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

  • * Corresponding author: Michael Herrmann

    * Corresponding author: Michael Herrmann
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  • 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.

    Mathematics Subject Classification: Primary: 37K60; Secondary: 37K40, 74H10.


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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 4.  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 $E[W]: = \frac{d_2}{3}W^3-\frac{d_1}{2}W^2$

    Figure 5.  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)

    Figure 6.  The auxiliary functions $\mu_1$ and $\mu_2$ from (52) in solid and dashed lines, respectively, for the square lattice with $\alpha = 0$

    Figure 7.  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

    Figure 8.  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 $z\in \mathbb{R}$

    Figure 9.  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

    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

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