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November  2019, 12(7): 2097-2113. doi: 10.3934/dcdss.2019135

Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction

Mathematics Department, Morgan State University, Baltimore, MD 21251, USA

Dedicated to Professor Norman Dancer on the occasion of his 70th birthday

Received  May 2018 Revised  June 2018 Published  December 2018

Fund Project: The author is supported by Simons Foundation, award 410289.

The paper is devoted to traveling waves in FPU type particle chains assuming that each particle interacts with several neighbors on both sides. Making use of variational techniques, we prove that under natural assumptions there exist monotone traveling waves with periodic velocity profile (periodic waves) as well as waves with localized velocity profile (solitary waves). In fact, we obtain periodic waves by means of a suitable version of the Mountain Pass Theorem. Then we get solitary waves in the long wave length limit.

Citation: 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
References:
[1]

P. W. Bates and C. Zhang, Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction, Discrete Contin. Dyn. Syst. A, 16 (2006), 235-252.  doi: 10.3934/dcds.2006.16.235.

[2]

H. BerestyckiJ. 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.

[3]

E. Dumas and D. Pelinovsky, Justification of the log-KdV equation in granular chains: The case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.  doi: 10.1137/140969270.

[4]

T. E. Faver and J. D. Wright, Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity, SIAM J Math. Anal., 50 (2018), 182-250.  doi: 10.1137/15M1046836.

[5]

M. Fečkan and V. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Appl. Anal., 89 (2010), 1387-1411.  doi: 10.1080/00036810903208130.

[6]

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

[7]

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.

[8]

G. Friesecke and A. Mikikis-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dyn. Diff. Equat., 27 (2015), 627-652.  doi: 10.1007/s10884-013-9343-0.

[9]

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

[10]

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

[11]

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

[12]

G. Friesecke and R. L. 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.

[13]

J. GaisonS. MoskowJ. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.  doi: 10.1137/130941638.

[14]

G. Galavotti, The Fermi-Pasta-Ulam Problem, A Status Report, Springer, Berlin, 2008. doi: 10.1007/978-3-540-72995-2.

[15]

L. Gasinski and N. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.

[16]

M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh A, 140 (2010), 753-785.  doi: 10.1017/S0308210509000146.

[17]

M. Herrmann and A. Mikikis-Leitner, KdV waves in atomic chains with nonlocal interaction, Discrete Contin. Dyn. Syst. A, 36 (2016), 2047-2067.  doi: 10.3934/dcds.2016.36.2047.

[18]

G. Iooss, Traveling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866.  doi: 10.1088/0951-7715/13/3/319.

[19]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[20]

A. Khan and D. Pelinovsky, Long-time stability of small FPU solitary waves, Discrete Contin. Dyn. Syst. A, 37 (2017), 2065-2075.  doi: 10.3934/dcds.2017088.

[21]

Y. LiZ.-Q. Wang and J. Zheng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré, Anal. Non Lin., 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.

[22]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, Ⅱ, Ann. Inst. H. Poincaré, Anal. Non Lin., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[23]

S. Liu, On superlinear problems without the Ambrosetti-Rabinowitz condition, Nonlin. Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.

[24]

D. Motreanu, V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[25]

A. Pankov, Periodic nonlinear Schrödinger equation with applications to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[26] A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005.  doi: 10.1142/9781860947216.
[27]

A. Pankov, Nonlinear Schrödinger equations on periodic metric graphs, Discrete Contin. Dyn. Syst. A, 38 (2018), 697-714.  doi: 10.3934/dcds.2018030.

[28]

A. Pankov and K. Pflüger, Traveling waves in lattice dynamical systems, Math. Meth. Appl. Sci., 23 (2000), 1223-1235.  doi: 10.1002/1099-1476(20000925)23:14<1223::AID-MMA162>3.0.CO;2-Y.

[29]

A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discrete Contin. Dyn. Syst. A, 30 (2011), 835-849.  doi: 10.3934/dcds.2011.30.835.

[30]

Z. Rapti, Multibreather stabilityin discrete Klein-Gordon equations: beyond nearest neighbor interaction, Phys. Lett. A, 377 (2013), 1543-1553.  doi: 10.1016/j.physleta.2013.04.035.

[31]

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

[32]

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.

