We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles.
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Figure 1. Cartoon of the square lattice. The vertical and the horizontal springs are described by the potential $V_1$, while all diagonal springs correspond to $V_2$. Center panel: Triangle lattice with identical springs and single potential function $V$. Right panel: Cartoon of the diamond lattice, which can be regarded as a square lattice without horizontal springs. The lattices have different symmetry groups and produce different coupling terms in the advance-delay-differential equation for lattice waves, see (3)
Figure 2. Left panel: Numerical approximations of $W_{\epsilon, 1}(\xi)$ (black) and $W_{\epsilon, 2}(\xi)$ (gray) for the square lattice with angle $\alpha = \frac{\pi}{8}$ and positive $\epsilon$. Right panel: The plot $W_{\epsilon, 2} $ versus $W_{\epsilon, 1}$ reveals that the two components of $W_{\epsilon}$ are not proportional, which means that $W_{\epsilon}$ is not unidirectional and our problem cannot be reduced to a one-dimensional one as in [5]
Figure 3. Left panel. Scaled velocity profile $W_{\epsilon}$ as function of $\xi$. Right panel. Cartoon of the atomistic velocities in the corresponding KdV wave, where $\zeta = \kappa_1i+\kappa_2j-c_{\epsilon}t$ denotes the phase with respect to the original variables. The unscaled profile is obtained from the scaled one by stretching the argument by ${1}/{\epsilon}$ and pressing the amplitude by $\epsilon^2$
Figure 10. The plots from Figure 7 for the diamond lattice. In the graph of $\lambda$ we find jumps at multiples of $\pi$, which is consistent with the fact that the lattice is symmetric with respect to the horizontal direction. For $\alpha = 0$ no KdV wave exists due to this singularity
Figure 11. The plots from Figure 8 for the diamond lattice
Figure 12. The plots from Figure 9 for the diamond lattice
Figure 13. The plots from Figure 7 for the triangle lattice
Figure 14. The plots from Figure 8 for the triangle lattice
Figure 15. The plots from Figure 9 for the triangle lattice
F. Chen,
Wandernde Wellen in FPU-Gittern, Master Thesis, Institute for Mathematics, Saarland University, Germany, 2013.
![]() |
|
F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017.
![]() |
|
E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940.
![]() |
|
A.-M. Filip
and S. Venakides
, Existence and modulation of traveling waves in particle chains, Comm. Pure Appl. Math., 52 (1999)
, 693-735.
doi: 10.1002/(SICI)1097-0312(199906)52:6<693::AID-CPA2>3.0.CO;2-9.![]() ![]() ![]() |
|
G. Friesecke
and K. Matthies
, Geometric solitary waves in a 2d mass spring lattice, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003)
, 105-114.
![]() ![]() |
|
G. Friesecke
and A. Mikikits-Leitner
, Cnoidal waves on Fermi-Pasta-Ulam lattices, J. Dynam. Differential Equations, 27 (2015)
, 627-652.
doi: 10.1007/s10884-013-9343-0.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. Friesecke
and J. A. D. Wattis
, Existence theorem for solitary waves on lattices, Comm. Math. Phys., 161 (1994)
, 391-418.
doi: 10.1007/BF02099784.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Herrmann
and A. Mikikits-Leitner
, KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst., 36 (2016)
, 2047-2067.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the FermiPasta-Ulam model near the KdV limit, in Infinite dimensional dynamical systems, vol. 64 of
Fields Inst. Commun., Springer, New York, 2013,185–192.
![]() ![]() |
|
A. Hoffman
and J. D. Wright
, Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Phys. D, 358 (2017)
, 33-59.
doi: 10.1016/j.physd.2017.07.004.![]() ![]() ![]() |
|
G. Iooss
, Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000)
, 849-866.
doi: 10.1088/0951-7715/13/3/319.![]() ![]() ![]() |
|
A. Pankov,
Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005.
![]() ![]() |
|
G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit for the Fermi-Pasta-Ulam model, in International Conference on Differential
Equations, vol. 1, World Scientific, 2000,390-404.
![]() ![]() |
|
H. Schwetlick
and J. Zimmer
, Kinetic relations for a lattice model of phase transitions, Arch. Rational Mech. Anal., 206 (2012)
, 707-724.
doi: 10.1007/s00205-012-0566-8.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
L. Truskinovsky
and A. Vainchtein
, Kinetics of martensitic phase transitions: Lattice model, SIAM J. Appl. Math., 66 (2005)
, 533-553.
doi: 10.1137/040616942.![]() ![]() ![]() |
|
A. Vainchtein, Y. Starosvetsky, J. Wright and R. Perline, Solitary waves in diatomic chains,
Phys. Rev. E, 93 (2016), 042210.
doi: 10.1103/PhysRevE.93.042210.![]() ![]() |
|
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.![]() ![]() |
Cartoon of the square lattice. The vertical and the horizontal springs are described by the potential
Left panel: Numerical approximations of
Left panel. Scaled velocity profile
Left panel: Graph of the potential energy for the limit ODE (11). Right panel: The unique homoclinic solution in
Left panel: The
The auxiliary functions
KdV-limit profiles for selected values of
Parameter test for the square lattice.
KdV-limit profiles for selected values of
The plots from Figure 7 for the diamond lattice. In the graph of
The plots from Figure 8 for the diamond lattice
The plots from Figure 9 for the diamond lattice
The plots from Figure 7 for the triangle lattice
The plots from Figure 8 for the triangle lattice
The plots from Figure 9 for the triangle lattice