 31, 127–134.
Rozhdestvensky, B.L., and Yanenko, N.N. (1983). Systems of Quasi-Linear Equations. American Mathematical Society Monograph 55. American Mathematical Society, Providence, RI.
Saut, J.-C. (1979). Sur quelques généalizations de l'équation de Korteweg-de Vries. J. Math. Pures Appl., 58, 21–61.
Saut, J.-C., and Temam, R. (1976). Remarks on the Korteweg-de Vries equation. Israel J. Math., 24, 78–87.
Ter-Krikorov, A.M. (1960). The existence of periodic waves which degenerate into a solitary wave. J. Appl. Math. Mech., 24, 930–939.
Ter-Krikorov, A.M. (1961). Surface solitary wave in fluid with vortices. Vych. Mat. Mat. Fiz., 1, 1077–1088 (in Russian).
Ter-Krikorov, A.M. (1963). Théorie exacte des ondes longues stationaries dans un liquide hétérogène. J. Mèc., 2, 351–376.
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).
Wenstein, M.I. (1986). Liapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math., 39, 51–68.
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.
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.
Zufiria, J.A. (1987). Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth. J. Fluid. Mech., 184, 183–206.
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Il'ichev, A.T., Semenov, A.Y. Stability of solitary waves in dispersive media described by a fifth-order evolution equation. Theoret. Comput. Fluid Dynamics 3, 307–326 (1992). https://doi.org/10.1007/BF00417931
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DOI: https://doi.org/10.1007/BF00417931