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Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation

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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

  1. Abdulloev K.O., Bogolubsky I.L., Makhankov V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56, 427–428 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  2. 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Bona J.L., Pritchard W.G., Scott L.R.: Solitary-wave interaction. Phys. Fluids 23, 438 (1980)

    Article  MATH  ADS  Google Scholar 

  4. 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)

    Google Scholar 

  5. Bryan A.C., Stuart A.E.G.: Solitons and the regularized long wave equation: a nonexistence theorem. Chaos, Solitons Fractals 7, 1881–1886 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cohen A.: Existence and regularity for solutions of the Korteweg–de Vries equation. Arch. Rat. Mech. Anal. 71, 143–175 (1979)

    Article  MATH  Google Scholar 

  7. Craig W., Guyenne P., Hammack J., Henderson D., Sulem C.: Solitary water wave interactions. Phys. Fluids 18, 057106 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  8. 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)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. El Dika K.: Asymptotic stability of solitary waves for the Benjamin–Bona–Mahoney equation. Discret. Contin. Dyn. Syst. 13, 583–622 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. El Dika K., Martel Y.: Stability of N solitary waves for the generalized BBM equations. Dyn. Partial Differ. Equ. 1, 401–437 (2004)

    MATH  MathSciNet  Google Scholar 

  12. 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

  13. 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)

    MATH  MathSciNet  Google Scholar 

  14. 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

  15. Hirota R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  MATH  ADS  Google Scholar 

  16. Kalisch H., Bona J.L.: Models for internal waves in deep water. Discrete. Continuous Dyn. Syst. 6, 1–20 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. 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)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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

  19. , : Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)

    Article  MATH  Google Scholar 

  20. Martel Y.: Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations. Am. J. Math. 127, 1103–1140 (2005)

    MATH  MathSciNet  Google Scholar 

  21. Martel Y.: Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38, 759–781 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Martel Y., Merle F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. 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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Martel Y., Merle F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. 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)

    MATH  MathSciNet  Google Scholar 

  26. 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)

    MATH  MathSciNet  Google Scholar 

  27. Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equation, submitted preprint. http://arxiv.org/abs/0709.2672

  28. Martel Y., Merle F.: Stability of two soliton collision for nonintegrable gKdV equations. Commun. Math. Phys. 286, 39–79 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Miura R.M.: The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  31. Miller J., Weinstein M.: Asymptotic stability of solitary waves for the regularized long wave equation. Commun. Pure Appl. Math. 49, 399–441 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mizumachi T.: Weak interaction between solitary waves of the generalized KdV equations. SIAM J. Math. Anal. 35, 1042–1080 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Mizumachi T.: Asymptotic stability of solitary wave solutions to the regularized long-wave equation. J. Differ. Equ. 200, 312–341 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Morrison P.J., Meiss J.D., Cary J.R.: Scattering of regularized-long-wave solitary waves. Phys. D 11, 324–336 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  35. Peregine D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321 (1966)

    Article  ADS  Google Scholar 

  36. Schuur, P.C.: Asymptotic Analysis of Solitons Problems. Lecture Notes in Mathematics, vol. 1232. Springer, Berlin, 1986

  37. Tao T.: Scattering for the quartic generalised Korteweg–de Vries equation. J. Differ. Equ. 232, 623–651 (2007)

    Article  MATH  Google Scholar 

  38. Wadati M., Toda M.: The exact N-soliton solution of the Korteweg–de Vries equation. J. Phys. Soc. Japan. 32, 1403–1411 (1972)

    Article  ADS  Google Scholar 

  39. 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)

    Article  MATH  Google Scholar 

  40. 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)

    Article  ADS  Google Scholar 

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Correspondence to Yvan Martel.

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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

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