385 research outputs found
Traveling wave solutions and numerical solutions for a mBBM equation
In this paper, some exact meromorphic solutions and generalized trigonometric solutions of the space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation are established by a new transformation and reliable methods. Moreover, some numerical solutions are obtained by using the optimal decomposition method (ODM), and their accuracy is shown in tables and images
The modified BenjaminBona-Mahony equation via the extended generalized Riccati equation mapping method
Abstract The generalized Riccati equation mapping is extended together with the ( ) expansion method and is a powerful mathematical tool for solving nonlinear partial differential equations. In this article, we construct twenty seven new exact traveling wave solutions including solitons and periodic solutions of the modified Benjamin-Bona-Mahony equation by applying the extended generalized Riccati equation mapping method. In this method, implemented as the auxiliary equation, where , r s and p are arbitrary constants and called the generalized Riccati equation. The obtained solutions are described in four different families including the hyperbolic functions, the trigonometric functions and the rational functions. In addition, it is worth mentioning that one of newly obtained solutions is identical for a special case with already published result which validates our other solutions. Mathematics Subject Classification: 35K99, 35P99, 35P05 Keywords: The modified Benjamin-Bona-Mahony equation, the generalized Riccati equation, the ( ) / G G ′ -expansion method, traveling wave solutions, nonlinear evolution equations
Generalized and Improved (G'/G)-Expansion Method for Nonlinear Evolution Equations
A generalized and improved (G'/G)-expansion method is proposed for finding more general type
and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty
and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-
Benjamin-Bona-Mahony �ZKBBM� equation and the strain wave equation in microstructured
solids. Abundant exact travelling wave solutions of these equations are obtained, which include
the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also
it is shown that the proposed method is efficient for solving nonlinear evolution equations in
mathematical physics and in engineering
Elliptic solutions to a generalized BBM equation
An approach is proposed to obtain some exact explicit solutions in terms of
the Weierstrass' elliptic function 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 . 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.
Existence, uniqueness and analyticity of space-periodic solutions to the regularised long-wave equation
We consider space-periodic evolutionary and travelling-wave solutions to the
regularised long-wave equation (RLWE) with damping and forcing. We establish
existence, uniqueness and smoothness of the evolutionary solutions for smooth
initial conditions, and global in time spatial analyticity of such solutions
for analytical initial conditions. The width of the analyticity strip decays at
most polynomially. We prove existence of travelling-wave solutions and
uniqueness of travelling waves of a sufficiently small norm. The importance of
damping is demonstrated by showing that the problem of finding travelling-wave
solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the
asymptotic convergence of the power series expansion of travelling waves for a
weak forcing.Comment: 29 pp., 4 figures, 44 reference
Generation of two-dimensional water waves by moving bottom disturbances
We investigate the potential and limitations of the wave generation by
disturbances moving at the bottom. More precisely, we assume that the wavemaker
is composed of an underwater object of a given shape which can be displaced
according to a prescribed trajectory. We address the practical question of
computing the wavemaker shape and trajectory generating a wave with prescribed
characteristics. For the sake of simplicity we model the hydrodynamics by a
generalized forced Benjamin-Bona-Mahony (BBM) equation. This practical problem
is reformulated as a constrained nonlinear optimization problem. Additional
constraints are imposed in order to fulfill various practical design
requirements. Finally, we present some numerical results in order to
demonstrate the feasibility and performance of the proposed methodology.Comment: 21 pages, 7 figures, 1 table, 69 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Asymptotic soliton like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficients
The paper deals with a problem of asymptotic soliton like solutions to the
Benjamin-Bona-Mahony (BBM) equaion with a small parameter at the highest
derivative and variable coefficients depending on the variables , as
well as a small parameter. There is proposed an algorithm of constructing the
solutions and there are proved theorems on accuracy with which the solutions
satisfy the BBM equation.Comment: 19 pages, 44 reference
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
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