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Asymptotics in shallow water waves

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  • In this paper we consider the initial value problem for a family of shallow water equations on the line $\mathbb{R}$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.
    Mathematics Subject Classification: 35Q35, 37K65, 35Q53, 37K10.

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  • [1]

    V. Arnold, Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.doi: 10.5802/aif.233.

    [2]

    I. Bondareva and M. Shubin, Growing asymptotic solutions of the Korteweg-de Vries equation and of its higher analogues, Dokl. Akad. Nauk SSSR, 267 (1982), 1035-1038.

    [3]

    I. Bondareva and M. Shubin, Uniqueness of the solution of the Cauchy problem for the Korteweg-de Vries equation in classes of increasing functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh, 102 (1985), 35-38.

    [4]

    I. Bondareva and M. Shubin, Equations of Korteweg-de Vries type in classes of increasing functions, J. Soviet Math., 51 (1990), 2323-2332.doi: 10.1007/BF01094991.

    [5]

    A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math Anal., 41 (2009), 1559-1588.doi: 10.1137/090748500.

    [6]

    R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett, 71 (1993), 1661-1664.doi: 10.1103/PhysRevLett.71.1661.

    [7]

    A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier, Grenoble, 50 (2000), 321-362.doi: 10.5802/aif.1757.

    [8]

    A. Constantin, On the scattering problem for Camassa-Holm equation, Proc. R. Soc. Lond. A, 457 (2001), 953-970.doi: 10.1098/rspa.2000.0701.

    [9]

    A. Constantin, The trajectories of particles in Stokes waves, Invent. math., 166 (2006), 523-535.doi: 10.1007/s00222-006-0002-5.

    [10]

    A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328.

    [11]

    A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.doi: 10.1007/PL00004793.

    [12]

    A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1545.doi: 10.1512/iumj.1998.47.1466.

    [13]

    A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equation, Arch. Rational Mech. Anal., 192 (2009), 165-186.doi: 10.1007/s00205-008-0128-2.

    [14]

    A. Constantin and H. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

    [15]

    C. De Lellis, T. Kappeler and P. Topalov, Low regularity solutions of the Camassa-Holm equation, Comm. Partial Differential Equations, 32 (2007), 87-126.doi: 10.1080/03605300601091470.

    [16]

    D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163.doi: 10.2307/1970699.

    [17]

    A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66.doi: 10.1016/0167-2789(81)90004-X.

    [18]

    D. Holm and M. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Applied Dynamical Systems, 2 (2003), 323-380.doi: 10.1137/S1111111102410943.

    [19]

    D. Holm and M. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in 1+1 nonlinear PDE, Phys. Lett. A, 308 (2003), 437-444.doi: 10.1016/S0375-9601(03)00114-2.

    [20]

    H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., 226 (2013), vi+60 pp.doi: 10.1090/S0065-9266-2013-00676-4.

    [21]

    T. Kappeler, P. Perry, M. Shubin and P. Topalov, Solutions of mKdV in classes of functions unbounded at infinity, J. Geom. Anal., 18 (2008), 443-477.doi: 10.1007/s12220-008-9013-3.

    [22]

    S. Lang, Differential Manifolds, Addison-Wesley Series in Mathematics, 1972.

    [23]

    H. McKean, Fredholm determinants and Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56 (2003), 638-680.doi: 10.1002/cpa.10069.

    [24]

    R. McOwen and P. Topalov, Groups of asymptotic diffeomorphisms, arXiv:1412.4732.

    [25]

    G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.doi: 10.1016/S0393-0440(97)00010-7.

    [26]

    G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, GAFA, 12 (2002), 1080-1104.doi: 10.1007/PL00012648.

    [27]

    V. Ovsienko and B. Khesin, Korteweg-de Vries superequations as an Euler equation, Functional Anal. Appl., 21 (1987), 81-82.

    [28]

    J. Toland, Stokes waves, Topological Methods in Nonlinear Analysis, 7 (1996), 1-48.

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