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We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using…
In this work, we study the nonlinear traveling waves in density stratified fluids with depth varying shear currents. Beginning the formulation of the water-wave problem due to [1], we extend the work of [4] and [18] to examine the interface…
In the present work we propose and analyze a fully coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for…
The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros…
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard…
Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider…
Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-geostrophic flow. The derivation uses a…
A large class of multidimensional nonlinear Schroedinger equations admit localized nonradial standing wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are…
In this work, a novel Boussinesq system is put forward. The system is naturally nonlinearly entropy/energy-stable, and is designed for problems with sharply varying bathymetric features. The system is flexible and allows tuning of the…
We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain…
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of…
Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian…
Eigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid…
The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is…
In this paper we give a survey on various multiscale methods for the numerical solution of second order hyperbolic equations in highly heterogeneous media. We concentrate on the wave equation and distinguish between two classes of…
Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically…
We consider a spectral problem associated with steady water waves of extreme form on the free surface of a rotational flow. It is proved that the spectrum of this problem contains arbitrary large negative eigenvalues and they are simple.…
The stability properties of line solitary wave solutions of the (2+1)-dimensional Boussinesq equation with respect to transverse perturbations and their consequences are considered. A geometric condition arising from a multi-symplectic…
The linear wave and geostrophic (vortex) solutions are shown to be a complete basis for physical variables $(u,v,w,\rho)$ in a rotating non-hydrostatic Boussinesq model with arbitrary stratification. As a consequence, the fluid can be…
In this paper, we study the generalized Boussinesq equation as a model for the water wave problem with surface tension. Initially, we investigate the initial value problem within Sobolev spaces, deriving conditions under which solutions are…