On asymptotically equivalent shallow water wave equations
Tài liệu tham khảo
Alber, 1994, The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s, Lett. Math. Phys., 32, 137, 10.1007/BF00739423
Alber, 1999, On billiard solutions of nonlinear PDE’s, Phys. Lett. A, 264, 171, 10.1016/S0375-9601(99)00784-7
Alber, 2001, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type, Comm. Math. Phys., 221, 197, 10.1007/PL00005573
Benjamin, 1972, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. Lond. A, 272, 47, 10.1098/rsta.1972.0032
Beale, 1991, Solitary water waves with capillary ripples at infinity, Commun. Pure Appl. Math., 64, 211, 10.1002/cpa.3160440204
Camassa, 1993, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661, 10.1103/PhysRevLett.71.1661
Champneys, 1997, A global investigations of solitary-wave solutions of a two-parameter model for water waves, J. Fluid Mech., 342, 199, 10.1017/S0022112097005193
Champneys, 2002, Do true elevation solitary waves exist? A numerical investigation, J. Fluid Mech., 454, 403, 10.1017/S0022112001007200
Craig, 1994, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19, 367, 10.1016/0165-2125(94)90003-5
A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23–37.
A. Degasperis, D.D. Holm, A.N.W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys. 133 (2002) 1463–1474. http://xxx.lanl.gov/abs/nlin.SI/0205023.
Dias, 1999, Nonlinear gravity and capillary-gravity waves, Ann. Rev. Fluid Mech., 31, 301, 10.1146/annurev.fluid.31.1.301
Dullin, 2001, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87, 4501, 10.1103/PhysRevLett.87.194501
Fenton, 1972, A ninth-order solution for the solitary wave, J. Fluid Mech., 53, 257, 10.1017/S002211207200014X
J.D. Fenton, Nonlinear wave theories, in: B. Le Méhauté, D.M. Hanes (Eds.), The Sea: Ocean Engineering Science, Part A, vol. 9, 1990, pp. 3–25.
Fokas, 1996, Asymptotic integrability of water waves, Phys. Rev. Lett., 77, 2347, 10.1103/PhysRevLett.77.2347
Fokas, 1995, On a class of physically important integrable equations, Physica D, 87, 145, 10.1016/0167-2789(95)00133-O
Fokas, 1981, Bäcklund transformations for hereditary symmetries, Nonlinear Anal. TMA, 5, 423, 10.1016/0362-546X(81)90025-0
Fuchssteiner, 1996, Some tricks from the symmetry-toolbox for nonlinear equations: generalization of the Camassa–Holm equation, Physica D, 95, 229, 10.1016/0167-2789(96)00048-6
Grimshaw, 1971, The solitary wave in water of variable depth, Part 2, J. Fluid Mech., 46, 611, 10.1017/S0022112071000739
Grimshaw, 1995, Weakly nonlocal solitary waves in a singularly perturbed Korteweg–de Vries equation, SIAM J. Appl. Math., 55, 124, 10.1137/S0036139993243825
Holm, 1998, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1, 10.1006/aima.1998.1721
D.D. Holm, M.F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003) 437–444 http://xxx.lanl.gov/abs/nlin.CD/0203007
D.D. Holm, M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dynam. Syst. 2 (2003) 323–380.
Johnson, 2002, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech., 455, 63, 10.1017/S0022112001007224
Kodama, 1985, On integrable systems with higher order corrections, Phys. Lett. A, 107, 245, 10.1016/0375-9601(85)90207-5
Y. Kodama, Normal forms for weakly dispersive wave equations, 112 (1985) 193–196
Kodama, 1987, On solitary-wave interaction, Phys. Lett. A, 123, 276, 10.1016/0375-9601(87)90227-1
Y. Kodama, A.V. Mikhailov, Obstacles to asymptotic integrability, in: A.S. Fokas, I.M. Gelfand (Eds.), Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, Birkhäuser, Boston, 1996, pp. 173–204.
Korteweg, 1895, On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves, Phil. Mag., 39, 422, 10.1080/14786449508620739
Li, 1997, An improved theory of long waves on the water surface, J. Appl. Math. Mech., 61, 177, 10.1016/S0021-8928(97)00024-5
E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible system, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000.
Marchant, 1990, The extended Korteweg–de Vries equation and the resonant flow over topography, J. Fluid Mech., 221, 263, 10.1017/S0022112090003561
Mikhailov, 2002, Perturbative symmetry approach, J. Phys. A, 35, 4775, 10.1088/0305-4470/35/22/309
Olver, 1984, Hamiltonian perturbation theory and water waves, Contemp. Math., 28, 231, 10.1090/conm/028/751987
Sun, 1999, Non-existence of truly solitary waves in water with small surface tension, Proc. Roy. Soc. Lond. A, 455, 2191, 10.1098/rspa.1999.0399
G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974.
Weidman, 1978, Experiments on strong interactions between solitary waves, J. Fluid Mech., 85, 417, 10.1017/S0022112078000713