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Periodic traveling--wave solutions of nonlinear dispersive evolution equations

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  • For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
    Mathematics Subject Classification: Primary: 35C07, 35C08, 35Q35, 35Q51, 35Q53, 35S10, 76B15, 76B25; Secondary: 35C10, 35Q86, 86A05.


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  • [1]

    L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Non-local models for nonlinear, dispersive waves, Physica D, 40 (1989), 360-392.doi: 10.1016/0167-2789(89)90050-X.


    J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Applied Analysis (Baton Rouge, LA 1996), Contemp. Math. 221, American Math. Soc., Providence, RI (1999), 1-29.doi: 10.1090/conm/221/03116.


    J. P. Albert, J. L. Bona and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Physica D, 24 (1987), 343-366.doi: 10.1016/0167-2789(87)90084-4.


    C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane, Acta Math., 167 (1991), 107-126.doi: 10.1007/BF02392447.


    J. Angulo Pava, "Nonlinear Dispersive Equations," Mathematical Surveys and Monographs, 156, American Math. Soc.: Providence, RI 2009.


    J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves, Advances in Differenticial Equations, 11 (2006), 1321-1374.


    T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.doi: 10.1017/S002211206700103X.


    T. B. Benjamin, The stability of solitary waves, Proc. Royal Soc. London Ser. A, 328 (1972), 153-183.doi: 10.1098/rspa.1972.0074.


    T. B. Benjamin, Lectures on nonlinear wave motion, Lect. Appl. Math., 15, (ed. A. Newell) American Math. Soc., Providence, RI (1974), 3-47.


    T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. London, Ser. A, 272 (1972), 47-78.doi: 10.1098/rsta.1972.0032.


    D. P. Bennett, R. W. Brown, S. E. Stansfield, J. D. Stroughair and J. L. Bona, The stability of internal solitary waves in stratified fluids, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351-379.doi: 10.1017/S0305004100061193.


    J. L. Bona, On the stability theory of solitary waves, Proc. Royal Soc. London, Ser. A, 344 (1975), 367-374.doi: 10.1098/rspa.1975.0106.


    J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res., 37 (1981), 21-30.doi: 10.1007/BF00382614.


    J. L. Bona, On solitary waves and their role in the evolution of long waves, in "Applications of Nonlinear Analysis in the Physical Sciences" (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman: London (1989), 183-205.


    J. L. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Cont. Dynamical Sys., 6 (2000), 1-20.doi: 10.3934/dcds.2000.6.1.


    J. L. Bona, Y. Liu and N. Nguyen, Stability of solitary waves in higher-order Sobolev spaces, Commun. Math. Sci., 2 (2004), 35-52.


    J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of KdV-type, Proc. Royal Soc. London, Ser. A, 411 (1987), 395-412.doi: 10.1098/rspa.1987.0073.


    J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves, J. Nonlinear Sci., 4 (1994), 449-470.doi: 10.1007/BF02430641.


    J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris, 72 (1871), 755-759.


    J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, 73 (1871), 256-260.


    H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity, 17 (2004), 2041-2056.doi: 10.1088/0951-7715/17/6/003.


    L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Math. Soc.: Providence, 2000.


    A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation, SIAM Review, 14 (1972), 582-643.doi: 10.1137/1014101.


    D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39 (1895), 422-443.doi: 10.1080/14786449508620739.


    P.-L. Lions, The concentration-compactness principle in the calculus of variations, Part I, Ann. Inst. H. Poincaré, Analysis Nonlinear, 1 (1984), 109-145.


    L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness Issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.doi: 10.1137/S0036141001385307.


    I. A. Svendsen and J. Buhr Hansen, Laboratory generation of waves of constant form, in the 14th Coastal Engr. Conf., Copenhagen (1974) Chapt. 17, pp. 321-339.


    I. A. Svendsen and J. Buhr Hansen, On the deformation of periodic long waves over a gently sloping bottom, J. Fluid Mech., 87 (1978), 433-448.doi: 10.1017/S0022112078001706.


    M. I. Weinstein, Liapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.doi: 10.1002/cpa.3160390103.

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