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

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