Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Dynamics and Exact Solutions of (1 + 1)-Dimensional Generalized Boussinesq Equation with Time-Space Dispersion Term

We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.

Diversity of Rogue Wave Solutions to the (1+1)-Dimensional Boussinesq Equation

A periodically homoclinic solution and some rogue wave solutions of (1+1)-dimensional Boussinesq equation are obtained via the limit behavior of parameters and different polynomial functions. Besides, the mathematics reasons for different spatiotemporal structures of rogue waves are analyzed using the extreme value theory of the two-variables function. The diversity of spatiotemporal structures not only depends on the disturbance parameter u0 but also has a relationship with the other parameters c0, α, β.

New method of local adjuvant therapy with bicarbonate Ringer’s solution for tumoral calcinosis: A case report

BACKGROUND Tumoral calcinosis is a condition characterized by deposits of calcium phosphate crystals in extra-articular soft tissues,occurring in hemodialysis patients.Calcium phosphate crystals are mainly composed of hydroxyapatite,which is highly infilt-rative to tissues,thus making complete resection difficult.An adjuvant method to remove or resolve the residual crystals during the operation is necessary.CASE SUMMARY A bicarbonate Ringer’s solution with bicarbonate ions(28 mEq/L)was used as the adjuvant.After resecting calcium phosphate deposits of tumoral calcinosis as much as possible,while filling with the solution,residual calcium phosphate deposits at the pseudocyst wall can be gently scraped by fingers or gauze in the operative field.A 49-year-old female undergoing hemodialysis for 15 years had swelling with calcium deposition for 2 years in the shoulders,bilateral hip joints,and the right foot.A shoulder lesion was resected,but the calcification remained and early re-deposition was observed.Considering the difficulty of a complete rection,we devised a bicarbonate dissolution method and excised the foot lesion.After resection of the calcified material,the residual calcified material was washed away with bicarbonate Ringer’s solution.CONCLUSION The bicarbonate dissolution method is a new,simple,and effective treatment for tumoral calcinosis in hemodialysis patients.

Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.

正压流体中具有beta效应的非线性Boussinesq方程及其多孤子解

本文主要考虑了具有beta效应的大尺度Rossby波控制方程的多孤子解。首先,从传统意义下的正压准地转位涡方程出发,利用多尺度分析法、坐标变换法、泰勒展开法以及分离变量法,推导出大尺度Rossby波的振幅满足带有beta效应的非线性Boussinesq方程。然后,基于这个方程通过适当的变量代换给出了模型方程的Hirota双形式,并利用Hirota双线性方法获得了非线性Boussinesq方程的1-孤子解、2-孤子解、3-孤子解及N-孤子解。基于模型方程得到推广的beta效应能诱导生成Rossby波,并分析孤子解的结果表明,随着非线性系数的增大,孤子解的振幅在逐渐变小。

Boussinesq方程的精确行波解

利用广义kudryashov方法讨论了Boussinesq方程的行波解,给出了其5个精确指数函数解的显式表达式,并利用Maple软件给出了该方程3个精确解的性态。

Adomian Modification Methods for the Solution of Chebyshev’s Differential Equations

The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.

ANALYTICAL SMOOTHING EFFECT OF SOLUTION FOR THE BOUSSINESQ EQUATIONS

In this article, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H^1 -solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equations admit exactly same smoothing effect properties of incompressible Navier-Stokes equations.

Multiple Soliton Solutions of Alice–Bob Boussinesq Equations

Three Alice-Bob Boussinesq(ABB) nonlocal systems with shifted parity■, delayed time reversal■ and ■ nonlocalities are investigated. The multi-soliton solutions of these models are systematically found from the ■ symmetry reductions of a coupled local Boussinesq system. The result shows that for ABB equations with ■ nonlocality, an odd number of solitons is prohibited. The solitons of the ■ nonlocal ABB and ■ nonlocal ABB equations must be paired, while any number of solitons is allowed for the ■ nonlocal ABB system. t-breathers, x-breathers and rogue waves exist for all three types of nonlocal ABB system.In particular, different from classical local cases, the first-order rogue wave can have not only four leaves but also five and six leaves.

A General Mapping Approach and New Travelling Wave Solutions to（2＋1）-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinese equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreover, it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.

MULTI-SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS AND KUPERSHMIDT EQUATIONS

Using the homogeneous balance method introduced by Wang Mingliang, the multi- solitary wave solutions are obtained for the variant Boussinesq equation and Kupershmidt equation. The Wang's result is a special case of above results for the variant Boussinesq equation.

Symmetry Reductions, Integrability and Solitary Wave Solutions to High-Order Modified Boussinesq Equations with Damping Term

Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.

Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations with Vacuum

We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on . At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that there is a global strong solution and is unique for the 2D Cauchy problem with the initial density which can allow vacuum conditions and even have compact support. Besides, the large time decay rates of the gradients of velocity, temperature and pressure can also be obtained which are also the same as those of the homogeneous case.

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schrodinger （NLS） equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright-bright, bright-dark, and dark-dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright-bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright-bright or bright-dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHRDINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation (S-B equations) which dealing with the stationary propagation of coupled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed. To examine its solitary wave solutions, a reduced set of ordinary differential equations are considered by a simple traveling wave transformation. It is then shown that several new solutions (either functional or parametrical) can be obtained systematically, in addition to rederiving all known ones by means of our simple and direct algebra method with the help of the computer algebra system Maple.