Soliton solution and bifurcation analysis of the KP–Benjamin–Bona–Mahoney equation with power law nonlinearity

This paper studies the Kadomtsev–Petviashvili–Benjamin–Bona–Mahoney equation with power law nonlinearity. The traveling wave solution reveals a non-topological soliton solution with a couple of constraint conditions. Subsequently, the dynamical system approach and the bifurcation analysis also reveals other types of solutions with their corresponding restrictions in place

Compactons and kink-like solutions of BBM-like equations by means of factorization

In this work, we study the Benjamin-Bona-Mahony like equations with a fully nonlinear dispersive term by means of the factorization technique. In this way we find the travelling wave solutions of this equation in terms of the Weierstrass function and its degenerated trigonometric and hyperbolic forms. Then, we obtain the pattern of periodic, solitary, compacton and kink-like solutions. We give also the Lagrangian and the Hamiltonian, which are linked to the factorization, for the nonlinear second order ordinary differential equations associated to the travelling wave equations.Comment: 10 pages, 8 figure

Exact closed form solutions of compound Kdv Burgers’ equation by using generalized (Gʹ/G) expansion method

In this investigation, the compound Korteweg-de Vries (Kd-V) Burgers equation with constant coefficients is considered as the model, which is used to describe the properties of ion-acoustic waves in plasma physics, and also applied for long wave propagation in nonlinear media with dispersion and dissipation. The aim of this paper to achieve the closed and dynamic closed form solutions of the compound KdV Burgers equation. We derived the completely new solutions to the considered model using the generalized (Gʹ/G)-expansion method. The newly obtained solutions are in form of hyperbolic and trigonometric functions, and rational function solutions with inverse terms of the trigonometric, hyperbolic functions. The dynamical representations of the obtained solutions are shown as the annihilation of three-dimensional shock waves, periodic waves, and multisoliton through their three dimensional and contour plots. The obtained solutions are also compared with previously exiting solutions with both analytically and numerically, and found that our results are preferable acceptable compared to the previous results.Publisher's Versio

Reliable Study of Nonhomogeneous BBM Equation with Time-Dependent Coefficients by the Modified Sine-Cosine Method

The modified sine-cosine method is an efficient and powerful mathematical tool in finding exact traveling wave solutions to nonlinear partial differential equations (NLPDEs) with time-dependent coefficients. In this paper, the proposed approach is applied to study a nonhomogeneous generalized form of Benjamin-Bona-Mahony (BBM) equation with time-dependent coefficients. Explicit traveling wave solutions of the equation are obtained under certain constraints on the coefficient functions

New Soliton Solutions for Systems of Nonlinear Evolution Equations by the Rational Sine-Cosine Method

In this paper, we construct new solitary solutions to nonlinear PDEs by the rational Sine and Cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the Benjamin-Bona-Mahony and the Gardner equations are obtained. New solutions to Broer-Kaup (BK) system are also obtained. Finally, the solution of a two-component evolutionary system of a homogeneous Kdv equations of order  has been investigated by the proposed method

Resolving Entropy Growth from Iterative Methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.Comment: 25 pages, 6 figure

On the Galilean invariance of some dispersive wave equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.Comment: 29 pages, 13 figures, 2 tables, 71 references. Other author papers can be downloaded at http://www.denys-dutykh.com