An official website of the United States government
It is well known that the Korteweg-deVries(KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other such results have been proved in this case. However, situ- ations in which the defocusing modified KdV (mKdV) equation is expected to be the modulation equation have been much less studied. As seen in numerical experiments, the kink solution of the mKdV seems essential in understanding the -FPUT recurrence. In this paper, we derive explicit approximation re- sults for solutions of the FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for solutions to be non-localized as to allow approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT.
more »
« less
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
-
We study the effects of localization on the long-time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun [4]. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of jointly hyperbolic-parabolic equations in terms of Hermite functions in which nth order approximations can be computed for solutions with nth-order moments. We then obtain existence of solutions to the mcNS system in weighted spaces and, based on the decay rates obtained for the various pieces of the solutions, determine the optimal choice of asymptotic approximation with respect to the various localization assumptions, which in certain cases can be evaluated explicitly in terms of Hermite functions.more » « less
-
In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive eigenvalue whose eigenvector lies almost tangent to the surface of the cylinder formed by the family of breathers. This causes solutions to slide along the cylinder without leaving its neighborhood for very long times.more » « less
-
In this paper we derive a bound on the integral of a product of two Hermite-Gaussian functions with a Gaussian weight. We prove that such integrals decay exponentially in the difference of the indices of the Hermite-Gaussian functions. Such integrals arise naturally in mathematical physics and applied mathematics. The estimate is applied to a variational problem related to a Strichartz functional.more » « less
-
null (Ed.)Abstract We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are $$n<\infty $$ n < ∞ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size $$\gamma $$ γ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.more » « less
Full Text Available