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Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities

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  • We prove the existence of periodic and solitary traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. The approach is based on variational techniques and concentration compactness.
    Mathematics Subject Classification: Primary: 37K60; Secondary: 82C20.

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  • [1]

    H. Berestycki, I, Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonlin. Differ. Equat. Appl., 2 (1995), 553-572.doi: 10.1007/BF01210623.

    [2]

    B. Bidégary-Fesquet and J.-C. Saut, On the propagation of an optical wave in a photorefrctive medium, Math. Models Methods Appl. Sci., 17 (2007), 1883-1904.doi: 10.1142/S0218202507002509.

    [3]

    E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, Los Alamos Sci. Lab. Rept., LA-1940 (1955); Reprinted in [10].

    [4]

    G. Iooss, Travelling waves in a chain of coupled nonlinear oscillators, Nonlinearity, 13 (2000), 849-866.doi: 10.1088/0951-7715/13/3/319.

    [5]

    G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.doi: 10.1007/BF02099784.

    [6]

    G. Friesecke and R. Pego, Solitary waves on FPU lattices, I: Qualitative properties, renormalization and continuous limit, Nonlinearity, 12 (1999), 1601-1627.doi: 10.1088/0951-7715/12/6/311.

    [7]

    G. Friesecke and R. Pego, Solitary waves on FPU lattices, II: Linear implies nonlinear stability, Nonlinearity, 15 (2002), 1343-1359.doi: 10.1088/0951-7715/15/4/317.

    [8]

    G. Friesecke and R. Pego, Solitary waves on FPU lattices, III: Howland-type Floquet theory, Nonlinearity, 17 (2004), 207-227.doi: 10.1088/0951-7715/17/1/013.

    [9]

    G. Friesecke and R. Pego, Solitary waves on FPU lattices, IV: Proof of stability at low energy, Nonlinearity, 17 (2004), 229-251.doi: 10.1088/0951-7715/17/1/014.

    [10]

    G. Gallavotti (Ed.), "The Fermi-Pasta-Ulam Problem, A Status Report," Springer, Berlin, 2008.

    [11]

    M. HerrmannUnimodal wave trains and solitons in convex FPU chains, preprint, arXiv:0901.3736.

    [12]

    P.-L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré, Anal. Nonlin., 1 (1984), 223-283.

    [13]

    A. Pankov, "Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices," Imperial College Press, London, 2005.doi: 10.1142/9781860947216.

    [14]

    A. Pankov and V. M. Rothos, Periodic and decaying solutions in DNLS with saturable nonlinearity, Proc. Roy. Soc. London, A, 464 (2008), 3219-3236.doi: 10.1098/rspa.2008.0255.

    [15]

    P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," Amer. Math. Soc., Providence, R. I., 1986.

    [16]

    D. Smets and M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149 (1997), 266-275.doi: 10.1006/jfan.1996.3121.

    [17]

    C. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rat. Mech. Anal., 125 (1993), 145-200.doi: 10.1007/BF00376812.

    [18]

    H. Schwetlick and J. Zimmer, Solitary waves in nonconvex FPU lattices, J. Nonlin. Sci., 17 (2007), 1-12.doi: 10.1007/s00332-005-0735-0.

    [19]

    M. Toda, "Theory of Nonlinear Lattices," Springer, Berlin, 1989.

    [20]

    C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder and C. Debz, Spatial optical $(2+1)$-dimensional scalar- and vector-solitons, Ann. Phys. (Leiptzig), 11 (2002), 573-629.doi: 10.1002/1521-3889(200209)11:8<573::AID-ANDP573>3.0.CO;2-G.

    [21]

    M. Willem, "Minimax Methods," Birkhäuser, Boston, 1996.

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