Existence theorem for solitary waves on lattices. (English) Zbl 0807.35121

Summary: We give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian \[ H = \sum_{n \in \mathbb{Z}} \left( {1 \over 2} p^ 2_ n + V(q_{n+1} - q_ n) \right), \] where \(V(\cdot)\) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda lattice \[ V(\varphi) = ab^{-1} (e^{-b \varphi} + b \varphi - 1), \] the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic lattices \[ V(\varphi) = {1 \over 2} \varphi^ 2 + {1 \over 3} a \varphi^ 3, \quad V(\varphi) = {1 \over 2} \varphi^ 2 + {1 \over 4} b \varphi^ 4, \] thereby shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but “generic” in this class of nonlinear lattices.
The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamilton’s principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions).


35Q51 Soliton equations
81T25 Quantum field theory on lattices
49J40 Variational inequalities
Full Text: DOI


[1] Eilbeck, J.C., Flesch, R.: Calculation of families of solitary waves on discrete lattices. Phys. Lett. A149, 200–202 (1990)
[2] Fermi, E., Pasta, J., Ulam, S.: Studies on nonlinear problems. Los Almos Scientific Laboratory report LA-1940, 1955. Reprinted in: Lect. in Appl. Math.15, 143–156 (1974)
[3] Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1. Ann. Inst. Henri Poincaré1, 109–145 (1984) · Zbl 0541.49009
[4] Flytzanis, N., Pnevmatikos, S., Peyrard, M.: Discrete lattice solitons: properties and stability. J. Phys. A22, 783–801 (1989)
[5] Peyrard, M., Pnevmatikos, S., Flytzanis, N.: Discreteness effects on non-topological kink soliton dynamics in nonlinear lattices. Physica D19, 268–281 (1986)
[6] Rosenau, P.: Dynamics of nonlinear mass spring chains near the continuum limit. Phys. Lett. A118, 222–227 (1986)
[7] Wattis J.A.D.: Approximations to solitary waves on lattices. II. Quasi-continuum approximations for fast and slow waves. J. Phys. A26, 1193–1209 (1993) · Zbl 0774.35069
[8] Hochstrasser, D., Mertens, F.G., Buttner, H.: An iterative method for the calculation of narrow solitary excitations on atomic chains. Physica D35, 259–266 (1989)
[9] Toda, M.: Theory of nonlinear lattices. Vol. 20 of Springer Series in Solid State Sciences. Berlin, Heidelberg, New York: Springer 1978 · Zbl 0406.35057
[10] Eilbeck, J.C.: Numerical studies of solitons on lattices. In: Remoissenet, M., Peyrard, M. (eds.), Nonlinear Coherent Structures in Physics and biology. In: Lect. Notes in Phys. Vol. 393, Berlin, Heidelberg, New York: Springer 1991, pp. 143–150
[11] Zabusky, N.J., Kruskal, M.D.: Interaction of ”solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett.15, 240–243 (1965) · Zbl 1201.35174
[12] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg de Vries equation. Phys. Rev. Lett.19, 1095–1097 (1967) · Zbl 1103.35360
[13] Sholl, D.S., Henry, B.I.: Perturbative calculation of superperiod recurrence times in nonlinear chains. Phys. Lett. A159, 21–27 (1991). Note p. 26, top of column 2
[14] Duncan, D.B., Wattis, J.A.D.: Approximations to solitary waves on lattices, use of identities and the lagrangian formulation. Chaos, Solitons and Fractals2, 505–518 (1992) · Zbl 0766.34009
[15] Russell, S.: Report on waves. Report of the 14th Meeting of the British Association for the Advancement of Science, 1844 pp. 311–390
[16] Flytzanis, N., Malomed, B.A., Wattis, J.A.D.: Analaysis of stability of solitons in one-dimensional lattices. Phys. Lett. A180, 107–112 (1993)
[17] Ball, J.M.: Loss of the constraint in convex variational problems. In: Analyse Mathématique et Applications. Paris: Gauthier-Villars 1988, pp. 39–53 · Zbl 0688.49005
[18] Ball, J.M., Carr, J., Penrose, O.: The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions. Commun. Math. Phys.104, 657–692 (1986) · Zbl 0594.58063
[19] Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math.23, 22–116 (1977) · Zbl 0938.81568
[20] Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys.85, 549–561 (1982) · Zbl 0513.35007
[21] Valkering, T.P.: Periodic permanent waves in an anharmonic chain with nearest neighbour interactions. J. Phys. A11, 1885–1897 (1978) · Zbl 0401.49037
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.