Abstract
We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves and shifts in their trajectories). To prove these results, we extend the method introduced in Martel and Merle (Description of two soliton collision for the quartic gKdV equation, submitted preprint. http://arxiv.org/abs/0709.2672; Commun Math Phys 286:39–79, 2009) for the generalized KdV equation, in particular in the quartic case. The main argument is the construction of an explicit approximate solution in the collision region.
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References
Abdulloev K.O., Bogolubsky I.L., Makhankov V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56, 427–428 (1976)
Benjamin T.B., Bona J.L., Mahony J.J.: Model equations for long waves nonlinear dispersion systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)
Bona J.L., Pritchard W.G., Scott L.R.: Solitary-wave interaction. Phys. Fluids 23, 438 (1980)
Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. Ser. A 411 1841, 395–412 (1987)
Bryan A.C., Stuart A.E.G.: Solitons and the regularized long wave equation: a nonexistence theorem. Chaos, Solitons Fractals 7, 1881–1886 (1996)
Cohen A.: Existence and regularity for solutions of the Korteweg–de Vries equation. Arch. Rat. Mech. Anal. 71, 143–175 (1979)
Craig W., Guyenne P., Hammack J., Henderson D., Sulem C.: Solitary water wave interactions. Phys. Fluids 18, 057106 (2006)
Eckhaus W., Schuur P.: The emergence of solutions of the Korteweg–de Vries equation from arbitrary initial conditions. Math. Meth. Appl. Sci. 5, 97–116 (1983)
Eilbeck J.C., McGuire G.R.: Numerical study of the regularized long-wave equation. II. Interaction of solitary waves. J. Comput. Phys. 23, 63–73 (1977)
El Dika K.: Asymptotic stability of solitary waves for the Benjamin–Bona–Mahoney equation. Discret. Contin. Dyn. Syst. 13, 583–622 (2005)
El Dika K., Martel Y.: Stability of N solitary waves for the generalized BBM equations. Dyn. Partial Differ. Equ. 1, 401–437 (2004)
Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems, I, Los Alamos Report LA1940 (1955); reproduced in Nonlinear Wave Motion, Newell, A.C.: ed., American Mathematical Society, Providence, R.I., pp. 143–156, 1974
Grillakis M., Shatah J., Strauss W.A.: Stability theory of solitary waves in the presence of symmetry, I. J. Differ. Equ. 74, 160–197 (1987)
Hammack, J., Henderson, D., Guyenne, P., Yi, M.: Solitary-wave collisions. Proceedings of the 23rd ASME Offshore Mechanics and Artic Engineering (A symposium to honor Theodore Yao-Tsu Wu), Vancouver, Canada, June 2004. Word Scientific, Singapore, 2004
Hirota R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)
Kalisch H., Bona J.L.: Models for internal waves in deep water. Discrete. Continuous Dyn. Syst. 6, 1–20 (2000)
Kenig C.E., Ponce G., Vega L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)
Kruskal, M.D.: The Korteweg–de Vries equation and related evolution equations. Nonlinear Wave Motion (Ed. A.C. Newell.) American Mathematical Society, Providence, R.I., 61–83, 1974
, : Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)
Martel Y.: Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations. Am. J. Math. 127, 1103–1140 (2005)
Martel Y.: Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38, 759–781 (2006)
Martel Y., Merle F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001)
Martel Y., Merle F.: Stability of blow up profile and lower bounds for blow up rate for the critical generalized KdV equation. Ann. Math. 155, 235–280 (2002)
Martel Y., Merle F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)
Martel Y., Merle F.: Refined asymptotics around solitons for the gKdV equations with a general nonlinearity. Discrete. Contin. Dyn. Syst. 20(2), 177–218 (2008)
Martel Y., Merle F.: Resolution of coupled linear systems related to the collision of two solitons for the quartic gKdV equation. Rev. Mat. Complut. 21, 327–349 (2008)
Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equation, submitted preprint. http://arxiv.org/abs/0709.2672
Martel Y., Merle F.: Stability of two soliton collision for nonintegrable gKdV equations. Commun. Math. Phys. 286, 39–79 (2009)
Martel Y., Merle F., Tsai T.-P.: Stability and asymptotic stability in the energy space of the sum of N solitons for the subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)
Miura R.M.: The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976)
Miller J., Weinstein M.: Asymptotic stability of solitary waves for the regularized long wave equation. Commun. Pure Appl. Math. 49, 399–441 (1996)
Mizumachi T.: Weak interaction between solitary waves of the generalized KdV equations. SIAM J. Math. Anal. 35, 1042–1080 (2003)
Mizumachi T.: Asymptotic stability of solitary wave solutions to the regularized long-wave equation. J. Differ. Equ. 200, 312–341 (2004)
Morrison P.J., Meiss J.D., Cary J.R.: Scattering of regularized-long-wave solitary waves. Phys. D 11, 324–336 (1984)
Peregine D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321 (1966)
Schuur, P.C.: Asymptotic Analysis of Solitons Problems. Lecture Notes in Mathematics, vol. 1232. Springer, Berlin, 1986
Tao T.: Scattering for the quartic generalised Korteweg–de Vries equation. J. Differ. Equ. 232, 623–651 (2007)
Wadati M., Toda M.: The exact N-soliton solution of the Korteweg–de Vries equation. J. Phys. Soc. Japan. 32, 1403–1411 (1972)
Weinstein M.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagations. Commun. Partial Differ. Equ. 12, 1133–1173 (1987)
Zabusky N.J., Kruskal M.D.: Interaction of “solitons” in a collisionless plasma and recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)
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Communicated by Y. Brenier
This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).
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Martel, Y., Merle, F. & Mizumachi, T. Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation. Arch Rational Mech Anal 196, 517–574 (2010). https://doi.org/10.1007/s00205-009-0244-7
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DOI: https://doi.org/10.1007/s00205-009-0244-7