Point Of Gradient Catastrophe Research Articles

Asymptotic Solutions of a Parabolic Equation Near Singular Points of A and B Types

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole–Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type \(A\) and the boundary singularities of type \(B\). The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.

Singularities of A and B types in asymptotic analysis of solutions of a parabolic equation

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases where the solution of the limit problem has a point of gradient catastrophe. The integrals determining the leading approximation correspond to the Lagrange singularity of type A3 and the boundary singularity of type B3. For another choice of the initial function, singular points corresponding to A2n+1 and B2n+1 with arbitrary n ≥ 1 are obtained.

Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlevé Equation

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials \(\pi _n(z)\) with the quartic exponential weight \(\exp \left[-N\left(\frac{1}{2}z^2 + \frac{1}{4}t z^4\right)\right]\), where \(t\in {\mathbb C}\) and \(N\rightarrow \infty \). We consider in detail the points \(t = -\frac{1}{12}\) and \(t = \frac{1}{15}\), where the recurrence coefficients of the orthogonal polynomial exhibit a behavior that involves special solutions of the first Painleve Riemann–Hilbert problem (RHP). Our principal concern is the description of their behavior in a neighborhood of special points in the \(t\)-plane (accumulating at the indicated values) where the corresponding Painleve function has poles. The nonlinear steepest descent method for the RHP is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing nonlinear Schrodinger equation near the point of gradient catastrophe that the present authors solved in 2013. Our approach is based on the technique developed in that earlier work. We also provide a numerical investigation of the “phase diagrams” in the \(t\)-plane where the recurrence coefficients exhibit different asymptotic behaviors (nonlinear Stokes’ phenomenon).

On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations.

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P_I) equation or its fourth-order analogue P_I^2. As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

AbstractThe semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$). We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$. The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$, with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.

Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.

Whitham Equations, Bergman Kernel and Lax–Levermore Minimizer

We study multiphase solutions of the Whitham equations. The Whitham equations describe the zero dispersion limit of the Cauchy problem for the Korteweg—de Vries (KdV) equation. The zero dispersion solution of the KdV equation is determined by the Lax—Levermore minimization problem. The minimizer is a measurable function on the real line. When the support of the minimizer consists of a finite number of disjoint intervals to be determined, the minimization problem can be reduced to a scalar Riemann Hilbert (RH) problem. For each fixed x and t ≥ 0, the end-points of the contour are determined by the solution of the Whitham equations. The Lax—Levermore minimizer and the solution of the Whitham equations are described in terms of a kernel related to the Bergman kernel. At t = 0 the support of the minimizer consists of one interval for any value of x, while for t > 0, the number of intervals is larger than one in some regions of the (x,t) plane where the multiphase solutions of the Whitham equations develop. The increase of the number of intervals happens whenever the solution of the Whitham equations has a point of gradient catastrophe. For a class of smooth monotonically increasing initial data, we show that the support of the Lax—Levermore minimizer increases or decreases the number of its intervals by one near each point of gradient catastrophe. This result justifies the formation and extinction of the multiphase solutions of the Whitham equations. Furthermore we characterize a class of initial data for which all the points of gradient catastrophe occur only a finite number of times and therefore the support of the Lax—Levermore minimizer consists of a finite number of disjoint intervals for any x and t ≥ 0. This corresponds to give an upper bound to the genus of the solution of the Whitham equations. Similar results are obtained for the semi-classical limit of the defocusing nonlinear Schrodinger equation.