Three Dimensional Reductions of Four-Dimensional Quasilinear Systems

In this paper we show that integrable four dimensional linearly degenerate equations of second order possess infinitely many three dimensional hydrodynamic reductions. Furthermore, they are equipped infinitely many conservation laws and higher commuting flows. We show that the dispersionless limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking Soliton equations introduced by O.I. Bogoyavlenski) are one and two component reductions (respectively) of one of these four dimensional linearly degenerate equations

Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions and their dispersive deformations

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. We classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions. Furthermore, we develop a theory of integrable dispersive deformations of these Hamiltonian systems following a scheme similar to that proposed by Dubrovin and his collaborators in 1 + 1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order

Information Modelling of Two-Dimensional Optical Parameters Measurement

A method for measurement and visualization of the complex transmission coefficient of 2-D micro- objects is proposed. The method is based on calculation of the transmission coefficient from the diffraction pattern and the illumination aperture function for monochromatic light. A phase-stepping method was used for diffracted light phase determination

Dispersionful Version of WDVV Associativity Equations

Non UBCUnreviewedAuthor affiliation: University of GoettingenPostdoctora

Multi-domain spectral approach with Sommerfeld condition for the Maxwell equations

International audienceWe present a multi-domain spectral approach with an exterior compactified domain for the Maxwell equations for monochromatic fields. The Sommerfeld radiation condition is imposed exactly at infinity being a finite point on the numerical grid. As an example, axisymmetric situations in spherical and prolate spheroidal coordinates are discussed, as well as the interaction of a radiating dipole with a nano-particle

Numerical scattering for the defocusing Davey–Stewartson II equation for initial data with compact support

International audienceIn this work we present spectral algorithms for the numerical scattering for the defocusing Davey–Stewartson (DS) II equation with initial data having compact support on a disk, i.e. for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solution to the d-bar system and applies the CGO conditions on the latter. This is especially efficient for small values of the modulus of the spectral parameter k. The second approach uses a fixed point iteration on a reformulated d-bar system containing the spectral parameter explicitly, a price paid to have simpler asymptotics. The approaches are illustrated for the example of the characteristic function of the disk and are shown to exhibit spectral convergence, i.e. an exponential decay of the numerical error with the number of collocation points. An asymptotic formula for large is given for the reflection coefficient

Numerical study of break-up in solutions to the dispersionless Kadomtsev–Petviashvili equation

International audienceWe present a numerical approach to study solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation on R x T. The dependence on the coordinate x is considered on the compactified real line, and the dependence on the coordinate y is assumed to be periodic. Critical behavior, the formation of a shock in the solutions, is of special interest. The latter permits the numerical study of Dubrovin's universality conjecture on the break-up of solutions to the Kadomtsev-Petviashvili equation. Examples from a previous paper on dKP solutions studied numerically on T-2 are addressed, and the influence of the periodicity or not in the x-coordinate on the break-up is studied

Spectral approach to Korteweg-de Vries equations on the compactified real line

International audienceWe present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit RungeKutta method of fourth order. Several examples are discussed: initial data bounded but not vanishing at infinity as well as data not satisfying the Faddeev condition, i.e. with a slow decay towards infinity. (C) 2022 IMACS

Numerical Approach to Painlevé Transcendents on Unbounded Domains

International audienceA multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation