×

zbMATH — the first resource for mathematics

Description of two soliton collision for the quartic gKdV equation. (English) Zbl 1300.37045
Summary: In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation \(\partial_t u + \partial_x (\partial_x^2 u + u^4)=0\), which behave as \(t\to -\infty\) like \[ u(t,x)=Q_{c_1}(x -c_1 t) + Q_{c_2}(x-c_2 t) + \eta(t,x), \] where \(Q_{c}(x-ct)\) is a soliton and \(\|\eta(t)\|_{H^1} \ll \|Q_{c_2}\|_{H^1}\ll \|Q_{c_1}\|_{H^1}\). The global behavior of \(u(t)\) is given by the following stability result: for all \(t\in \mathbb{R}\), \(u(t,x)=Q_{ c_1(t)}(x-y_1(t)) + Q_{ c_2(t)}(x-y_2(t)) + \eta(t,x)\), where \(\|\eta(t)\|_{H^1} \ll \|Q_{c_2}\|_{H^1}\) and \(\lim_{t\to +\infty} c_1(t)=c_1^{+}\), \(\lim_{t\to +\infty} c_2(t)=c_2^{+}\). In the case where \(u(t)\) is a pure 2-soliton solution as \(t\to -\infty\) (i.e. \(\mathrm{lim}_{t\to -\infty} |\eta(t)|_{H^1}=0)\), we obtain \( c_1^{+}>c_1\), \(c_2^{+}< c_2 \) and for the residual part, \(\mathrm{lim}_{t\to +\infty} |\eta(t)|_{H^1}> 0\). Therefore, in contrast with the integrable KdV equation (or mKdV equation), no global pure 2-soliton solution exists and the collision is inelastic. A different notion of global 2-soliton is then proposed.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] T. B. Benjamin, ”The stability of solitary waves,” Proc. Roy. Soc. \((\)London\()\) Ser. A, vol. 328, pp. 153-183, 1972.
[2] J. L. Bona, ”On the stability theory of solitary waves,” Proc. Roy. Soc. London Ser. A, vol. 344, iss. 1638, pp. 363-374, 1975. · Zbl 0328.76016
[3] J. L. Bona, W. G. Pritchard, and L. R. Scott, ”Solitary-wave interaction,” Phys. Fluids, vol. 23, pp. 438-441, 1980. · Zbl 0425.76019
[4] T. Cazenave and P. -L. Lions, ”Orbital stability of standing waves for some nonlinear Schrödinger equations,” Comm. Math. Phys., vol. 85, iss. 4, pp. 549-561, 1982. · Zbl 0513.35007
[5] A. Cohen, ”Existence and regularity for solutions of the Korteweg-de Vries equation,” Arch. Rational Mech. Anal., vol. 71, iss. 2, pp. 143-175, 1979. · Zbl 0415.35069
[6] R. Côte, ”Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior,” J. Funct. Anal., vol. 241, iss. 1, pp. 143-211, 2006. · Zbl 1157.35091
[7] R. Côte, ”Construction of solutions to the \(L^2\)-critical KdV equation with a given asymptotic behaviour,” Duke Math. J., vol. 138, iss. 3, pp. 487-531, 2007. · Zbl 1130.35112
[8] W. Craig, P. Guyenne, J. Hammack, D. Henderson, and C. Sulem, ”Solitary water wave interactions,” Phys. Fluids, vol. 18, iss. 5, p. 057106, 2006. · Zbl 1185.76463
[9] W. Eckhaus and P. Schuur, ”The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions,” Math. Methods Appl. Sci., vol. 5, iss. 1, pp. 97-116, 1983. · Zbl 0518.35074
[10] E. Fermi, J. Pasta, and S. Ulam, Studies of nonlinear problems, I, Los Alamos Report LA1940. · Zbl 0353.70028
[11] J. Hammack, D. Henderson, P. Guyenne, and Y. Ming, ”Solitary-wave collisions,” in Proceedings of the 23rd ASME Offshore Mechanics and Artic Engineering, Singapore, 2004. · Zbl 0353.70028
[12] M. Huaruagucs-Courcelle and D. H. Sattinger, ”Inversion of the linearized Korteweg-de Vries equation at the multi-soliton solutions,” Z. Angew. Math. Phys., vol. 49, iss. 3, pp. 436-469, 1998. · Zbl 0904.35076
[13] R. Hirota, ”Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Phys. Rev. Lett., vol. 27, pp. 1192-1194, 1971. · Zbl 1168.35423
[14] H. Kalisch and J. L. Bona, ”Models for internal waves in deep water,” Discrete Contin. Dynam. Systems, vol. 6, iss. 1, pp. 1-20, 2000. · Zbl 1021.76006
[15] C. E. Kenig, G. Ponce, and L. Vega, ”Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Comm. Pure Appl. Math., vol. 46, iss. 4, pp. 527-620, 1993. · Zbl 0808.35128
[16] M. D. Kruskal, ”The Korteweg-de Vries equation and related evolution equations,” in Nonlinear Wave Motion, Newell, A. C., Ed., Providence, R.I.: Amer. Math. Soc., 1974, vol. 15, pp. 61-83. · Zbl 0292.35017
[17] P. D. Lax, ”Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., vol. 21, pp. 467-490, 1968. · Zbl 0162.41103
