Existence of exponentially spatially localized breather solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach. (English) Zbl 1493.37088
The authors consider an infinite particle system on the line interacting with a nearest neighbor potential \(V\) obeying a number of assumptions. The main goal of this work is to establish the existence of exponentially localized breather solutions. To achieve this goal, the problem is reformulated as a fixed point equation in an appropriate weighted sequence space. Then the authors show that Schauder’s fixed point theorem can be applied and furthermore, they obtain energy bounds for the solutions.
Reviewer: Luen-Chau Li (University Park)
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 47H10 | Fixed-point theorems |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
nonlinear lattice; nearest neighbor interaction; breather solutions; Schauder’s fixed point theoremReferences:
| [1] | Braun, O. M.; Kivshar, Y. S., The Frenkel-Kontorova Model: Concepts, Methods, and Applications (2004), Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 1140.82001 |
| [2] | Flach, S.; Gorbach, A. V., Discrete breathers—Advances in theory and applications, Phys. Rep., 467, 1-116 (2008) · doi:10.1016/j.physrep.2008.05.002 |
| [3] | Hennig, D.; Tsironis, G. P., Wave transmission in nonlinear lattices, Phys. Rep., 307, 333-432 (1999) · doi:10.1016/s0370-1573(98)00025-8 |
| [4] | Kevredikis, P. G., The Nonlinear Discrete Schrödinger Equation: Mathematical Analysis, Numerial Computations, and Physical Perspectives (2009), Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 1169.35004 |
| [5] | Eilbeck, J. C.; Johansson, M.; Vázquez, L.; MacKay, R. S.; Zorzano, M. P., The discrete nonlinear Schrödinger equation-20 years on, Localization and Energy Transfer in Nonlinear Systems, 44-67 (2003), World Scientific: World Scientific, Singapore · Zbl 1032.81007 |
| [6] | Frenkel, Y.; Kontorova, T., On the theory of plastic defomation and twinning, J. Phys., 1, 137-149 (1939) · JFM 65.1469.02 |
| [7] | Berman, G. P.; Izrailev, F. M., The Fermi-Pasta-Ulam problem: Fifty years of progress, Chaos, 15, 015104 (2005) · Zbl 1080.37077 · doi:10.1063/1.1855036 |
| [8] | Dauxois, T.; Peyrard, M.; Ruffo, S., The Fermi-Pasta-Ulam ‘numerical experiment’: History and pedagogical perspectives, Eur. J. Phys., 26, S3-S11 (2005) · doi:10.1088/0143-0807/26/5/s01 |
| [9] | Dauxois, T., Fermi, Pasta, Ulam and a mysterious lady, Phys. Today, 61, 1, 55-57 (2008) · doi:10.1063/1.2835154 |
| [10] | Toda, M., Theory of Nonlinear Lattices (1989), Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 0694.70001 |
| [11] | Ablowitz, M. J.; Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys., 17, 1011-1018 (1976) · Zbl 0322.42014 · doi:10.1063/1.523009 |
| [12] | Ablowitz, M. J.; Ladik, J. F., A nonlinear difference scheme and inverse scattering, Stud. Appl. Math., 55, 213-229 (1976) · Zbl 0338.35002 · doi:10.1002/sapm1976553213 |
| [13] | MacKay, R. S.; Aubry, S., Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7, 1623-1643 (1994) · Zbl 0811.70017 · doi:10.1088/0951-7715/7/6/006 |
| [14] | Bambusi, D., Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 9, 433-457 (1996) · Zbl 0925.70230 · doi:10.1088/0951-7715/9/2/009 |
| [15] | Bambusi, D., Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators, Commun. Math. Phys., 324, 515-547 (2013) · Zbl 1286.37064 · doi:10.1007/s00220-013-1817-8 |
| [16] | Bambusi, D.; Paleari, S.; Penati, T., Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein-Gordon lattices, Appl. Anal., 89, 1313-1334 (2010) · Zbl 1203.37121 · doi:10.1080/00036811003627518 |
| [17] | Komech, A. I.; Kopylova, E. A.; Kunze, M., Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations, Appl. Anal., 85, 1487-1508 (2006) · Zbl 1121.39015 · doi:10.1080/00036810601074321 |
| [18] | Paleari, S.; Penati, T., Long time stability of small amplitude breathers in a mixed FPU-KG model, Z. Angew. Math. Phys., 67, 148 (2016) · Zbl 1368.37071 · doi:10.1007/s00033-016-0738-8 |
| [19] | Pelinovsky, D.; Sakovich, A., Multi-site breathers in Klein-Gordon lattices: Stability, resonances and bifurcations, Nonlinearity, 25, 3423-3451 (2012) · Zbl 1323.37044 · doi:10.1088/0951-7715/25/12/3423 |
| [20] | Koukouloyannis, V.; Kevrekidis, P. G., On the stability of multibreathers in Klein-Gordon chains, Nonlinearity, 22, 2269-2285 (2009) · Zbl 1179.37103 · doi:10.1088/0951-7715/22/9/011 |
| [21] | Livi, R.; Spicci, M.; MacKay, R. S., Breathers on a diatomic FPU chain, Nonlinearity, 10, 1421-1434 (1997) · Zbl 0917.70022 · doi:10.1088/0951-7715/10/6/003 |
