Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} + \mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint operator, and $f$ is a real-valued function with $f(0) = 0$. The SpectraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that is possible for two solitary waves to interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime

Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.Comment: 14 pages, 11 figure

Exact Traveling Wave Solutions for the BBM Equation, Schamel Equation and Modified Kawahara Equation and their Geometric Interpretations

In this paper, we employ the extended mapping method to obtain the exact traveling wavesolutions of the Benjamin-Bona-Mahony (BBM) equation, the Schamel equation and the modified Kawahara equation. Our results show that these solutionsinclude periodic wavesolutions and solitary wave solutions.The geometric interpretation for some of these solstionare introduced. The solitary wave solutions are obtained as a limiting case.Keywords: Traveling wave solutions, Benjamin-Bona-Mahony (BBM) equation, the Schamelequation, the modified Kawahara equation, mapping method, geometric interpretations

Elliptic solutions to a generalized BBM equation

An approach is proposed to obtain some exact explicit solutions in terms of the Weierstrass' elliptic function $\wp$ to a generalized Benjamin-Bona-Mahony (BBM) equation. Conditions for periodic and solitary wave like solutions can be expressed compactly in terms of the invariants of $\wp$. The approach unifies recently established ad-hoc methods to a certain extent. Evaluation of a balancing principle simplifies the application of this approach.Comment: 11 pages, 2 tables, submitted to Phys. Lett.

Slow solitary waves in multi-layered magnetic structures

The propagation of slow sausage surface waves in a multi-layered magnetic configuration is considered. The magnetic configuration consists of a central magnetic slab sandwiched between two identical magnetic slabs (with equilibrium quantities different from those in the central slab) which in turn are embedded between two identical semi-infinite regions. The dispersion equation is obtained in the linear approximation. The nonlinear governing equation describing waves with a characteristic wavelength along the central slab much larger than the slab thickness is derived. Solitary wave solutions to this equation are obtained in the case where these solutions deviate only slightly from the algebraic soliton of the Benjamin-Ono equation