Christo I. Christov

An energy-consistent dispersive shallow-water model

The flow of inviscid liquid in a shallow layer with free surface is revisited in the framework of the Boussinesq approximation. The unnecessary approximations connected with the moving frame are removed and a Boussinesq model is derived which is Galilean invariant to the leading asymptotic order. The Hamiltonian structure of the new model is demonstrated. The conservation and/or balance laws for wave mass, energy and wave momentum (pseudo-momentum) are derived. A new localized solution is obtained analytically and compared to the classical Boussinesq sech. Numerical simulation of the collision of two solitary waves is conducted and the impact of Galilean invariance on phase shift is discussed.

INELASTIC INTERACTION OF BOUSSINESQ SOLITONS

Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

Strong coupling of Schrödinger equations: conservative scheme approach

The system of coupled nonlinear Schrodingers equations (CNLSE) is considered and the physical meaning of the coupling terms is identified. The attention is focused on the case of real-valued parameter of linear cross-diffusion. A new analytical solution for the coupled case is found and used as initial condition for the interaction and evolution of two pulses. Conservative numerical scheme and algorithm are devised for the time evolution of solitons in CNLSE. The results show that the coupling term brings into play localized solutions with rotating polarization which in many instances behave as breathers. Both elastic and inelastic collisions are uncovered numerically.

Fourier-Galerkin method for 2D solitons of Boussinesq equation

We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special complete orthonormal system of functions in L^2(-~,~) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible.

Nonlinear Duality Between Elastic Waves and Quasi-particles

Some systems governed by a set of partial differential equations present the necessary ingredients (nonlinearity and dispersion) in appropriate doses so as to become the arena of the propagation and interactions of solitary waves. In general such systems are not exactly integrable in the sense of soliton theory. But some of their nearly solitonic solutions can nonetheless be apprehended as quasi-particles in a certain dynamics that depends on the original system. The present chapter considers this reductive representation of nonlinear dynamical solutions for physical systems issued from solid mechanics, and more particularly elasticity with a microstructure of various origin. A whole collection of “point-mechanics” emerges thus, among which the simpler ones are Newtons and Lorentz-Einstein’s. This quasi-particle representation is intimately related to the existence of conservation laws for the system under study and the recent recognition of the essential role played by fully material balance laws in the continuum mechanics of inhomo-geneous and defective elastic bodies.

NOVEL NUMERICAL APPROACH TO SOLITARY{WAVE SOLUTIONS IDENTIFICATION OF BOUSSINESQ AND KORTEWEG{DE VRIES EQUATIONS

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.

A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractional-time step which allows satisfaction of the boundary conditions without introducing artificial conditions for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The splitting algorithm for the 3D case is verified through an analytic solution test. The stability of the method at high values of Reynolds number is illustrated by accurate numerical solutions for the flow in a lid-driven rectangular cavity with aspect ratio 1 and 2, as well as for the flow after a back-facing step.

A fourier-series method for solving soliton problems

A Fourier–Galerkin method with an earlier proposed complete orthonormal system of functions in

Maxwell-Lorentz electrodynamics as a manifestation of the dynamics of a viscoelastic metacontinuum

L^2 ( - \infty ,\infty )

Interacting localized waves for the regularized long wave equation via a Galerkin spectral method

as the set of trial functions is developed and displayed for the problem of calculating the shape of the one-soliton solution of the Korteweg–de Vries equation. The convergence of the method is investigated through comparison with the analytic solution, which appears to be very good. The truncation and discretization errors are assessed pointwise. The technique developed is also applied to the soliton problem for the so-called Kuramoto–Sivashinsky equation and the obtained soliton shape is compared to the existing difference solution. The quantitative agreement between the Fourier-series-method result and the numerical one is good. In the present paper, however, the soliton solution is obtained for a significantly wider range of phase velocities, which suggests that the spectrum might be continuous. The new technique can also be applied to a variety of other problems involving identification of homoclinic solutions.