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April  2016, 36(4): 2047-2067. doi: 10.3934/dcds.2016.36.2047

KdV waves in atomic chains with nonlocal interactions

Michael Herrmann 1, and  Alice Mikikits-Leitner 2,

1. 

Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstraße 62, 48149 Münster, Germany
2. 

Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85747 Garching, Germany

Received  March 2015 Revised  June 2015 Published  September 2015

We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.
Keywords: KdV limit of lattice waves, Hamiltonian lattices with nonlocal coupling., Asymptotic analysis.
Mathematics Subject Classification: Primary: 37K60, 37K40; Secondary: 74H1.
Citation: Michael Herrmann, Alice Mikikits-Leitner. KdV waves in atomic chains with nonlocal interactions. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2047-2067. doi: 10.3934/dcds.2016.36.2047
References:
[1]

M. Chirilus-Bruckner, C. Chong, O. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901. doi: 10.3934/dcdss.2012.5.879.

[2]

G. Friesecke and A. Mikikits-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, 2014, To appear in J. Dyn. Diff. Equat., available via Springer online first.

[3]

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.

[4]

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.

[5]

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

[6]

G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy, Nonlinearity, 17 (2004), 229-251. doi: 10.1088/0951-7715/17/1/014.

[7]

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

[8]

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

[9]

M. Herrmann, K. Matthies, H. Schwetlick and J. Zimmer, Subsonic phase transition waves in bistable lattice models with small spinodal region, SIAM J. Math. Anal., 45 (2013), 2625-2645. doi: 10.1137/120877878.

[10]

M. Herrmann and J. D. M. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, SIAM J. Math. Anal., 42 (2010), 1483-1504. doi: 10.1137/080743147.

[11]

A. Hoffman and C. E. Wayne, Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice, Nonlinearity, 21 (2008), 2911-2947. doi: 10.1088/0951-7715/21/12/011.

[12]

A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351. doi: 10.1007/s10884-009-9134-9.

[13]

A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of Fields Inst. Commun., Springer, New York, 2013, 185-192. doi: 10.1007/978-1-4614-4523-4_7.

[14]

T. Mizumachi, $N$-soliton states of the Fermi-Pasta-Ulam lattices, SIAM J. Math. Anal., 43 (2011), 2170-2210. doi: 10.1137/100792457.

[15]

T. Mizumachi, Asymptotic stability of $N$-solitary waves of the FPU lattices, Arch. Ration. Mech. Anal., 207 (2013), 393-457. doi: 10.1007/s00205-012-0564-x.

[16]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.

[17]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 390-404.

[18]

N. J. Zabusky and M. D. Kruskal, Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

show all references

References:
[1]

M. Chirilus-Bruckner, C. Chong, O. Prill and G. Schneider, Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 879-901. doi: 10.3934/dcdss.2012.5.879.

[2]

G. Friesecke and A. Mikikits-Leitner, Cnoidal waves on Fermi-Pasta-Ulam lattices, 2014, To appear in J. Dyn. Diff. Equat., available via Springer online first.

[3]

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.

[4]

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.

[5]

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

[6]

G. Friesecke and R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy, Nonlinearity, 17 (2004), 229-251. doi: 10.1088/0951-7715/17/1/014.

[7]

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

[8]

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

[9]

M. Herrmann, K. Matthies, H. Schwetlick and J. Zimmer, Subsonic phase transition waves in bistable lattice models with small spinodal region, SIAM J. Math. Anal., 45 (2013), 2625-2645. doi: 10.1137/120877878.

[10]

M. Herrmann and J. D. M. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, SIAM J. Math. Anal., 42 (2010), 1483-1504. doi: 10.1137/080743147.

[11]

A. Hoffman and C. E. Wayne, Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice, Nonlinearity, 21 (2008), 2911-2947. doi: 10.1088/0951-7715/21/12/011.

[12]

A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351. doi: 10.1007/s10884-009-9134-9.

[13]

A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of Fields Inst. Commun., Springer, New York, 2013, 185-192. doi: 10.1007/978-1-4614-4523-4_7.

[14]

T. Mizumachi, $N$-soliton states of the Fermi-Pasta-Ulam lattices, SIAM J. Math. Anal., 43 (2011), 2170-2210. doi: 10.1137/100792457.

[15]

T. Mizumachi, Asymptotic stability of $N$-solitary waves of the FPU lattices, Arch. Ration. Mech. Anal., 207 (2013), 393-457. doi: 10.1007/s00205-012-0564-x.

[16]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.

[17]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 390-404.

[18]

N. J. Zabusky and M. D. Kruskal, Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

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