39 research outputs found

### Modulation of Camassa--Holm equation and reciprocal transformations

We derive the modulation equations or Whitham equations for the Camassa--Holm
(CH) equation. We show that the modulation equations are hyperbolic and admit
bi-Hamiltonian structure. Furthermore they are connected by a reciprocal
transformation to the modulation equations of the first negative flow of the
Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by
the Casimir of the second Poisson bracket of the KdV averaged flow. We show
that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation
equations is quite different: indeed the KdV averaged bi-Hamiltonian structure
can always be related to a semisimple Frobenius manifold while the CH one
cannot

### 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

### Integrable operators, $\overline{\partial}$-Problems, KP and NLS hierarchy

We develop the theory of integrable operators $\mathcal{K}$ acting on a
domain of the complex plane with smooth boundary in analogy with the theory of
integrable operators acting on contours of the complex plane. We show how the
resolvent operator is obtained from the solution of a
$\overline{\partial}$-problem in the complex plane. When such a
$\overline{\partial}$-problem depends on auxiliary parameters we define its
Malgrange one form in analogy with the theory of isomonodromic problems. We
show that the Malgrange one form is closed and coincides with the exterior
logarithmic differential of the Hilbert-Carleman determinant of the operator
$\mathcal{K}$. With suitable choices of the setup we show that the
Hilbert-Carleman determinant is a $\tau$-function of the Kadomtsev-Petviashvili
(KP) or nonlinear Schr\"odinger hierarchies.Comment: 24 pages, no figure

### Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

We consider the Laguerre partition function, and derive explicit generating func-tions for connected correlators with arbitrary integer powers oftraces in terms of products ofHahn polynomials. It was recently proven in [22] that correlators have a topological expansionin terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed fromour formul\ue6. As a second result we identify the Laguerre partition function with only positivecouplings and a special value of the parameter\u3b1= 121/2 with the modified GUE partitionfunction, which has recently been introduced in [28] as a generating function for Hodge inte-grals. This identification provides a direct and new link between monotone Hurwitz numbersand Hodge integrals

### Soliton shielding of the focusing nonlinear Schrodinger equation

We consider a gas of $N$ solitons of the Focusing Nonlinear Schr\"odinger
(FNLS) equation in the limit $N\to\infty$ with apoint spectrum chosen to
interpolate a given spectral soliton density over a domain of the complex
spectral plane. We call this class of initial data, deterministic soliton gas.
We show that when the domain is a disc and the soliton density is an analytic
function, then the corresponding deterministic soliton gas surprisingly yields
the one-soliton solution with point spectrum the center of the disc. We call
this effect {\it soliton shielding}. When the domain is an ellipse, the soliton
shielding reduces the spectral data to the soliton density concentrating
between the foci of the ellipse. The physical solution is asymptotically
step-like oscillatory, namely, the initial profile is a periodic elliptic
function in the negative $x$--direction while it vanishes exponentially fast in
the opposite direction.Comment: 5 pages, add references and missprint correction

### Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

We express the topological expansion of the Jacobi Unitary Ensemble in terms
of triple monotone Hurwitz numbers. This completes the combinatorial
interpretation of the topological expansion of the classical unitary invariant
matrix ensembles. We also provide effective formulae for generating functions
of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson
polynomials, generalizing the known relations between one point correlators and
Wilson polynomials.Comment: 27 page

### Shock formation in the dispersionless Kadomtsevâ€“Petviashvili equation

The dispersionless Kadomtsev-Petviashvili (dKP) equation (u(t) + uu(x))(x)= u(yy) is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation u(t) + uu(x) = 0. We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral