Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K_m−1 of arithmetic genus m−1, from which we establish the associated Baker-Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.     

Stabilization of the Hierarchy in Brane-World Scenarios

We consider a class of warped brane models with topology M ₄ × × S¹/Z ₂, where is a D₂-dimensional compact manifold, and two branes are placed at the orbifold fixed points. In a scenario where supersymmetry is broken not far below the cutoff scale, the hierarchy between the electroweak and the Planck scales is generated by a combination of the redshift and the large volume effects. We evaluate the effective potential induced by bulk scalar fields in these models and show that it can stabilize the moduli and the hierarchy without fine-tuning, provided that the internal space is flat. We also comment on the relation between these models and the five-dimensional scalar-tensor models that describe them classically when the compactification scale is small.     

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.     

A New Loop Algebra and Its Corresponding Multi-component Integrable Hierarchy

A type of new loop algebra $\tilde{G}_M$ is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.     

The Restricted Boussinesq Flows,Their Lax Representations and r-Matrix

We present the Lax representations for the restricted Boussinesq flows resulted from the binary nonlinearization,of Lax systems of Boussinesq equation. Furthermore we show this Lax operator satisfies the T-matrix relation. Thus 3N involutive integrals for restricted Boussinesq flows are obtained.     

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.     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics.Various nonlinear excitations of the Boussinesq equation have been found by many methods.However,it is very difcult to find interaction solutions among diferent types of nonlinear excitations.In this peper,two equivalent very simple methods,the truncated Painlev′e analysis and the generalized tanh function expansion approaches,are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

Solutions to the Boltzmann Equation in the Boussinesq Regime

We consider a gas in a horizontal slab in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number is small and the temperature difference between the walls as well as the velocity field is of order , while the gravitational force is of order ². We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order ², to the density, velocity and temperature obtained from the smooth solution of the Oberbeck–Boussinesq equations assumed to exist for .     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics. Various nonlinear excitations of the Boussinesq equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this peper, two equivalent very simple methods, the truncated Painlevé analysis and the generalized tanh function expansion approaches, are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

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.     

Magnetic curves in Sol3

Magnetic curves with respect to the canonical contact structure of the space Sol₃ are investigated.     

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     

Model flames in the Boussinesq limit: Rising bubbles

Using the Boussinesq buoyancy approximation, we study a bubble of reaction products rising in the reactant fluid under the influence of gravity. Reaction on the surface of the bubble (the flame) results in an increase of the volume of the bubble. We consider fluids with low Prandtl and high Froude numbers (heat diffusion dominates over viscous dissipation, and burning dominates over gravitational effects). We show that, under these conditions, all initially small bubbles follow the same growth pattern, regardless of the flame speed, the reaction type, the gravity, the viscosity, the initial size, and, to some extent, the initial shape of the bubble. In the initial stage of this similarity solution a bubble grows radially in an essentially motionless fluid until it reaches some critical size, which is determined by the laminar flame speed, the gravitational acceleration, and the Atwood number. Once the bubble reaches the critical size, convection becomes significant and the bubble evolves into a more complicated, mushroom-like shape. The similarity solution is expressed using the critical bubble size for the unit length and the critical size divided by the laminar flame speed as the unit time.     

A Note on the Modified KP Hierarchy and its (Yet another) Dispersionless Limit

The modified KP hierarchies of Kashiwara and Miwa are formulated in Lax formalism by Dickey. Their solutions are parametrised by flag varieties. Its dispersionless limit is considered.     

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.     

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.     

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.