## Seminars and Colloquia by Series

Monday, March 6, 2017 - 11:00 , Location: Skiles 005 , Dr. Jiayin Jin , Georgia Tech , Organizer: Jiayin Jin
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.
Monday, March 6, 2017 - 11:00 , Location: Skiles 005 , Dr. Jiayin Jin , Georgia Tech , Organizer: Jiayin Jin
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.
Friday, March 3, 2017 - 11:00 , Location: Skiles 005 , Diego Del Castillo-Negrete , Oak Ridge National Lab. , Organizer: Rafael de la Llave
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.
Friday, March 3, 2017 - 11:00 , Location: Skiles 005 , Diego Del Castillo-Negrete , Oak Ridge National Lab. , Organizer: Rafael de la Llave
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.
Monday, January 30, 2017 - 11:00 , Location: Skiles 005 , T.M-Seara , Univ. Polit. Catalunya , Organizer: Rafael de la Llave
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.
Monday, January 30, 2017 - 11:00 , Location: Skiles 005 , T.M-Seara , Univ. Polit. Catalunya , Organizer: Rafael de la Llave
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.
Friday, January 13, 2017 - 11:00 , Location: Skiles 005 , Florian Kogelbauer , ETH (Zurich) , Organizer: Rafael de la Llave
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.
Friday, January 13, 2017 - 11:00 , Location: Skiles 005 , Florian Kogelbauer , ETH (Zurich) , Organizer: Rafael de la Llave
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.
Wednesday, November 30, 2016 - 11:00 , Location: Skiles 005 , Prof. Eugene Wayne , Boston University , Organizer: Chongchun Zeng
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.
Wednesday, November 16, 2016 - 11:00 , Location: 006 , , McMaster University , Organizer: Livia Corsi
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.