Dynamics of breathers in the Gardner hierarchy
universality of the variational characterization
DOI:
https://doi.org/10.22199/issn.0717-6279-5354Keywords:
Higher order Gardner equation, hierarchy breather, integrabilityAbstract
We present a new variational characterization of breather solutions of any equation of the focusing Gardner hierarchy. This hierarchy is characterized by a nonnegative index n, and 2n + 1 represents the order of the corresponding PDE member. In this paper, we first show the existence of such breathers, and that they are solutions of the (2n+1)th-order Gardner equation. Then we prove a variational universality property, in the sense that all these breather solutions satisfy the same fourth order stationary elliptic ODE, regardless the order of the hierarchy member. This fact also characterizes them as critical points of the same Lyapunov functional, that we also construct here. As by product of our approach, we find breather solutions of the hierarchy of (2n+1)th-order mKdV equations, as well as a respective characterization of them as solutions of a fourth order stationary elliptic ODE. We also extend part of these results to the periodic setting, presenting new breather solutions for the 5th and 7th mKdV members of the hierarchy. Finally, we prove ill-posedness results for the whole Gardner hierarchy, by using appropiately their breather solutions.
References
M. Ablowitz and P. Clarkson, Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series, 149. Cambridge: Cambridge University Press, 1991.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
M. A. Alejo and E. Cardoso, “Nonlinear Stability of higher order mKdV breathers”, arXiv: 1804.02285, 2018.
M. A. Alejo and E. Cardoso, “On the ill-posedness of the 5th-order Gardner equation”, SPJM 13, pp. 383-390, 2019.
M. A. Alejo, “On the ill-posedness of the Gardner equation”, Journal of Mathematical Analysis and Applications, vol. 396, no. 1, pp. 256-260, 2012. https://doi.org/10.1016/j.jmaa.2012.06.018
M. A. Alejo and C. Kwak, “Global solutions and stability properties of the 5th order Gardner equation”, Journal of Dynamics and Differential Equations, vol. 35, pp. 575-621, 2023. https://doi.org/10.1007/s10884-021-10022-4
M. A. Alejo and C. Muñoz, “Nonlinear stability of mKdV breathers”, Communications in Mathematical Physics, vol. 37 pp. 2050-2080, 2013.
M. A. Alejo and C. Muñoz, “Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers”, Analysis and PDE, vol. 8-3, pp. 629-674, 2015.
M. A. Alejo, “Nonlinear stability of Gardner breathers”, Journal of Differential Equations, vol. 264, no. 2, pp. 1192-1230, 2018. https://doi.org/10.1016/j.jde.2017.09.035
M. A. Alejo, L. Fanelli and C. Muñoz, “Stability and instability of breathers in the U(1) Sasa-Satusuma and Nonlinear Schrödinger models”, Nonlinearity, vol. 34, pp. 3429-3484, 2021.
M. A. Alejo, C. Muñoz and J. M. Palacios, “On the variational structure of breather solutions I: the Sine-Gordon case”, Journal of Mathematical Analysis and Applications, vol.453, no. 2, pp. 1111-1138, 2017. https://doi.org/10.1016/j.jmaa.2017.04.056
M. A. Alejo, C. Muñoz and J. M. Palacios, “On the variational structure of breather solutions II: periodic mKdV case”, Electronic Journal of Differential Equations, vol. 2017, No. 56, pp. 1-26, 2017.
T. B. Benjamin, “The stability of solitary waves”, Proceedings of the Royal Society A. vol. 328, pp. 153-183, 1972. https://doi.org/10.1098/rspa.1972.0074
P. F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. New York: Springer-Verlag, 1971.
P. Clarkson, N. Joshi and M. Mazzorocco, “The Lax pair for the mKdV hierarchy”, Seminaires & Congress, vol. 14, pp. 53-64, 2006.
A. Alvarez, F. Romero, J. Cuevas and J.F.R. Archilla, “Moving breather collisions in Klein-Gordon chains of oscillators”, The European Physical Journal B, vol. 70, no. 2. https://doi.org/10.1140/epjb/e2009-00256-6
J. Cuevas, Q. Hoq, H. Susanto and PG. Kevrekidis, “Interlaced solitons and vortices in coupled DNLS lattices”, Physica D, vol. 238, pp. 2216, 2009. https://doi.org/10.1016/j.physd.2009.09.002
J.F.R. Archilla, J. Cuevas and F.R. Romero, “Effect of breather existence on reconstructive transformations in mica muscovite”, AIP Conference Proceedings, vol. 982, pp. 788, 2008.
