189 research outputs found

### 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

### Critical asymptotic behavior for the Korteweg\u2013de Vries equation and in random matrix theory

We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random matrix ensembles with their coun- terparts for the Korteweg\u2013de Vries equation, emphasizing the similarities between both subjects

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