# American Institute of Mathematical Sciences

May 2018, 38(5): 2305-2332. doi: 10.3934/dcds.2018095

## KdV-like solitary waves in two-dimensional FPU-lattices

 1 University of Münster, Institute for Analysis and Numerics, Einsteinstr. 62, 48149 Münster, Germany 2 Technische Universität Braunschweig, Institute for Computational Mathematics, Universitätsplatz 2, 38106 Braunschweig, Germany

* Corresponding author: Michael Herrmann

Received  March 2017 Revised  January 2018 Published  March 2018

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.

Citation: Fanzhi Chen, Michael Herrmann. KdV-like solitary waves in two-dimensional FPU-lattices. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2305-2332. doi: 10.3934/dcds.2018095
##### References:
 [1] F. Chen, Wandernde Wellen in FPU-Gittern, Master Thesis, Institute for Mathematics, Saarland University, Germany, 2013. [2] F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. [3] E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. [4] 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. [5] G. Friesecke and K. Matthies, Geometric solitary waves in a 2d mass spring lattice, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 105-114. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] M. Herrmann and A. Mikikits-Leitner, KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst., 36 (2016), 2047-2067. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] G. Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866. doi: 10.1088/0951-7715/13/3/319. [22] A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. [23] 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. [24] 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. [25] 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. [26] L. Truskinovsky and A. Vainchtein, Kinetics of martensitic phase transitions: Lattice model, SIAM J. Appl. Math., 66 (2005), 533-553. doi: 10.1137/040616942. [27] 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. [28] 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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##### References:
 [1] F. Chen, Wandernde Wellen in FPU-Gittern, Master Thesis, Institute for Mathematics, Saarland University, Germany, 2013. [2] F. Chen, Traveling waves in two-dimensional FPU lattices, PhD Thesis, Institute for Applied Mathematics, University of Münster, Germany, 2017. [3] E. Fermi, J. Pasta and S. Ulam, Studis on nonlinear problems, Los Alamos Scientific Laboraty Report, 1940. [4] 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. [5] G. Friesecke and K. Matthies, Geometric solitary waves in a 2d mass spring lattice, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 105-114. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] M. Herrmann and A. Mikikits-Leitner, KdV waves in atomic chains with nonlocal interactions, Discrete Contin. Dyn. Syst., 36 (2016), 2047-2067. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] G. Iooss, Travelling waves in the Fermi-Pasta-Ulam lattice, Nonlinearity, 13 (2000), 849-866. doi: 10.1088/0951-7715/13/3/319. [22] A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London, 2005. [23] 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. [24] 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. [25] 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. [26] L. Truskinovsky and A. Vainchtein, Kinetics of martensitic phase transitions: Lattice model, SIAM J. Appl. Math., 66 (2005), 533-553. doi: 10.1137/040616942. [27] 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. [28] 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 $V_1$, 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)
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]
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 $\epsilon^2$
Left panel: Graph of the potential energy for the limit ODE (11). Right panel: The unique homoclinic solution in $\mathsf{L}_{\rm{even}}^2(\mathbb{R})$, which corresponds to the region between the two zeros of $E[W]: = \frac{d_2}{3}W^3-\frac{d_1}{2}W^2$
Left panel: The ${\mathop{\rm sinc}\nolimits}$ function. Right panel: Lower bound $\tfrac{1}{6}\min\{|z|, 2\}^2$ (dashed) and upper bound $\tfrac{1}{3}\min\{|z|, 2\}^2$ (dashed) for $S_1 = 1-{\mathop{\rm sinc}\nolimits} ^2$ (solid)
The auxiliary functions $\mu_1$ and $\mu_2$ from (52) in solid and dashed lines, respectively, for the square lattice with $\alpha = 0$
KdV-limit profiles for selected values of $\alpha$ in the square lattice, where the first and the second component of $W_0$ are represented by the solid and the dashed lines, respectively
Parameter test for the square lattice. $T(z)$ (solid) and $g(z) = 0.3\cdot (\min\{z, 2\})^2$ (dashed) for several values of $\alpha$. Assumption 7 requires $T(z)\geq g(z)$ for all $z\in \mathbb{R}$
KdV-limit profiles for selected values of $\alpha$ in the square lattice, where the first and the second component of $W_0$ are represented by the solid and the dashed lines, respectively
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
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
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