Benney-Luke Equation Research Articles

Applications of the exp(-Φ(ξ))-Expansion Method to Find Exact Traveling Wave Solutions of the Benney-Luke Equation in Mathematical Physics

In this article, we construct the traveling wave solutions involving parameters of nonlinear evolutions equations via the Benney-Luke equation using the exp(-Φ(ξ))-expansion method. The traveling wave solutions are expressed in terms of hyperbolic, trigonometric and rational functions. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The proposed method is direct, concise elementary and effective and can be used for many other nonlinear evolutions equations.

Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

The linear stability of the solitary waves for the one‐dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.

Strichartz Estimates for Some 2D Water Wave Models

In this paper we establish Strichartz type estimates associated with a class of semigroup operators in Rn, which for n = 2 correspond to some 2D water wave models. We also establish a nonlinear scattering result for solutions of the generalized Benney-Luke equation for higher order nonlinearity and small data initial in the energy space.

Conservation laws and non-decaying solutions for the Benney–Luke equation

A long wave multi-dimensional approximation of shallow-water waves is the bi-directional Benney–Luke (BL) equation. It yields the well-known Kadomtsev–Petviashvili (KP) equation in a quasi one-directional limit. A direct perturbation method is developed; it uses underlying conservation laws to determine the slow evolution of parameters of two space-dimensional, non-decaying solutions to the BL equation. These non-decaying solutions are perturbations of recently studied web solutions of the KP equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the KP and the BL equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the BL equation are also studied.

Some New Exact Traveling Wave Solutions for the Generalized Benney-Luke (GBL) Equation with Any Order

In the paper, a auxiliary equation expansion method and its algorithm is proposed by studying a second order nonlinear ordinary differential equation with a four-degree term. The method is applied to the generalized Ben-ney-Luke (GBL) equation with any order. As a result, some new exact traveling wave solutions are obtained which singular solutions, triangular periodic wave solutions and jacobian elliptic function solutions .This algorithm can also be applied to other nonlinear wave equations in mathematical physics.

On the evolution of perturbations to solutions of the Kadomtsev–Petviashvilli equation using the Benney–Luke equation

The Benney–Luke equation, which arises as a long wave asymptotic approximation of water waves, contains the Kadomtsev–Petviashvilli (KP) equation as a leading-order maximal balanced approximation. The question analyzed is how the Benney–Luke equation modifies the so-called web solutions of the KP equation. It is found that the Benney–Luke equation introduces dispersive radiation which breaks each of the symmetric soliton-like humps well away from the interaction region of the KP web solution into a tail of multi-peaked oscillating profiles behind the main solitary hump. Computation indicates that the wave structure is modified near the center of the interaction region. Both analytical and numerical techniques are employed for working with non-periodic, non-decaying solutions on unbounded domains.

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2 T , the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney-Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.

A remark on the Cauchy problem for the generalized Benney-Luke equation

In this article, we address the well posedness of the Cauchy problem associated with the generalized Benney--Luke equation in $\mathbb R^{1+2}:$ \begin{multline*} \Phi_{tt} - \Delta \Phi + a \Delta^2 \Phi - b \Delta \Phi_{tt} + \theta\Big ( \Phi_t \big [\partial_{x} \big [ \big (\partial_{x} \Phi \big )^{p} \big ]+ \partial_{y} \big [ \big (\partial_{y}\Phi \big )^{p} \big ] \big ] \\ + 2 \big [ \big (\partial_{x} \Phi \big )^{p}\Phi_{xt}+ \big (\partial_{y} \Phi \big )^{p}\Phi_{yt} \big ] \Big ) + \beta \nabla \cdot \big (|\nabla \Phi|^m \nabla \Phi \big )=0, \end{multline*} under a reasonable ``physical" initial condition, which is imposed from the formal derivation of the Benney-Luke water wave model.

Cauchy problem for the generalized Benney-Luke equation

This paper is concerned with the study of the Cauchy problem associated with an n-dimensional generalized Benney-Luke equation, utt−mΔu−Δutt+Δ2u+α(2∇u∙∇ut+utΔu)+β∇(∣∇u∣p∇u)=0, where n=1,2,3,4. We prove the existence and the uniqueness of the global solution of the Cauchy problem for the β⩽0 case by using energy conservation law and give the existence and the nonexistence of the global solution of the Cauchy problem for the β>0 case by constructing the stable set and the unstable set.

