## Multiple periodic solutions for lattice dynamical systems with superquadratic potentials.(English)Zbl 1318.37026

Summary: In this paper, we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equations $\ddot {q}_i={\varPhi}_{i-1}'(q_{i-1}-q_i)-{\varPhi}_i'(q_i-q_{i+1}), \quad i\in \mathbb Z,$ where $${\varPhi}_i$$ denotes the interaction potential between two neighboring particles and $$q_i(t)$$ is the state of the $$i$$-th particle. Supposing $${\varPhi}_i$$ is superquadratic at infinity, for all $$T>0$$, we obtain a nonzero $$T$$-periodic solution of finite energy which may be nonconstant in some range of period. If in addition $${\varPhi}_i(x)$$ is even in $$x$$, we also obtain infinitely many geometrically distinct solutions for any period $$T>0$$. In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group $$\mathbb Z$$ and the continuous compact group $$S^1$$ under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by T. Bartsch and Y. Ding [Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)] to a more general class of symmetry.

### MSC:

 37K60 Lattice dynamics; integrable lattice equations 34C25 Periodic solutions to ordinary differential equations 70F45 The dynamics of infinite particle systems 34A33 Ordinary lattice differential equations

Zbl 1117.58007
Full Text:

### References:

 [1] Ackermann, N., On a periodic Schrödinger equation with nonlocal part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037 [2] Amborsetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063 [3] Arioli, G.; Gazzola, F., Existence and numerical approximation of periodic motions of an infinite lattice of particles, Z. Angew. Math. Phys., 46, 898-912, (1995) · Zbl 0838.34046 [4] Arioli, G.; Gazzola, F., Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26, 1103-1114, (1996) · Zbl 0867.70004 [5] Arioli, G.; Gazzola, F.; Terracini, S., Multibump periodic motions of an infinite lattice of particles, Math. Z., 223, 627-642, (1996) · Zbl 0871.34028 [6] Arioli, G.; Szulkin, A., Periodic motions of an infinite lattice of particles: the strongly indefinite case, Ann. Sci. Math. Québec., 22, 97-119, (1998) · Zbl 1100.37503 [7] Arioli1, G.; Koch, H.; Terracini, S., Two novel methods and multi-mode periodic solutions for the Fermi-pasta-Ulam model, Comm. Math. Phys., 255, 1-19, (2005) · Zbl 1076.70008 [8] Bartsch, T.; Ding, Y. H., On a nonlinear Schrödinger equations, Math. Ann., 313, 15-37, (1999) · Zbl 0927.35103 [9] Bartsch, T.; Ding, Y. H., Solutions of nonlinear Dirac equations, J. Differential Equations, 226, 210-249, (2006) · Zbl 1126.49003 [10] Bartsch, T.; Ding, Y. H., Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279, 1267-1288, (2006) · Zbl 1117.58007 [11] Cerami, G., Un criterio di esistenza per i punti critici su varietá illimitate, Rend. Ist. Lomb. Sci. Lett., 112, 332-336, (1978) · Zbl 0436.58006 [12] Coti-Zelati, V.; Rabinowitz, P., Homoclinic orbits for second order Hamiltionian systems possessing superquadratic potentials, J. Amer. Math. Sor., 4, 693-727, (1991) · Zbl 0744.34045 [13] Ding, Y. H., Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8, 453-480, (2006) · Zbl 1104.70013 [14] Ding, Y. H., Variational methods for strongly indefinite problems, Interdiscip. Math. Sci., vol. 7, (2007), World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1133.49001 [15] Ding, Y. H.; Lee, Cheng, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222, 137-163, (2006) · Zbl 1090.35077 [16] Ding, Y. H.; Luan, S. X.; Willem, M., Solutions of a system of diffusion equations, J. Fixed Point Theory Appl., 2, 117-139, (2007) · Zbl 1134.35094 [17] Fermi, E.; Pasta, J.; Ulam, S., Studies of nonlinear problems, (Collected Works of E. Fermi, vol. II, (1965), University of Chicago Press), 978, (1955), also in: [18] Friesecke, G.; Wattis, J. A.D., Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161, 391-418, (1994) · Zbl 0807.35121 [19] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3, 441-472, (1998) · Zbl 0947.35061 [20] Makita, P. D., Periodic and homoclinic travelling waves in infinite lattices, Nonlinear Anal., 74, 2071-2086, (2011) · Zbl 1213.37106 [21] Pankov, A. A.; Pflüfer, K., Travelling waves in lattice dynamical systems, Math. Meth. Appl. Sci., 23, 1223-1235, (2000) · Zbl 0963.37072 [22] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., vol. 65, (1986), American Mathematical Society Providence, RI · Zbl 0609.58002 [23] Ruf, B.; Srikanth, P. N., On periodic motions of lattices of Toda type via critical point theory, Arch. Ration. Mech. Anal., 126, 369-385, (1994) · Zbl 0809.34056 [24] Smets, D.; Willem, M., Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149, 266-275, (1997) · Zbl 0889.34059 [25] Struwe, M., Variational methods. applications to nonlinear partial differential equations and Hamiltonian systems, (2000), Springer-Verlag Berlin · Zbl 0939.49001 [26] Tang, C. L.; Guo, B. L., Multiple periodic solutions for two-dimensional lattice dynamic systems, Nonlinear Anal., 65, 1306-1317, (2006) · Zbl 1107.34036 [27] Toda, M., Theory of nonlinear lattices, (1989), Springer-Verlag Berlin · Zbl 0694.70001 [28] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
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.