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

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

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.
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.

[1]

Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135

[2]

Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710

[3]

Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991

[4]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[5]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[6]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[7]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[8]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[9]

E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure and Applied Analysis, 2016, 15 (2) : 457-475. doi: 10.3934/cpaa.2016.15.457

[10]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001

[11]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[12]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111

[13]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051

[14]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

[15]

Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975

[16]

Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054

[17]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

[18]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[19]

Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537

[20]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]