A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.     

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.     

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.     

Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing

This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying with Dirichlet boundary conditions, where ε is a small positive parameter, (t) is a real analytic quasi-periodic function in t with frequency vector ω=(ω₁,ω₂…,ω_m) and the nonlinearity h is a real analytic odd function of the form It is shown that, under a suitable hypothesis on (t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

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...     

一类广义Boussinesq方程和Boussinesq-Burgers方程新的显式精确解

利用一种推广的代数方法，求解了一类广义Boussinesq方程(B(m，n)方程)和Boussinesq-Burgers方程(B-B方程).获得了其多种形式的显式精确解，包括孤波解、三角函数解、有理函数解、Jacobi椭圆函数周期解和Weierstrass椭圆函数周期解，进一步丰富了这两类方程的解. 关键词： Boussinesq方程 Boussinesq-Burgers方程 推广的代数方法 显式精确解     

用三角函数法获得非线性Boussinesq方程的广义孤子解

找到一个合适的代换——三角函数法，将非线性Boussinesq微分方程转换为非线性代数方程组.用吴消元法求解该非线性代数方程组，从而获得一般形式Boussinesq微分方程的广义孤子解. 关键词： Boussinesq方程 吴消元法 非线性代数方程组 孤子解     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

(2+1)维Boussinesq方程的Backlund变换与精确解

借助于符号计算软件Maple，对方程的种子解作适当的未知函数替换，然后利用Backlund 变 换通过具体的符号演算获得了(2+1)维Boussinesq方程的一系列精确解.这些解包括类孤子解 和有理解,其中有的解中含有任意函数，当任意函数取特殊函数时，这些解具有丰富的结构 ，有些结构可能对物理现象的研究是有意义的. 关键词： (2+1)维Boussinesq方程 Backlund 变换 精确解 类孤子解     

Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

Symmetry Reductions and Integrability for the Modified Boussinesq Equation from EPU Problem

In this paper, we discuss symmetry reductions of the modified Boussinesq equation arising in the Fermi-Pasta-Ulam problem, which is used to investigate the behaviour of systems which are primarily linear but a nonlinearity is introduced as a perturbation by using the direct method due to Clarkson and Kruskal. A symmetry group theoretical explanation for the results is provided. We conclude that the modified Boussinesq equation is not integrable.     

Explicit N-fold Darboux transformation for the classical Boussinesq system and multi-soliton solutions

In this Letter, a systematic method is presented to construct the N-fold Darboux transformation with multi-parameters for a spectral problem associated with a classical Boussinesq system. As an application, we obtain the new soliton solutions of the classical Boussinesq system.     

Intersections in Genus 3 and the Boussinesq Hierarchy

In this Letter we prove that the enlarged Witten conjecture is true in the case of the Boussinesq hierarchy for correlators .     

Non-autonomous discrete Boussinesq equation:Solutions and consistency

A non-autonomous 3-component discrete Boussinesq equation is discussed. Its spacing parameters pn and qm are related to independent variables n and m, respectively. We derive bilinear form and solutions in Casoratian form. The plain wave factor is defined through the cubic roots of unity. The plain wave factor also leads to extended non-autonomous discrete Boussinesq equation which contains a parameter δ. Tree-dimendional consistency and Lax pair of the obtained equation are discussed.     

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.     

Newly modified method and its application to the coupled Boussinesq equation in ocean engineering with its linear stability analysis

Investigating the dynamic characteristics of nonlinear models that appear in ocean science plays an important role in our lifetime. In this research, we study some features of the paired Boussinesq equation that appears for two-layered fluid flow in the shallow water waves. We extend the modified expansion function method (MEFM) to obtain abundant solutions, as well as to find new solutions. By using this newly modified method one can obtain novel and more analytic solutions comparing to MEFM. Also, numerical solutions via the Adomian decomposition scheme are discussed and favorable comparisons with analytical solutions have been done with an outstanding agreement. Besides, the instability modulation of the governing equations are explored through the linear stability analysis function. All new solutions satisfy the main coupled equation after they have been put into the governing equations.     

Painleve properties and exact solutions for the high-dimensional Schwartz Boussinesq equation

The usual （1＋1）-dimensional Schwartz Boussinesq equation is extended to the （1＋1）-dimensional space-time symmetric form and the general （n＋1）-dimensional space-time symmetric form. These extensions are Painleve integrable in the sense that they possess the Painleve property. The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space-time symmetric form are obtained by the Painleve-Backlund transformation.     

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.