# American Institute of Mathematical Sciences

July  2015, 35(7): 3103-3131. doi: 10.3934/dcds.2015.35.3103

## Asymptotics in shallow water waves

 1 Northeastern University, 360 Huntington Avenue, Boston, MA 02115, United States, United States

Received  August 2014 Revised  September 2014 Published  January 2015

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.
Citation: Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
##### References:
 [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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##### References:
 [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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