From the solution of the Tsarev system to the solution of the Whitham equations

We study the Cauchy problem for the Whitham modulation equations for monotone increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2,... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so called hodograph transform introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each $g>0$, we construct the unique solution of the Tsarev system, which matches the genus $g+1$ and $g-1$ solutions on the transition boundaries. Next we characterize initial data such that the solution of the Whitham equations has genus $g\leq N$, $N>0$, for all real $t\geq 0$ and $x$.Comment: Latex2e 41 pages, 5 figure

The KdV hierarchy: universality and a Painleve transcendent

We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where \e\to 0. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution 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 Painlev\'e 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

Numerical study of a multiscale expansion of KdV and Camassa-Holm equation

We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equationComment: 17 pages, 13 figure

On the long-time asymptotic behavior of the modified korteweg-de vries equation with step-like initial data

We study the long-time asymptotic behavior of the solution q(x; t), of the modified Korteweg-de Vries equation (MKdV) with step-like initial datum q(x, 0). For the exact step initial data q(x,0)=c_+ for x>0 and q(x,0)=c_- for x<0, the solution develops an oscillatory region called the dispersive shock wave region that connects the two constant regions c_- and c_+ at x=-infinity and x=+infinity. We show that the dispersive shock wave is described by a modulated periodic traveling wave solution of the MKdV equation where the modulation parameters evolve according to a Whitham modulation equation. The oscillatory region is expanding within a cone in the (x,t) plane. For step-like initial data we show that the solution decomposes for long times into three main regions: (1) a region where solitons and breathers travel with positive velocities on a constant background c_+, (2) an expanding oscillatory region (that generically contains breathers); (3) a region of breathers traveling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the exact step initial data. The phase shift depends on the solitons, the breathers, and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers, and radiation

Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular β -Ensemble and Double Confluent Heun Equation

We consider the discrete defocusing nonlinear Schrodinger equation in its integrable version, which is called defocusing Ablowitz-Ladik lattice. We consider periodic boundary conditions with period N and initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular beta-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation