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

Asymptotics in shallow water waves

Robert McOwen 1, and  Peter Topalov 1,

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.
Keywords: spatial asymptotics, well-posedness of non-linear PDEs, Camassa-Holm equation., Shallow water waves, groups of asymptotic diffeomorphisms.
Mathematics Subject Classification: 35Q35, 37K65, 35Q53, 37K1.
Citation: Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

R. McOwen and P. Topalov, Groups of asymptotic diffeomorphisms,, , ().   Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

R. McOwen and P. Topalov, Groups of asymptotic diffeomorphisms,, , ().   Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

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