We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N gg 1$ particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $beta^{-1}$. Given a fixed ${1leq m ll N}$, we prove that the
first $m$ integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $beta$, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.
A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the descrip
tion of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit.
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $eto 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solu
tion to the hyperbolic transport equation which corresponds to $e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painleve transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using t
he Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.
In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscil
latory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painleve II equation. We prove our results using the Riemann-Hilbert approach.
