Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation

We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions relevant in General Relativity, which arises from the commutation of multidimensional Hamiltonian vector fields.Comment: 15 pages, submitted to Phisics Letters A. This paper replaces nlin.SI/051204

On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

We make use of the nonlinear Riemann Hilbert problem of the dispersionless Kadomtsev Petviashvili equation, i) to construct the longtime behaviour of the solutions of its Cauchy problem; ii) to characterize a class of implicit solutions; iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.Comment: 33 pages, 10 figures, few formulas update

One-loop self-energy correction in a strong binding field

A new scheme for the numerical evaluation of the one-loop self-energy correction to all orders in Z \alpha is presented. The scheme proposed inherits the attractive features of the standard potential-expansion method but yields a partial-wave expansion that converges more rapidly than in the other methods reported in the literature.Comment: 8 pages, 4 table

On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking

We study the (n+1)-dimensional generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation, a universal equation describing the propagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions, and arising in several physical contexts, like acoustics, plasma physics and hydrodynamics. For n=2, this equation is integrable, and it has been recently shown to be a prototype model equation in the description of the two dimensional wave breaking of localized initial data. We construct an exact solution of the n+1 dimensional model containing an arbitrary function of one variable, corresponding to its parabolic invariance, describing waves, constant on their paraboloidal wave front, breaking simultaneously in all points of it. Then we use such solution to build a uniform approximation of the solution of the Cauchy problem, for small and localized initial data, showing that such a small and localized initial data evolving according to the (n+1)-dimensional dKP equation break, in the long time regime, if and only if n=1,2,3; i.e., in physical space. Such a wave breaking takes place, generically, in a point of the paraboloidal wave front, and the analytic aspects of it are given explicitly in terms of the small initial data.Comment: 20 pages, 10 figures, few formulas adde

New reductions of integrable matrix PDEs: Sp(m)Sp(m)-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretization of the Manakov model (self-focusing vector NLS equation) is obtained.Comment: 33 pages; (v4) to appear in JMP; This paper states clearly that the elementary function solutions of (a vector/matrix generalization of) the derivative NLS equation can be expressed as the partial $x$-derivatives of elementary functions. Explicit soliton solutions are given in the author's talks at http://poisson.ms.u-tokyo.ac.jp/~tsuchida

Soliton dynamics in deformable nonlinear lattices

We describe wave propagation and soliton localization in photonic lattices which are induced in a nonlinear medium by an optical interference pattern, taking into account the inherent lattice deformations at the soliton location. We obtain exact analytical solutions and identify the key factors defining soliton mobility, including the effects of gap merging and lattice imbalance, underlying the differences with discrete and gap solitons in conventional photonic structures.Comment: 5 pages, 4 figure

Vector solitons in nearly-one-dimensional Bose-Einstein condensates

We derive a system of nonpolynomial Schroedinger equations (NPSEs) for one-dimensional wave functions of two components in a binary self-attractive Bose-Einstein condensate loaded in a cigar-shaped trap. The system is obtained by means of the variational approximation, starting from the coupled 3D Gross-Pitaevskii equations and assuming, as usual, the factorization of 3D wave functions. The system can be obtained in a tractable form under a natural condition of symmetry between the two species. A family of vector (two-component) soliton solutions is constructed. Collisions between orthogonal solitons (ones belonging to the different components) are investigated by means of simulations. The collisions are essentially inelastic. They result in strong excitation of intrinsic vibrations in the solitons, and create a small orthogonal component ("shadow") in each colliding soliton. The collision may initiate collapse, which depends on the mass and velocities of the solitons.Comment: 7 pages, 6 figures; Physical Review A, in pres