Special Session 118: Nonlinear Wave systems: Analysis and Computation

High Order Mimetic Difference Methods and Applications

Jose E Castillo Computational Science Research Center at San Diego State University USA Co-Author(s):

Abstract: Mimetic difference operators have been used more and more frequently to construct numerical schemes for solving partial differential equations with variable degree of success. There are many researches currently active in this area pursuing different approaches to achieve this goal and many algorithms have been developed along these lines. Loosely speaking, mimetic methods have discrete structures that mimic vector calculus identities and theorems. These make the numerical schemes based on mimetic difference operators more faithful to the physics of the problem under investigation. Specific approaches to discretization have achieved this compatibility following different paths, and with diverse degrees of generality in relation to the problems solved and the order of accuracy obtainable. In this session advances of High Order Mimetic differences methods and applications will be presented.

Pattern formation near a Turing-fold bifurcation: tipping evasion, quasi-periodicity & chaos

Arjen Doelman Leiden University Netherlands Co-Author(s):    Dock Staal (Leiden University) & Todd Kapitula (Calvin College)

Abstract: Recent simulations of ecosystem models (of reaction-diffusion type) have shown that a Turing bifurcation preceding a saddle-node bifurcation may enable an ecosystem to evade tipping: under slowly changing circumstances, an ecosystem may form patterns instead of collapsing into a less desirable state. Thus, the formation of patterns may increase the resilience of the ecosystem. In this talk we present the system of coupled modulation equations that governs the dynamics of small amplitude patterns near a co-dimension 2 Turing-fold bifurcation. First we show that there is a critical coefficient -- that can be determined explicitly and that corresponds to the Landau coefficient in the classical (co-dimension 1) Ginzburg-Landau setting -- that decides whether the system tips or evades tipping by forming (stable) patterns. Moreover, we show that the boundary of the region in (wavenumber,parameter)-space of stable periodic solutions -- i.e. the Busse balloon -- is much richer than that in the classical Ginzburg-Landau case. Instead of only a sideband instability, patterns can also be destabilized by several Turing- and Turing-Hopf-type mechanisms. The associated bifurcations may first lead to the formation of stable stationary quasi-periodic patterns and subsequently to time-periodic and eventually irregular dynamics.

On the proximal dynamics between integrable and non-integrable members of a generalized Korteweg-de Vries family of equations

Nikos I Karachalios Department of Mathematics, University of Thessaly Greece Co-Author(s):    Dionyssios Mantzavinos and Jeffrey Oregero

Abstract: We discuss current results concerning the distance between solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations in appropriate Sobolev spaces. This family of equations includes, as special cases, the standard gKdV equation with power nonlinearities as well as weakly nonlinear perturbations of the KdV equation. The distance estimates are based on a crucial size estimate for local gKdV solutions. Consequently, these estimates predict that the dynamics of the gKdV and KdV equations remain close over long time intervals for initial amplitudes approaching unity, while providing an explicit rate of deviation for larger amplitudes. Furthermore, it is demonstrated that in the case of power nonlinearities and large solitonic initial data, the deviation between the integrable and non-integrable dynamics can be drastically reduced by incorporating suitable rotation effects via a rescaled KdV equation. As a result, the integrable dynamics stemming from the rescaled KdV equation may persist within the gKdV family of equations over remarkably long timescales. Finally, we comment on similar results regarding Nonlinear Schr\odinger (NLS) equations.

Nonlinear Waves Beyond the Laplacian Klein-Gordon and Nonlinear Schrodinger Settings

Panayotis Kevrekidis University of Massachusetts, Amherst USA Co-Author(s):

Abstract: In the present work, we will explore a number of variants of the standard Klein-Gordon and NLS problems and study both kinks/dark solitons but also occasionally pulses/bright solitons in them. We will be motivated by the recent remarkable experimental realization of not only biharmonic, but also arbitrary fractional index Laplacian operators in nonlinear optics and we will see how the tails of the structures behave for different regimes of the fractional exponent $\alpha$. This, in turn, will guide us to explore not only the soliton-soliton interactions, but also their stability. We will seek to present a unifying picture of the relevant interactions as a function of $\alpha$, starting from the biharmonic setting and interpolating between that and the Laplacian one, as well as moving to the sub-Laplacian index case. Finally, we will touch upon dissipative variant of the problem, presently of interest, including the quartic Lugiato-Lefever equation and related problems that experiments are currently getting to.

New methods and results for Benjamin-Ono and related equations

Peter D Miller University of Michigan USA Co-Author(s):    Peter D. Miller

Abstract: We describe some recent results on asymptotic properties of solutions of the Benjamin-Ono equation that have been obtained due to a breakthrough yielding an explicit formula for the solution of the Cauchy problem for this equation. Only a few integrable equations have been proved to have such a formula, which can be regarded as a rare and practically useful compact representation of all the parts of an inverse-scattering transform solution.

Existence and instability of periodic waves for KdV-Burgers equations with a source

Ramon G. Plaza Universidad Nacional Autonoma de Mexico Mexico Co-Author(s):    Raffaele Folino, Anna Naumkina

Abstract: In this talk I will discuss the existence and (in)stability of periodic waves for equations of KdV-Burgers type with a source. In particular, I will discuss the existence of small-amplitude periodic waves for the KdV-Burgers-Fisher equation (with a source of Fisher-KPP type), which arise from a local bifurcation on the wave speed. Moreover, these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. I will also discuss how to conclude the orbital (nonlinear) instability of the waves departing from the spectral instability result for a general family of equations. This is joint work with Anna Naumkina and Raffaele Folino (IIMAS, UNAM).

Solitons in the 4th order NLS

Svetlana Roudenko Florida International University USA Co-Author(s):

Abstract: The 4th order (or bi-harmonic) NLS equation has recently been attracting attention of scientists as the experimental quartic solitons, aka light bullets, have been obtained. We discuss solutions to this equation, examining more carefully the ground states in one- and two-dimensional settings. We look at the soliton dynamics, including stability, branching and long-time behavior. This talk is based on joint works with Iryna Petrenko (FIU, Miami) and Christian Klein & Nikola Stoilov (IMB, Dijon).