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Series: CDSNS Colloquium

We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifold. Furthermore, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

The study of nonlocal transport in physically relevant systems requires
the formulation of mathematically well-posed and physically meaningful
nonlocal models in bounded spatial domains. The main problem faced by
nonlocal partial differential equations in general,
and fractional diffusion models in particular, resides in the treatment
of the boundaries. For example, the naive truncation of the
Riemann-Liouville fractional derivative in a bounded domain is in
general singular at the boundaries and, as a result, the incorporation
of generic, physically meaningful boundary conditions is not feasible.
In this presentation we discuss alternatives to address the problem of
boundaries in fractional diffusion models. Our main goal is to present
models that are both mathematically well posed
and physically meaningful. Following the formal construction of the
models we present finite-different methods to evaluate the proposed
non-local operators in bounded domains.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

We present a general mechanism to establish the existence of diffusing
orbits in a large class of nearly integrable Hamiltonian systems. Our
approach relies on successive applications of the `outer dynamics'
along homoclinic orbits to a normally hyperbolic invariant manifold.
The information on the outer dynamics is encoded by a geometrically
defined map, referred to as the `scattering map'.
We find pseudo-orbits of the scattering map that keep moving in some
privileged direction.
Then we use the recurrence property of the `inner dynamics', restricted
to the normally hyperbolic invariant manifold, to return to those
pseudo-orbits.
Finally, we apply topological methods to show the existence of true
orbits that follow the successive applications of the two dynamics.
This method differs, in several crucial aspects, from earlier works.
Unlike the well known `two-dynamics' approach, the method relies
heavily on the outer dynamics alone.
There are virtually no assumptions on the inner dynamics, as its
invariant objects (e.g., primary and secondary tori, lower dimensional
hyperbolic tori and their
stable/unstable manifolds, Aubry-Mather sets) are not used at all.
The method applies to unperturbed Hamiltonians of arbitrary degrees of
freedom that are not necessarily convex.
In addition, this mechanism is easy to verify (analytically or
numerically) in concrete examples, as well as to establish diffusion in
generic systems.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

We use invariant manifold
results on Banach spaces to conclude the existence of spectral
submanifolds (SSMs) in a class of nonlinear, externally forced beam
oscillations .
Reduction of the governing PDE to the SSM provides an exact
low-dimensional model which we compute explicitly. This model captures
the correct asymptotics of the full, infinite-dimensional
dynamics. Our approach is general enough to admit extensions to other
types of continuum vibrations. The model-reduction procedure we employ
also gives guidelines for a mathematically
self-consistent modeling of damping in PDEs describing structural vibrations.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

The nonlinear Schroedinger
equation (NLS) can be derived as a formal approximating equation for the
evolution of wave packets in a wide array of nonlinear dispersive PDE’s
including the propagation of waves on the surface of an inviscid
fluid. In
this talk I will describe recent work that justifies this approximation
by exploiting analogies with the theory of normal forms for ordinary
differential equations.

Series: CDSNS Colloquium

It was shown by V.E. Zakharov that the equations for water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. This latter is joint work with C. Sulem.