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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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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