Sixth-order Boussinesq Equation Research Articles

Symmetries and conservation laws for a sixth-order Boussinesq equation

This paper considers a generalization depending on an arbitrary function f(u) of a sixth-order Boussinesq equation which arises in shallow water waves theory. Interestingly, this equation admits a Hamiltonian formulation when written as a system. A classification of point symmetries and conservation laws in terms of the function f(u) is presented for both, the generalized Boussinesq equation and the equivalent Hamiltonian system.

Global existence and asymptotic behavior of solution for the sixth order Boussinesq equation with damped term

We investigate the small-data Cauchy problem for the sixth order Boussinesq equation with damped term. The decay estimates of solutions to the corresponding linear equation are given by using the dyadic decomposition and some properties of Bessel functions. Then, we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.

An application of generalized tanh function method for the sixth-order Boussinesq (sB) equation and (1+1) dimensional dispersive long wave equation

An application of generalized tanh function method for the sixth-order Boussinesq (sB) equation and (1+1) dimensional dispersive long wave equation

Explicit Travelling Wave Solutions of Two Nonlinear Evolution Equations

In this paper, we applied the sine-cosine method and the rational functions in exp(ksi) method for the modified Kawachara equation and the Damped Sixth-order Boussinesq Equation, respectively. New solitons solutions and periodic solutions are explicitly obtained with the aid of symbolic computation.

A bilinear estimate with application to the sixth-order Boussinesq equation

In this work, we study the Cauchy problem of the sixth-order Boussinesq equation. We shall use the $[k; Z]$-multiplier norm method to get a bilinear estimate on nonlinear term, and the local well-posedness on $H^s(\mathbb{R})$ is obtained for $s\gt-\frac{3}{4}$.

Global rough solutions to the sixth-order Boussinesq equation

We show that the Cauchy problem for the sixth-order Boussinesq equation with cubic nonlinearity is globally well-posed in Hs(R) provided 3/2<s<2.

Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.

Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation

We consider the initial value problem associated to the nonlinear periodic sixth-order Boussinesq equation. For given initial data in Sobolev spaces Hs(T), the local well-posedness of this equation is established for s>−1/2.

Existence and asymptotic behavior of solution of Cauchy problem for the damped sixth-order Boussinesq equation

We consider the existence, both locally and globally in time, as well as the asymptotic behavior of solutions for the Cauchy problem of the sixth-order Boussinesq equation with damping term. Under rather mild conditions on the nonlinear term and initial data, we prove that the above-mentioned problem admits a unique local solution, which can be continued to a global solution, and the problem is globally well-posed. Finally, under certain conditions, we prove that the global solution decays exponentially to zero in the infinite time limit.

Global existence and blow-up for the generalized sixth-order Boussinesq equation

In this paper we prove local well-posedness in L2(R) and H1(R) for the generalized sixth-order Boussinesq equation utt=uxx+βuxxxx+uxxxxxx+(|u|αu)xx. Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) [14]. We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.

Multiple (G′/G)-expansion method and its applications to nonlinear evolution equations in mathematical physics

In this paper, an extended multiple (G′/G)-expansion method is proposed to seek exact solutions of nonlinear evolution equations. The validity and advantages of the proposed method is illustrated by its applications to the Sharma–Tasso–Olver equation, the sixth-order Ramani equation, the generalized shallow water wave equation, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation, the sixth-order Boussinesq equation and the Hirota–Satsuma equations. As a result, various complexiton solutions consisting of hyperbolic functions, trigonometric functions, rational functions and their mixture with parameters are obtained. When some parameters are taken as special values, the known double solitary-like wave solutions are derived from the double hyperbolic function solution. In addition, this method can also be used to deal with some high-dimensional and variable coefficients’ nonlinear evolution equations.

Local well-posedness for the sixth-order Boussinesq equation

In this work we study the local well-posedness of the initial value problem for the nonlinear sixth-order Boussinesq equation u t t = u x x + β u x x x x + u x x x x x x + ( u 2 ) x x , where β = ± 1 . We prove the local well-posedness with initial data in non-homogeneous Sobolev spaces H s ( R ) for negative indices of s ∈ R .

Exact Travelling Solutions For The Sixth-order Boussinesq Equation

In this paper, we establish some distinct exact solutions for a nonlinear evolution equation. The sin-cosine method and the rational Exp-Function and the rational hyperbolic function method are used to construct the solitary travelling wave solutions of the sixth-order Boussinesq equation . These solutions may be important of significance for the explanation of some practical physical problem.

Non-integrable variants of Boussinesq equation with two solitons

Three variants of the Boussinesq equation, namely, the (2 + 1)-dimensional Boussinesq equation, the (3 + 1)-dimensional Boussinesq equation, and the sixth-order Boussinesq equation are studied. The Hirota bilinear method is used to construct two soliton solutions for each equation. The study highlights the fact that these equations are non-integrable and do not admit N-soliton solutions although these equations can be put in bilinear forms.

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation, which arises from fluid dynamics. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders, educing the related homotopy series solutions. Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation. Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations. The auxiliary parameter has an effect on the convergence of homotopy series solutions. Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.

On the validity and reliability of the ( [formula omitted]/ G)-expansion method by using higher-order nonlinear equations

In this study, we demonstrate the validity and reliability of the so-called ( G ′ / G)-expansion method via symbolic computation. For illustrative examples, we choose the sixth-order Boussinesq equation and the ninth-order Korteweg-de-Vries equation. As a result, the power of the employed method is confirmed.

Multiple-soliton solutions for the ninth-order KdV equation and sixth-order Boussinesq equation

Two nonlinear dispersive equations, namely, the ninth-order KdV equation and the sixth-order Boussinesq equation, are formally derived by generalizing the bilinear forms of the KdV and Boussinesq equations, respectively. The two equations are approached by using the tanh–coth method to obtain single soliton solutions, and by the Hirota bilinear method, to determine the multiple-soliton solutions. The study highlights the fact that both equations are completely integrable and admits N-soliton solutions.

Higher-order Boussinesq equations for two-way propagation of shallow water waves

Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a / h 0 , and long-wavelength parameter, β = ( h 0 / l ) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ / ρ g h 0 2 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | ≫ β , and (2) | 1 3 − τ | = O ( β ) . Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3 . Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.

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 class of model equations for bi-directional propagation of capillary–gravity waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter α=a/h0 and wavelength parameter β=(h0/l)2, where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number τ=Γ/ρgh02, where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity.The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) β≪(1/3−τ)⩽1/3 and (ii) (1/3−τ)=O(β). The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for τ=1/3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.