Nonlinear Dispersive Partial Differential Equations

  • Castilla, Alejandro Castro (PI)
  • Staubach, Wolfgang (Other Faculty/Researcher)
  • Valido Flores, Antonio A. (Postdoctoral scholar (PhD degree holder))
  • Israelsson, Anders (Other participant)

Project: Research project

Grant Program

Faculty Development Competitive Research Grant Program 2019-2021

Project Description

During the last decades the study of Nonlinear Dispersive Partial Differential Equations has proved to be fundamental in our understanding of various important physical phenomena emerging in a broad range of stages [CSS], from the atomic or photonic [DDEG] to the oceanic scale [AAS], and which span a great array of scientific disciplines such as telecommunication engineering [HKFMFDM], biomolecule dynamics [Sco], plasma physics [BSN], or non-linear optics [Kib], to name a few. A prominent example of this is the so-called Bose-Einsten condensate (BEC) [And], which has attracted special attention due to it is one of the most important manifestations of the underlying quantum nature of the surrounding matter at the mesoscopic level. The basic experimental scenario of the BEC turns to be theoretically described by the paradigmatic Nonlinear Schrödinger equation (NLSE) [VF], which also finds application in the explanation of relative fundamental physical phenomena, for instance superconductivity. Remarkably, the study of both the theoretical and experimental implications of this equation in the BEC context served as the ideal catalyst for the establishment of new physical concepts and ideas, as well as it put the ground for the development of innovative atomic technology [CW], that today constitutes most of the elemental toolkit for the experimental manipulation and detection of large ensembles of atoms without perturbing their fundamental quantum nature. Despite all these progresses, there are still many open questions on the theoretical side of nonlinear dispersive partial differential equations, which have counterparts on the experimental ground. For instance, in fiber optic systems we know little about the intricate interplay between the non-linear effects and the unavoidable dissipative mechanisms, the latter is mainly responsible for the degradation of the telecommunication properties of such setups. On the other side, it would be interesting to come up with theoretically wellunderstood experimental setups that could serve as testbeds for fundamental nonlinear phenomena, like the famous rogue waves. It is the purpose of this work to address these questions.
StatusActive
Effective start/end date1/31/19 → 12/31/21

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partial differential equations
condensates
telecommunication
plasma physics
nonlinear optics
nonlinear equations
manipulators
fiber optics
emerging
superconductivity
engineering
photonics
degradation
catalysts
atoms

Keywords

  • Nonlinear Partial Differential Equations
  • Dispersive Partial Differential Equations
  • Harmonic Analysis
  • Fourier Analysis