[33] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, omerville, MA, 2010. 
[34]

J. A. D. Wattis, Approximations to solitary waves on lattices: Ⅲ. The monoatomic lattice with second-neighbour interaction, J. Phys. A: Math. Gen., 29 (1996), 8139-8157.  doi: 10.1088/0305-4470/29/24/035.

show all references

References:
[1]

P. W. Bates and C. Zhang, Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction, Discrete Contin. Dyn. Syst. A, 16 (2006), 235-252.  doi: 10.3934/dcds.2006.16.235.

[2]

H. BerestyckiJ. 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.

[3]

E. Dumas and D. Pelinovsky, Justification of the log-KdV equation in granular chains: The case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.  doi: 10.1137/140969270.

[4]

T. E. Faver and J. D. Wright, Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity, SIAM J Math. Anal., 50 (2018), 182-250.  doi: 10.1137/15M1046836.

[5]

M. Fečkan and V. Rothos, Traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions, Appl. Anal., 89 (2010), 1387-1411.  doi: 10.1080/00036810903208130.

[6]

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

[7]

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.

[8]

G. Friesecke and A. Mikikis-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dyn. Diff. Equat., 27 (2015), 627-652.  doi: 10.1007/s10884-013-9343-0.

[9]

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

[10]

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

[11]

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

[12]

G. Friesecke and R. L. 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.

[13]

J. GaisonS. MoskowJ. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.  doi: 10.1137/130941638.

[14]

G. Galavotti, The Fermi-Pasta-Ulam Problem, A Status Report, Springer, Berlin, 2008. doi: 10.1007/978-3-540-72995-2.

[15]

L. Gasinski and N. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.

[16]

M. Herrmann, Unimodal wavetrains and solitons in convex Fermi-Pasta-Ulam chains, Proc. Roy. Soc. Edinburgh A, 140 (2010), 753-785.  doi: 10.1017/S0308210509000146.

[17]

M. Herrmann and A. Mikikis-Leitner, KdV waves in atomic chains with nonlocal interaction, Discrete Contin. Dyn. Syst. A, 36 (2016), 2047-2067.  doi: 10.3934/dcds.2016.36.2047.

[18]

G. Iooss, Traveling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866.  doi: 10.1088/0951-7715/13/3/319.

[19]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[20]

A. Khan and D. Pelinovsky, Long-time stability of small FPU solitary waves, Discrete Contin. Dyn. Syst. A, 37 (2017), 2065-2075.  doi: 10.3934/dcds.2017088.

[21]

Y. LiZ.-Q. Wang and J. Zheng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré, Anal. Non Lin., 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.

[22]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, Ⅱ, Ann. Inst. H. Poincaré, Anal. Non Lin., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[23]

S. Liu, On superlinear problems without the Ambrosetti-Rabinowitz condition, Nonlin. Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.

[24]

D. Motreanu, V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[25]

A. Pankov, Periodic nonlinear Schrödinger equation with applications to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[26] A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005.  doi: 10.1142/9781860947216.
[27]

A. Pankov, Nonlinear Schrödinger equations on periodic metric graphs, Discrete Contin. Dyn. Syst. A, 38 (2018), 697-714.  doi: 10.3934/dcds.2018030.

[28]

A. Pankov and K. Pflüger, Traveling waves in lattice dynamical systems, Math. Meth. Appl. Sci., 23 (2000), 1223-1235.  doi: 10.1002/1099-1476(20000925)23:14<1223::AID-MMA162>3.0.CO;2-Y.

[29]

A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discrete Contin. Dyn. Syst. A, 30 (2011), 835-849.  doi: 10.3934/dcds.2011.30.835.

[30]

Z. Rapti, Multibreather stabilityin discrete Klein-Gordon equations: beyond nearest neighbor interaction, Phys. Lett. A, 377 (2013), 1543-1553.  doi: 10.1016/j.physleta.2013.04.035.

[31]

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

[32]

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.

[33] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, omerville, MA, 2010. 
[34]

J. A. D. Wattis, Approximations to solitary waves on lattices: Ⅲ. The monoatomic lattice with second-neighbour interaction, J. Phys. A: Math. Gen., 29 (1996), 8139-8157.  doi: 10.1088/0305-4470/29/24/035.

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