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On the critical nongauge invariant nonlinear Schrödinger equation
Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities
1. | Mathematics Department, Morgan State University, 1700 E Cold Spring Lane, Baltimore, MD 21251, United States |
2. | School of Mathematics, Physics and Computational Sciences, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece |
References:
[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. Herrmann, Unimodal wave trains and solitons in convex FPU chains,, preprint, ().
|
[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] |
show all references
References:
[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. Herrmann, Unimodal wave trains and solitons in convex FPU chains,, preprint, ().
|
[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] |
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