C. S. Gardner, M.D. Kruskal and R. Miura, “Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion”, Journal of Mathematical Physics, vol. 9, no. 8, pp. 1204-1209, 1968. https://doi.org/10.1063/1.1664701
L. Greenberg, “An oscillation method for fourth order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues”, SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 1021-1042, 1991. https://doi.org/10.1137/0522067
R. Grimshaw, A. Slunyaev and E. Pelinovsky, “Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive cubic nonlinearity”. Chaos, vol. 20, no. 1, 2010, 01310201- 01310210. https://doi.org/10.1063/1.3279480
J. F. Gomes, G.S. Franca and A.H. Zimerman, “Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation”, Journal of Physics A-mathematical and Theoretical, vol. 45, 015207. 2012.
A. Grünrock, “On the hierarchies of higher order mKdV and KdV equations”, Central European Journal of Mathematics, vol. 8, no. 3, pp. 500-536, 2010.
J. Holmer, G. Perelman and M. Zworski, “Effective dynamics of double solitons for perturbed mKdV”, Communications in Mathematical Physics, vol. 305, pp. 363-425, 2011.
C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg—de Vries equation via the contraction principle”, Communications on Pure and Applied Mathematics, vol. 46, pp. 527-620, 1993. https://doi.org/10.1002/CPA.3160460405
C. E. Kenig, G. Ponce and L. Vega, “On the ill-posedness of some canonical dispersive equations”, Duke Mathematical Journal, vol. 106, mo. 3, pp. 617-633, 2001. https://doi.org/ 10.1215/S0012-7094-01-10638-8
P. G. Kevrekidis, A. Khare, A. Saxena and G. Herring, “On some classes of mKdV periodic solutions”, Journal of Physics A: Mathematical and General, vol. 37, pp. 10959-10965, 2004. https://doi.org/10.1088/0305-4470/37/45/014
P. G. Kevrekidis, A. Khare and A. Saxena, “Breather lattice and its stabilization for the modified Korteweg-de Vries equation”, Physical Review E, vol. 68, pp. 0477011-0477014, 2003. https://doi.org/10.1103/PhysRevE.68.047701
S. Kwon, “Well posedness and Ill-posedness of the Fifth-order modified KdV equation”, Electronic Journal of Differential Equations, vol. 2008, no. 1, pp. 1-15, 2008.
G. L. Lamb, Elements of Soliton Theory, Pure and Applied Mathematics. New York: Wiley, 1980.
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Communications on Pure and Applied Mathematics, vol. 21, pp. 467-490, 1968. https://doi.org/10.1002/cpa.3160210503
F. Linares, “A higher order modified Korteweg-de Vries equation”, Computational and Applied Mathematics, vol.14, no. 3, pp. 253-267, 1995.
J. H. Maddocks and R.L. Sachs, “On the stability of KdV multi-solitons”, Computational and Applied Mathematics, vol. 46, pp. 867-901, 1993.
Y. Matsuno, Bilinear transformation Method. vol. 174. Academic Press, 1984.
Y. Matsuno, “Bilinearization of Nonlinear Evolution Equations: Higher Order mKdV”, Journal of the Physical Society of Japan, vol. 49, no. 2, 1980.
Y. Matsuno, Personal communication.
R. M. Miura, “Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation”, Journal of Mathematical Physics, vol. 9, no. 8, pp. 1202-1204, 1968. https://doi.org/10.1063/1.1664700
C. Muñoz and G. Ponce, “Breathers and the dynamics of solutions to the KdV type equations”. arXiv: 1803.05475, 2019.
P. J. Olver, Evolution equations possessing infinitely many symmetries”, Journal of Mathematical Physics, vol. 18, pp. 1212, 1977.
M. Wadati, “The modified Korteweg-de Vries Equation”, Journal of the Physical Society of Japan, vol. 34, No.5, pp. 1289-1296, 1973.
M. I. Weinstein, “Lyapunov stability of ground states of nonlinear dispersive evolution equations”, Communications on Pure and Applied Mathematics, vol. 39, pp. 51-68, 1986. https://doi.org/10.1002/cpa.3160390103
Published
How to Cite
Issue
Section
Copyright (c) 2023 Miguel A Alejo
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