Instability of solitary waves for a generalized Benney–Luke equation

We study linear instability of solitary wave solutions of a one-dimensional generalized Benney–Luke equation, which is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension. Further, we implement a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions of the model equation which is used to validate the theory presented.

Solitons and periodic travelling waves for the 2D-generalized Benney–Luke equation

This article is related with one direction periodic travelling waves solutions for the 2D generalized Benney–Luke equation i.e, solution of the form Φ(x,y,t)= u(x−ct, y), with ζ =x−ct periodic. We establish the existence of a family of x-periodic travelling wave solutions for c 2<min {1, a/b}. We show that a special sequence of this family is uniformly bounded in norm and converges to a lump soliton in (solitary wave of finite energy) in an appropriate sense.

The Cauchy problem for Benney-Luke and generalized Benney-Luke equations

We examine the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for the Benney--Luke (BL) and generalized Benney--Luke (gBL) equations. As a consequence we prove that the initial-value problems are globally well-posed in the energy space.

Towards a rigorous derivation of the fifth order KP equation

The Kadomtsev–Petviashvili equations (KP) are universal models for dispersive, weakly nonlinear, almost one-dimensional long waves of small amplitude. In [L. Paumond, A rigorous link between KP and a Benney–Luke equation, Differential Integral Equations 16 (9) (2003) 1039–1064], we proved a rigorous link between KP and a Benney–Luke equation. Here, we derive a new model still valid when the Bond number is equal to 1/3. Following the work [R.L. Pego, J.R. Quintero, Two-dimensional solitary waves for a Benney–Luke equation, Physica D 132 (4) (1999) 476–496], we show the existence of solitary waves for this equation and their convergence towards the solitary waves of the fifth order KP equation. We also link rigorously the dynamics of the two equations.

Numerical comparisons of two long-wave limit models

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.

A rigorous link between KP and a Benney-Luke equation

Kadomtsev-Petviashvili (KP) equations model weakly nonlinear dispersive waves, which are essentially unidimensional, when weak transverse effects are taken into account. These equations can be formally obtained by asymptotic methods from the Euler equations, but there is no rigorous justification yet. In this paper we consider an intermediate equation (BL) derived first by Benney and Luke. (BL) reduces formally to (KP) if we seek waves propagating essentially in one direction, with a weak variations in time and in the transverse direction measured by a small parameter ${\varepsilon}$. We show rigorously that the $L^2(\mathbb R^2)$-norm of the difference between the amplitude of the wave given by (KP) and the one given by (BL) is of order $\mathcal{O}({\varepsilon}^{3/4})$ during a growing with ${\varepsilon}$ time.

Nonlinear Stability of a One-Dimensional Boussinesq Equation

We study nonlinear orbital stability and instability of the set of ground state solitary wave solutions of a one-dimensional Boussinesq equation or one-dimensional Benney–Luke equation. It is shown that a solitary wave (traveling wave with finite energy) may be orbitally stable or unstable depending on the range of the wave's speed of propagation.

Analyticity and decay properties of the solitary waves to the Benney-Luke equation

We prove that the Benney-Luke solitary waves, as well as their derivatives, are analytic functions and decay at infinity with an optimal algebraic rate.

Two-dimensional solitary waves for a Benney–Luke equation

We prove the existence of finite-energy solitary waves for isotropic Benney–Luke equations that arise in the study of the evolution of small amplitude, three-dimensional water waves when the horizontal length scale is long compared with the depth. The family of Benney–Luke equations discussed in this paper includes the effect of surface tension and a variety of equivalent forms of dispersion. These equations reduce formally to the Korteweg–de Vries (KdV) equation and to the Kadomtsev–Petviashvili (KP-I or KP-II) equation in the appropriate limits. Existence of finite-energy solitary waves or lumps is proved via the concentration-compactness method. When surface tension is sufficiently strong (Bond number larger than 1/3), we prove that a suitable family of Benney–Luke lump solutions converges to a nontrivial lump solution for the KP-I equation.