Ill-posed Boussinesq Equation Research Articles

Stability analysis and multi-wave structures of the ill-posed Boussinesq equation arising in nonlinear physical science

Stability analysis and multi-wave structures of the ill-posed Boussinesq equation arising in nonlinear physical science

Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

A Note on the $$\exp (-\varphi (z))$$ Expansion Method

In this paper, we prove the equivalence between the $$\exp (-\varphi (z))$$ expansion method and the tanh method. The two methods give the same exact traveling wave solutions of nonlinear partial differential equations. We use the ill-posed Boussinesq equation as an example to demonstrate this fact.

Response Solution to Ill-Posed Boussinesq Equation with Quasi-Periodic Forcing of Liouvillean Frequency

In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: $$\begin{aligned} \begin{aligned} y_{tt}(t,x)=\mu y_{xxxx}+y_{xx}+\left( y^{3}+\varepsilon f(\omega t,x)\right) _{xx},\,\, x\in [0, \pi ],\,\, \mu >0, \end{aligned} \end{aligned}$$subject to the hinged boundary conditions $$\begin{aligned} \begin{aligned} y(t,0)=y(t,\pi )=y_{xx}(t,0)=y_{xx}(t,\pi )=0, \end{aligned} \end{aligned}$$where $$\omega =(1,\alpha )$$ with $$\alpha $$ being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allow $$\alpha $$ to be any irrational numbers, and thus the frequency $$\omega =(1,\alpha )$$ is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine.

Auxiliary equation method for ill-posed Boussinesq equation

This paper comprises the extraction of many new general exact soliton solutions of the ill-posed Boussinesq equation, which has a number of applications in mathematical physics, by employing auxiliary equation method. The method successfully yields bell shape soliton, singular soliton, topological (dark) soliton, non-topological (bright) soliton and periodic soliton solutions of nonlinear partial differential equations solutions. These solutions are greater in number as compared to those attained by the exiting techniques. Parameters, contained in the solutions are given appropriate values for intelligible physical facets and vivid graphical view. Our results justify the efficacy, validity and favorability of auxiliary equation method for promising results while solving many other nonlinear partial differential equations.

The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science

In manuscript, with the help of the Wolfram Mathematica 9, we employ the modified exponential function method in obtaining some new soliton solutions to the ill-posed Boussinesq equation arising in nonlinear media. Results obtained with use of technique, and also, surfaces for soliton solutions are given. We also plot the 3D and 2D of each solution obtained in this study by using the same program in the Wolfram Mathematica 9.

Dynamics of solitons to the ill-posed Boussinesq equation

In this paper, we analyze the dynamic behavior of the ill-posed Boussinesq equation (IPBE) that arises in nonlinear lattices and also in shallow water waves. Some solitary wave solutions are obtained by using the solitary wave ansatz method and the Bernoulli sub-Ode. By applying the technique of nonlinear self-adjoint, a quasi self-adjoint substitution for the IPBE is constructed. The classical symmetries of the equation are constructed. Then, we used along with the obtained nonlinear self-adjoint substitution to construct a set of new conservation laws (Cls).

The investigation of exact solutions of nonlinear partial differential equations by using exp(−Φ(ξ)) method

In this work, the methodology of the exp(−Φ(ξ)) method is given. The effectiveness of the method is demonstrated by applications to two nonlinear partial differential equations (NPDEs), namely, the Zakharov Kuznetsov–Benjamin Bona Mahony (ZK-BBM) equation and ill-posed Boussinesq equation. The obtained exact solutions are consist of trigonometric, hyperbolic and rational functions.

Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Abstract In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

Symmetry reductions and exact solutions to the ill-posed Boussinesq equation

In this paper, using the Lie symmetry analysis method, we study the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. The similarity reductions and exact solutions for the equation are obtained. Then the exact analytic solutions are considered by the power series method, and the physical significance of the solutions is considered from the transformation group point of view.

Thermal diffusion of Boussinesq solitons

We consider the problem of the soliton dynamics in the presence of an external noisy force for the Boussinesq type equations. A set of ordinary differential equations (ODEs) of the relevant coordinates of the system is derived. We show that for the improved Boussinesq (IBq) equation the set of ODEs has limiting cases leading to a set of ODEs which can be directly derived either from the ill-posed Boussinesq equation or from the Korteweg-de Vries (KdV) equation. The case of a soliton propagating in the presence of damping and thermal noise is considered for the IBq equation. A good agreement between theory and simulations is observed showing the strong robustness of these excitations. The results obtained here generalize previous results obtained in the frame of the KdV equation for lattice solitons in the monatomic chain of atoms.

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.

Approximation of ill-posed boussinesq equations

In this paper we compare the dynamics on the centre manifold of the solutions of an ill-posed Boussinesq equation with a well-posed version. We show that the dynamics in the centre manifold of the ill-posed equation tracks the dynamics of the well-posed equation. Our results give partial justification to the long-wave perturbation theory.

A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques

We consider an ill-posed Boussinesq equation which arises in shallow water waves and nonlinear lattices. This equation has growing and decaying modes in the linear as well as nonlinear regimes and its linearized growth rate σ for short-waves of wavenumber k is given by σ∼ k 2. Previous numerical studies have addressed numerical difficulties and construction of approximate solutions for ill-posed problems with short-wave instability up to σ∼ k, e.g. Kelvin–Helmholtz ( σ∼ k) and Rayleigh–Taylor (σ∼ k ) instabilities. These same issues are addressed and critically examined here for the present problem which has more severe short-wave instability. In order to develop numerical techniques for constructing good approximate solutions of this equation, we use a finite difference scheme to investigate the effect of this short-wave instability on the numerical accuracy of the exact solitary wave solution of this equation. Computational evidence is presented which indicates that numerical accuracy of the solutions is lost very quickly due to severe growth of numerical errors, roundoff as well as truncation. We use both filtering and regularization techniques to control growth of these errors and to provide better approximate solutions of this equation. In the filtering technique, numerical experiments with three types of spectral filters of increasing order of regularity are performed. We examine the role of regularity of these filters on the accuracy of the numerical solutions. Numerical evidence is provided which indicates that the regularity of a filter plays an important role in improving the accuracy of the solutions. In the regularization technique, the ill-posed equation is regularized by adding a higher order term to the equation. Two types of higher order terms are discussed: (i) one that diminishes the growth rate of all modes below a cutoff wavenumber and sets the growth rate of all modes above it to zero; and (ii) the other one diminishes the growth rate of all modes and the growth rate asymptotically approaches to zero as the wavenumber approaches infinity. We have argued in favor of the first type of regularization and numerical results using a finite difference scheme are presented. Numerical evidence is provided which suggests that regularization in combination with the most regular ( C 2 here) spectral filter for small values of the regularization parameter can provide good approximate solutions of the ill-posed Boussinesq equation for longer time than possible otherwise. Some of the ideas presented here can possibly be utilized for solving other ill-posed problems with severe short-wave instabilities and may have an important role to play in numerical studies of their solutions.