Abstract : Dispersive partial differential equations $\left({\rm PDEs}\right)$ play a fundamental role in many fields such as the nonlinear optics, water wave theory, quantum mechanics, etc. From the perspective of computational mathematics, it is significant to design efficient numerical methods to solve dispersive PDEs with in-depth numerical analysis and provide an intuitive view for physical phenomena. The proposed minisymposium invites experts in this field to review recent advances in numerical methods for dispersive PDEs and applications.
[05510] Numerical studies of two regularized versions of the cubic NLS
Format : Online Talk on Zoom
Author(s) :
Christof Sparber (University of Illinois Chicago)
Abstract : We consider two types of regularization for the focusing, cubic nonlinear Schrödinger equation (NLS) posed in two and/or three spatial dimensions. One type of regularization is given by a defocusing quintic nonlinearity, while the other is given by a second order elliptic differential operator, describing off-axis variations of the NLS in the context of laser physics. While the non-regularized NLS is know to exhibit finite-time blow-up, these augmented equations are proved to be globally well-posed. In both cases we numerically investigate the long time behavior of solutions using a time-splitting method. In particular, we are interested in the orbital (in-)stability of least action ground states in the radially symmetric case. This is joint work with Christian Klein, Remi Carles, and Jack Arbunich.
[05557] Dirac equations for the modeling of electron dynamics on strained graphene surfaces
Format : Talk at Waseda University
Author(s) :
Emmanuel Lorin de la Grandmaison (Carleton University)
Abstract : This talk is devoted to the modelling of the dynamics of electrons on strained graphene surfaces. A hierarchy of mathematical models will be derived, and some numerical experiments illustrating the scattering of wave packets on locally deformed graphene will be proposed
[02732] Low regularity exponential-type integrators for the "good" Boussinesq equation
Format : Talk at Waseda University
Author(s) :
Chunmei Su (Tsinghua University)
Hang Li (Tsinghua University )
Abstract : We introduce a series of semi-discrete low regularity exponential-type time integrators for the “good” Boussinesq equation. Compared to the existing numerical methods, the temporal convergence of ours can be achieved under weaker regularity assumptions on the exact solutions. The methods are constructed based on twisted variables and some harmonic analysis techniques in approximating the exponential integral. The methods are explicit and easy to be implemented efficiently when combined with pseudospectral method for spatial discretization.
[04017] Computational methods for stationary states of nonlinear Schrödinger/Gross-Pitaevskii equations
Format : Talk at Waseda University
Author(s) :
Wei Liu (National University of Singapore)
Abstract : I will present some recent advances in the computation of stationary-state solutions to the nonlinear Schrödinger/Gross-Pitaevskii equations, primarily in the context of Bose-Einstein condensation. The (normalized) energy ground/excited states and action ground states will be mainly considered. Based on the analysis of variational characterizations and stabilities/instabilities for these stationary-state solutions, efficient and accurate numerical methods utilizing novel artificial dynamical flows and/or optimization techniques will be developed, with further extensions to challenging high-spin or fast-rotating models.
[05499] Scattering and uniform in time error estimates for splitting method in NLS
Format : Online Talk on Zoom
Author(s) :
Rémi Carles (CNRS & Univ Rennes)
Chunmei Su (Tsinghua University)
Abstract : We consider the nonlinear Schrödinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie–Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory and requires several technical modifications compared to previous error estimates for splitting methods.
[05587] Resonances as a computational tool
Format : Online Talk on Zoom
Author(s) :
Katharina Schratz (Sorbonne University)
Abstract : A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.
[01752] Error estimates of numerical methods for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
Format : Talk at Waseda University
Author(s) :
Weizhu Bao (National University of Singapore)
Chushan Wang (National University of Singapore)
Abstract : We prove optimal error bounds of time-splitting methods and the exponential wave integrator for the nonlinear Schr\"{o}dinger equation (\text{(NLSE)}) with low regularity potential and nonlinearity, including purely bounded potential and locally Lipschitz nonlinearity. Arising from different physical applications, low regularity potential and nonlinearity are introduced into the NLSE such as the discontinuous potential or non-integer power nonlinearity, which make the error estimates of classical numerical methods very subtle and challenging.
