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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

M. Herrmann, Unimodal wave trains and solitons in convex FPU chains,, preprint, ().   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

M. Herrmann, Unimodal wave trains and solitons in convex FPU chains,, preprint, ().   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

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