Similar oscillations on both sides of a shock. Part I. Even-odd alternative dispersions and general assignments of Fourier dispersions towards a unfication of dispersion models

We consider assigning different dispersions for different dynamical modes, particularly with the distinguishment and alternation of opposite signs for alternative Fourier components. The Korteweg-de Vries (KdV) equation with periodic boundary condition and longest-wave sinusoidal initial field, as used by N. Zabusky and M. D. Kruskal, is chosen for our case study with such alternating-dispersion of the Fourier modes of (normalized) even and odd wavenumbers. Numerical results verify the capability of our new model to produce two-sided (around the shock) oscillations, as appear on both sides of some ion-acoustic and quantum shocks, not admitted by models such as the KdV(-Burgers) equation, but also indicate even more, including singular zero-dispersion limit or non-convergence to the classical shock (described by the entropy solution), non-thermalization (of the Galerkin-truncated models) and applicability to other models (showcased by the modified KdV equation with cubic nonlinearity). A unification of various dispersive models, keeping the essential mathematical elegance (such as the variational principle and Hamiltonian formulation) of each, for phenomena with complicated dispersion relation is thus suggested with a further explicit example of two even-order dispersions (from the Hilbert transforms) extending the Benjamin-Ono model. The most general situation can be simply formulated by the introduction of the dispersive derivative, the indicator function and the Fourier transform, resulting in an integro-differential dispersion equation. Other issues such as the real-number order dispersion model and the transition from non-thermalization to thermalization and, correspondingly, from regularization to non-regularization for untruncated models are also briefly remarked

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

The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems

The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior can be described in terms of distinguished Painlev\'e transcendents, which are universal in the sense that they are, to some extent, independent of the equation and the initial data. We will consider several examples of well-known integrable equations that are expected to show this type of Painlev\'e behavior near critical points. The Riemann-Hilbert method is a useful tool to obtain rigorous results for such equations. We will explain the main lines of this method and we will discuss the universality conjecture from a Riemann-Hilbert point of view.Comment: review paper, 22 pages, 4 figure