Maintaining Constant Pulse-Duration in Highly Dispersive Media Using Nonlinear Potentials

A method is shown for preventing temporal broadening of ultrafast optical pulses in highly dispersive and fluctuating media for arbitrary signal-pulse profiles. Pulse pairs, consisting of a strong-field control-pulse and a weak-field signal-pulse, co-propagate, whereby the specific profile of the strong-field pulse precisely compensates for the dispersive phase in the weak pulse. A numerical example is presented in an optical system consisting of both resonant and gain dispersive effects. Here, we show signal-pulses that do not temporally broaden across a vast propagation distance, even in the presence of dispersion that fluctuates several orders of magnitude and in sign (for example, within a material resonance) across the pulse’s bandwidth. Another numerical example is presented in normal dispersion telecom fiber, where the length at which an ultrafast pulse does not have significant temporal broadening is extended by at least a factor of 10. Our approach can be used in the design of dispersion-less fiber links and navigating pulses in turbulent dispersive media. Furthermore, we illustrate the potential of using cross-phase modulation to compensate for dispersive effects on a signal-pulse and fill the gap in the current understanding of this nonlinear phenomenon.

Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications

We study the Γ-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressed.

Conservación de invariantes de la ecuación de Schrödinger no lineal por el método LDG

Conservation of the energy and the Hamiltonian of a general non linear Schr¨odinger equation is analyzed for the finite element method “Local Discontinuous Galerkin” spatial discretization. Conservation of the discrete analogue of these quantities is also proved for the fully discrete problem using the modified Crank-Nicolson method as time marching scheme. The theoretical results are validated on a series of problemsfor different nonlinear potentials.

Quasilinear elliptic equations with sub-natural growth and nonlinar potentials

Necessary and sufficient conditions for the existence of finite energy and weak solutions are given. Sharp global pointwise estimates of solutions are obtained as well. We also discuss the uniqueness and regularity properties of solutions. As a consequence, characterization of solvability of the equations with singular natural growth in the gradient terms is deduced. Our main tools are Wolff potential estimates, dyadic models, and related integral inequalities. Special nonlinear potentials of Wolff type ssociated with "sublinear" problems are constructed to obtain sharp bounds of solutions. We also treat equations with the fractional Laplacians. Our approach is applicable to more general quasilinear A-Laplace operators as well as the fully nonlinear k-Hessian operators.