The KP equation, which is an integrable nonlinear evolution equation in 2 + 1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the...

In this paper we propose a generalization of multiscale finite element methods (MsFEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the oversampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we des...

In previous work, Doebner and I introduced a group of nonlinear gauge transformations for quantum mechanics, acting in a certain family of nonlinear Schrödinger equations. Here the idea for a further generalization is presented briefly. It makes possible the treatment of the logarithmic amplitude and the phase of the wave function on an equal footing, suggesting a more radical reinterpretation ...

New manifestly gauge-invariant forms of two-dimensional generalized dispersive long wave and NizhnikVeselov-Novikov systems of integrable nonlinear equations are presented. It is shown how in different gauges from such forms famous two-dimensional generalization of dispersive long wave system of equations, Nizhnik-Veselov-Novikov and modified Nizhnik-Veselov-Novikov equations and other known an...

The Schrödinger equation with the nonlinear term −b(ln |Ψ|2)Ψ is derived by the natural generalization of the hydrodynamical model of quantum mechanics. The nonlinear term appears to be logically necessary because it enables explanation of the classical limit of the wave function, the collaps of the wave function and solves the Schrödinger cat paradox.

The dynamics of recombination in genetics leads to an interesting nonlinear differential equation, which has a natural generalization to a measure valued version. The latter can be solved explicitly under rather general circumstances. It admits a closed formula for the semigroup of nonlinear positive operators that emerges from the forward flow and is, in general, embedded in a multi-parameter ...

In this paper, a normal form for nonlinear control systems with a single-input by studying its k-jet is found. It is a generalization of linear Brunovsky form to nonlinear control systems. A set of invari-ants which characterizes the homogeneous parts of a control system is also found. Then the problem of convergence and uniqueness of the normal forms is studied.

The behaviour of a scalar optical beam at the boundary between two dissimilar nonlinear media is of fundamental interest in nonlinear photonics. Here, we report the first systematic generalization of our Kerr analyses to a wider class of power-law materials. Universal refraction laws will be given, and theorysimulation agreement demonstrated.

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in L or H. The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described abo...

New families of exact analytical Helmholtz solitons are presented for various material nonlinearities, and simulations confirm their stability. A novel generalization of Snell’s Law for Kerr solitons is also reported. Helmholtz angular corrections to paraxial theory, exceeding 100%, are uncovered. © 2007 Optical Society of America OCIS codes: (190.4400) Nonlinear Optics, Materials; (190.9540) N...