650 research outputs found
Partial differential systems with nonlocal nonlinearities: Generation and solutions
We develop a method for generating solutions to large classes of evolutionary
partial differential systems with nonlocal nonlinearities. For arbitrary
initial data, the solutions are generated from the corresponding linearized
equations. The key is a Fredholm integral equation relating the linearized flow
to an auxiliary linear flow. It is analogous to the Marchenko integral equation
in integrable systems. We show explicitly how this can be achieved through
several examples including reaction-diffusion systems with nonlocal quadratic
nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic
nonlinearity. In each case we demonstrate our approach with numerical
simulations. We discuss the effectiveness of our approach and how it might be
extended.Comment: 4 figure
KdV equation under periodic boundary conditions and its perturbations
In this paper we discuss properties of the KdV equation under periodic
boundary conditions, especially those which are important to study
perturbations of the equation. Next we review what is known now about long-time
behaviour of solutions for perturbed KdV equations
Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear
evolution partial differential equations. It is based on approximations of
generalized Lax pairs. Contrary to other reduced-order methods, like Proper
Orthogonal Decomposition, the basis on which the solution is searched for
evolves in time according to a dynamics specific to the problem. It is
therefore well-suited to solving problems with progressive front or wave
propagation. Another difference with other reduced-order methods is that it is
not based on an off-line / on-line strategy. Numerical examples are shown for
the linear advection, KdV and FKPP equations, in one and two dimensions
Regularization by noise and stochastic Burgers equations
We study a generalized 1d periodic SPDE of Burgers type: where , is
the 1d Laplacian, is a space-time white noise and the initial condition
is taken to be (space) white noise. We introduce a notion of weak
solution for this equation in the stationary setting. For these solutions we
point out how the noise provide a regularizing effect allowing to prove
existence and suitable estimates when . When we obtain
pathwise uniqueness. We discuss the use of the same method to study different
approximations of the same equation and for a model of stationary 2d stochastic
Navier-Stokes evolution.Comment: clarifications and small correction
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