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August  2011, 30(3): 835-849. doi: 10.3934/dcds.2011.30.835

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

Alexander Pankov 1, and  Vassilis M. Rothos 2,

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

Received  December 2009 Revised  December 2010 Published  March 2011

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.
Keywords: Fermi-Pasta-Ulam lattice, mountain pass theorem, concentration compactness., traveling wave.
Mathematics Subject Classification: Primary: 37K60; Secondary: 82C2.
Citation: Alexander Pankov, Vassilis M. Rothos. Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 835-849. doi: 10.3934/dcds.2011.30.835
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]

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

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]

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

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