[18] Y. Li and D. H. Sattinger, ”Soliton collisions in the ion acoustic plasma equations,” J. Math. Fluid Mech., vol. 1, iss. 1, pp. 117-130, 1999. · Zbl 0934.35148
[19] Y. Martel, ”Asymptotic \(N\)-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations,” Amer. J. Math., vol. 127, iss. 5, pp. 1103-1140, 2005. · Zbl 1090.35158
[20] Y. Martel, ”Linear problems related to asymptotic stability of solitons of the generalized KdV equations,” SIAM J. Math. Anal., vol. 38, iss. 3, pp. 759-781, 2006. · Zbl 1126.35055
[21] Y. Martel and F. Merle, ”Asymptotic stability of solitons for subcritical generalized KdV equations,” Arch. Ration. Mech. Anal., vol. 157, iss. 3, pp. 219-254, 2001. · Zbl 0981.35073
[22] Y. Martel and F. Merle, ”Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation,” Ann. of Math., vol. 155, iss. 1, pp. 235-280, 2002. · Zbl 1005.35081
[23] Y. Martel and F. Merle, ”Asymptotic stability of solitons of the subcritical gKdV equations revisited,” Nonlinearity, vol. 18, iss. 1, pp. 55-80, 2005. · Zbl 1064.35171
[24] Y. Martel and F. Merle, ”Asymptotic stability of solitons of the gKdV equations with general nonlinearity,” Math. Ann., vol. 341, iss. 2, pp. 391-427, 2008. · Zbl 1153.35068
[25] Y. Martel and F. Merle, ”Refined asymptotics around solitons for gKdV equations,” Discrete Contin. Dyn. Syst., vol. 20, iss. 2, pp. 177-218, 2008. · Zbl 1137.35062
[26] Y. Martel and F. Merle, ”Stability of two soliton collision for nonintegrable gKdV equations,” Comm. Math. Phys., vol. 286, iss. 1, pp. 39-79, 2009. · Zbl 1179.35291
[27] Y. Martel and F. Merle, ”Note on coupled linear systems related to two soliton collision for the quartic gKdV equation,” Rev. Mat. Complut., vol. 21, iss. 2, pp. 327-349, 2008. · Zbl 1188.35164
[28] Y. Martel, F. Merle, and T. Tsai, ”Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations,” Comm. Math. Phys., vol. 231, iss. 2, pp. 347-373, 2002. · Zbl 1017.35098
[29] R. M. Miura, ”The Korteweg-de Vries equation: a survey of results,” SIAM Rev., vol. 18, iss. 3, pp. 412-459, 1976. · Zbl 0333.35021
[30] T. Mizumachi, ”Weak interaction between solitary waves of the generalized KdV equations,” SIAM J. Math. Anal., vol. 35, iss. 4, pp. 1042-1080, 2003. · Zbl 1054.35083
[31] R. L. Pego and M. I. Weinstein, ”Asymptotic stability of solitary waves,” Comm. Math. Phys., vol. 164, iss. 2, pp. 305-349, 1994. · Zbl 0805.35117
[32] G. Perelman, ”Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations,” Comm. Partial Differential Equations, vol. 29, iss. 7-8, pp. 1051-1095, 2004. · Zbl 1067.35113
[33] I. Rodnianski, W. Schlag, and A. D. Soffer, ”On asymptotic stability of \(N\)-soliton states of NLS,” Rev. Math. Phys., vol. 15, pp. 877-903, 2003.
[34] P. C. Schuur, Asymptotic Analysis of Soliton Problems, New York: Springer-Verlag, 1986, vol. 1232. · Zbl 0643.35003
[35] L. Y. Shih, ”Soliton-like interaction governed by the generalized Korteweg-de Vries equation,” Wave Motion, vol. 2, iss. 3, pp. 197-206, 1980. · Zbl 0443.76019
[36] T. Tao, ”Scattering for the quartic generalised Korteweg-de Vries equation,” J. Differential Equations, vol. 232, iss. 2, pp. 623-651, 2007. · Zbl 1171.35107
[37] M. Wadati and M. Toda, ”The exact \(N\)-soliton solution of the Korteweg-de Vries equation,” J. Phys. Soc. Japan, vol. 32, pp. 1403-1411, 1972.
[38] P. D. Weidman and T. Maxworthy, ”Experiments on strong interactions between solitary waves,” J. Fluids Mech., vol. 85, pp. 417-431, 1978.
[39] M. I. Weinstein, ”Modulational stability of ground states of nonlinear Schrödinger equations,” SIAM J. Math. Anal., vol. 16, iss. 3, pp. 472-491, 1985. · Zbl 0583.35028
[40] M. I. Weinstein, ”Lyapunov stability of ground states of nonlinear dispersive evolution equations,” Comm. Pure Appl. Math., vol. 39, iss. 1, pp. 51-67, 1986. · Zbl 0594.35005
[41] N. J. Zabusky, ”Solitons and energy transport in nonlinear lattices,” Comput. Phys. Comm., vol. 5, pp. 1-10, 1973.
[42] N. J. Zabusky and M. D. Kruskal, ”Interaction of “solitons” in a collisionless plasma and recurrence of initial states,” Phys. Rev. Lett., vol. 15, pp. 240-243, 1965. · Zbl 1201.35174
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.