16 research outputs found
Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis
Considered here is a class of Boussinesq systems of Nwogu type. Such systems
describe propagation of nonlinear and dispersive water waves of significant
interest such as solitary and tsunami waves. The initial-boundary value problem
on a finite interval for this family of systems is studied both theoretically
and numerically. First, the linearization of a certain generalized Nwogu system
is solved analytically via the unified transform of Fokas. The corresponding
analysis reveals two types of admissible boundary conditions, thereby
suggesting appropriate boundary conditions for the nonlinear Nwogu system on a
finite interval. Then, well-posedness is established, both in the weak and in
the classical sense, for a regularized Nwogu system in the context of an
initial-boundary value problem that describes the dynamics of water waves in a
basin with wall-boundary conditions. In addition, a new modified Galerkin
method is suggested for the numerical discretization of this regularized system
in time, and its convergence is proved along with optimal error estimates.
Finally, numerical experiments illustrating the effect of the boundary
conditions on the reflection of solitary waves by a vertical wall are also
provided
Local well-posedness of the higher order nonlinear Schr\"odinger equation on the half-line: single boundary condition case
We establish local well-posedness for the higher-order nonlinear
Schr\"odinger equation, formulated on the half-line. We consider the scenario
of associated coefficients such that only one boundary condition is required,
which is assumed to be Dirichlet type. Our functional framework centers around
fractional Sobolev spaces. We treat both high regularity and low regularity
solutions: in the former setting, the relevant nonlinearity can be handled via
the Banach algebra property; in the latter setting, however, delicate
Strichartz estimates must be established. This task is especially challenging
in the framework of nonhomogeneous initial-boundary value problems, as it
involves proving boundary-type Strichartz estimates that are not common in the
study of initial value problems.
The linear analysis, which is the core of this work, crucially relies on a
weak solution formulation defined through the novel solution formulae obtained
via the Fokas method. In this connection, we note that the higher-order
Schr\"odinger equation comes with an increased level of difficulty due to the
presence of more than one spatial derivative. This feature manifests itself via
several complications throughout the analysis, including (i) analyticity issues
related to complex square roots, which require careful treatment of branch cuts
and deformations of integration contours; (ii) singularities that emerge upon
changes of variables in the Fourier analysis arguments; (iii) complicated
oscillatory kernels in the weak solution formula for the linear
initial-boundary value problem, which require a subtle analysis of the
dispersion in terms of the regularity of the boundary data. The present work
provides a first, complete treatment via the Fokas method of a nonhomogeneous
initial-boundary value problem for a partial differential equation associated
with a multi-term linear differential operator.Comment: 30 pages, 2 figure
The Korteweg-de Vries equation on an interval
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 051507 (2019) and may be found at https://doi.org/10.1063/1.5080366.The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega