Paper detail

Numerical study of the long wavelength limit of the Toda lattice

We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the focusing nonlinear Schrödinger equation in $1+1$ dimensions. The full Toda system is then studied for finite small values of the dispersion parameter $ε$ in the vicinity of the shocks of the dispersionless Toda equations. We determine the scaling in $ε$ of the difference between the Toda solution for small $ε$ and the singular solution of the dispersionless Toda system. In the hyperbolic case, the same scaling proportional to $ε^{2/7}$ is found as in the small dispersion limit of the Korteweg-de Vries and the defocusing nonlinear Schrödinger equations. In the elliptic case, we obtain the same scaling proportional to $ε^{2/5}$ as in the semiclassical limit for the focusing nonlinear Schrödinger equation. We also study the formation of dispersive shocks for times much larger than the break-up time in the hyperbolic case. In the elliptic case, an $L_{\infty}$ blow-up is observed instead of a dispersive shock for finite times greater than the break-up time. The $ε$-dependence of the blow-up time is determined.