We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.

In this paper the concept of eigenvalues and eigenvectors of nonlinear systems, both continuousand discrete-time, is suggested. It represents a generalization of the concept known from linear control theory. Some basic properties, like invariance of eigenvalues under a (nonlinear) change of coordinates, possibility to transform the system to the diagonal form and, respectively, to the feedforwa...

Contributions to the fifth-order nonlinear optical susceptibility 5 of a collection of homogeneously broadened two-level atoms that scale as N2 at 3 2 and N2 at 3 2, where at 3 is the lower-order atomic hyperpolarizability and N is the atomic number density, are predicted theoretically. These “cascaded” contributions are a consequence of local-field effects. We determine them from a fifth-order...

We describe new classes of nonlinear Galilean–invariant equations of Burgers and Korteweg–de Vries type and study symmetry properties of these equations.

Generally, blind separation of sources from their nonlinear mixtures is rather difficult. This nonlinear mapping, constituted by unsupervised linear mixing followed by unknown and invertible nonlinear distortion, is found in many signal processing cases. We propose using a kernel density estimator incorporated within an equivariant gradient algorithm to separate the nonlinear mixed sources. The...

We derive coupled nonlinear Schrödinger equation (CNLSE) for arbitrary polarized light propagation in single-mode fiber. Considering a weak nonlinearity which is connected with Kerr effect, we give explicit expressions for nonlinear constants via integrals of Bessel functions. The method of projecting we use allows a direct generalization to multi-mode fiber case.

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. Moreover, as a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen. AMS Subject Classification (2000...

We prove that a nonlinear version of the Grothendieck-Katz conjecture (essentially in the form given by Ekhedahl, Shepherd-Barron and Taylor) is equivalent to the original Grothendieck-Katz conjecture together with a certain differential algebraic geometric/model-theoretic statement: a type over C(t) with “p-curvature 0 for almost all p” is nonorthogonal to the constants.

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable...

We consider the problem of performing the preliminary “symmetry classification” of a class of quasi-linear PDE’s containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a “geometrical” characterization of the relevant system of equations determining these symmetries. Two detailed examples w...