| [22] | James, G., Existence of breathers on FPU lattices, C. R. Acad. Sci., Ser I, 332, 581-586 (2001) · Zbl 1116.37303 · doi:10.1016/s0764-4442(01)01894-8 |
| [23] | Iooss, G.; James, G., Localized waves in nonlinear oscillator chains, Chaos, 15, 015113 (2005) · Zbl 1080.37080 · doi:10.1063/1.1836151 |
| [24] | Mirollo, R.; Rosen, N., Existence, uniqueness, and non-uniqueness of single-wave-form solutions to Josephson junction systems, SIAM J. Appl. Math., 60, 1471-1501 (2000) · Zbl 0958.34034 · doi:10.1137/s003613999834385x |
| [25] | Friesecke, G.; Wattis, J. A. D., Existence theorem for travelling waves on lattices, Commun. Math. Phys., 161, 391-418 (1994) · Zbl 0807.35121 · doi:10.1007/bf02099784 |
| [26] | Willem, M.; Smets, D., Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149, 266-275 (1997) · Zbl 0889.34059 · doi:10.1006/jfan.1996.3121 |
| [27] | Weinstein, M. I., Excitation thresholds for nonlinear localised modes on lattices, Nonlinearity, 12, 673-691 (1999) · Zbl 0984.35147 · doi:10.1088/0951-7715/12/3/314 |
| [28] | Pankov, A., Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19, 27-40 (2006) · Zbl 1220.35163 · doi:10.1088/0951-7715/19/1/002 |
| [29] | Zhang, G., Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 50, 013105 (2009) · Zbl 1200.37072 · doi:10.1063/1.3036182 |
| [30] | Pankov, A., Solitary waves on nonlocal Fermi-Pasta-Ulam lattices: Exponential localisation, Nonlinear Anal.: Real World Appl., 50, 603-612 (2019) · Zbl 1510.34031 · doi:10.1016/j.nonrwa.2019.06.007 |
| [31] | Cuevas, J.; Eilbeck, J. C.; Karachalios, N. I., Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity, Discrete Contin. Dyn. Syst., 21, 445-475 (2008) · Zbl 1148.37039 · doi:10.3934/dcds.2008.21.445 |
| [32] | Cuevas, J.; Karachalios, N. I.; Palmero, F., Lower and upper estimates on the excitation threshold for breathers in discrete nonlinear Schrödinger lattices, J. Math. Phys., 50, 11, 112705 (2009) · Zbl 1304.34022 · doi:10.1063/1.3263142 |
| [33] | Karachalios, N. I.; Sánchez-Rey, B.; Kevrekidis, P. G.; Cuevas, J., Breathers for the discrete nonlinear Schrödinger equation with nonlinear hopping, J. Nonlinear Sci., 23, 2, 205-239 (2013) · Zbl 1323.37047 · doi:10.1007/s00332-012-9149-y |
| [34] | Pankov, A. A., Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices (2005), Imperial College Press: Imperial College Press, London · Zbl 1088.35001 |
| [35] | Hennig, D., Localised time-periodic solutions of discrete nonlinear Klein-Gordon systems with convex on-site potentials, J. Fixed Point Theory Appl., 23, 31-38 (2021) · Zbl 1476.34049 · doi:10.1007/s11784-021-00866-0 |
| [36] | Schauder, J., Der fixpunktsatz in funktionalraümen, Stud. Math., 2, 171-180 (1930) · JFM 56.0355.01 · doi:10.4064/sm-2-1-171-180 |
| [37] | Gendelman, O. V.; Savin, A. V., Normal heat conductivity of the one-dimensional lattice with periodic potential of nearest-neighbor interaction, Phys. Rev. Lett., 84, 2381 (2000) · doi:10.1103/physrevlett.84.2381 |
| [38] | Lepri, S.; Livi, R.; Politi, A., Thermal conduction in classical low-dimensional lattices, Phys. Rep., 377, 1-80 (2003) · doi:10.1016/S0370-1573(02)00558-6 |
| [39] | Karachalios, N. I.; Yannacopoulos, A. N., Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differ. Equations, 217, 88-123 (2005) · Zbl 1084.35092 · doi:10.1016/j.jde.2005.06.002 |
| [40] | Hennig, D.; Karachalios, N. I., Dynamics of nonlocal and local discrete Ginzburg-Landau equations: Global attractors and their congruence, Nonlinear Anal., 215, 112647 (2022) · Zbl 1483.34022 · doi:10.1016/j.na.2021.112647 |
| [41] | Wattis, J. A. D.; Pickering, A.; Gordoa, P. R., Combined breathing-kink modes in the FPU lattice, Physica D, 240, 547-553 (2011) · Zbl 1208.37042 · doi:10.1016/j.physd.2010.11.002 |
| [42] | Arioli, G.; Koch, H., Traveling wave solutions for the FPU chain: A constructive approach, Nonlinearity, 33, 1705-1722 (2020) · Zbl 1444.34077 · doi:10.1088/1361-6544/ab6a78 |
| [43] | Cretegny, T.; Dauxois, T.; Ruffo, S.; Torcini, A., Localization and equipartition of energy in β-FPU chain: Chaotic breathers, Physica D, 121, 109-126 (1998) · doi:10.1016/s0167-2789(98)00107-9 |
| [44] | Ankiewicz, A.; Akhmediev, N.; Soto-Crespo, J. M., Discrete rogue waves of the Ablowitz-Ladik and Hirota equations, Phys. Rev. E, 82, 026602 (2010) · doi:10.1103/PhysRevE.82.026602 |
| [45] | Akhmediev, N.; Ankiewicz, A., Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation, Phys. Rev. E, 83, 046603 (2011) · doi:10.1103/PhysRevE.83.046603 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
