Search

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Bäcklund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Bäcklund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Bäcklund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis–Procesi (DP) equation, and construct a Lax pair of it. In addition, the Bäcklund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a Z n -DP equation. Considering the superposition principle, we obtain the solution of an associated Z n -DP equation by using Bäcklund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we performed Lie symmetry analysis and applied G ′ / G expansion method on the nonlinear dissipative–dispersive equation. The purpose of this research is to find the vector fields and transform the nonlinear dissipative–dispersive equation into simpler forms. The Maple software was used to obtain the vector field and similarity reductions for nonlinear dissipative–dispersive equations. In addition, we obtained exact solutions based on the G ′ / G expansion method and power series method, including the hyperbolic functions, the trigonometric functions and the rational functions. The method we used is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations. Furthermore, the invariant subspaces of the nonlinear dissipative–dispersive equation were identified using the refined invariant subspaces method. The invariant subspaces of solutions to linear ordinary differential equations were used to prove that nonlinear dissipative–dispersive equation admits subspaces. The exact solutions were obtained by using generalized separated variables. [ABSTRACT FROM AUTHOR]

We generalize the constrained modified KP (mKP) hierarchy to the multi-component version whose tau function can be expressed as the vacuum expectation value of Clifford operators using free fermions. Meanwhile, we extend the relevant theory of the Boson–Fermion correspondence in single component constrained mKP hierarchy to the multi-component version. This leads us to obtain the rational and soliton solutions of the multi-component constrained mKP hierarchy utilizing the Boson–Fermion correspondence. Furthermore, we give five sets of differential Fay identities and two sets of difference Fay identities of the multi-component mKP hierarchy. [ABSTRACT FROM AUTHOR]

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge-Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out. [ABSTRACT FROM AUTHOR]

The nonlinear dynamical behaviors of single-walled carbon nanotubes (SWCNTs) with the initial axial stress are investigated based on Donnell's cylindrical shell model. By using the bifurcation theory of dynamical systems on a vibration equation of SWCNTs, the existence of centers, saddles, families of periodic orbits, homoclinic orbits and/or heteroclinic orbits is shown. Under different initial conditions, various sufficient conditions to guarantee the existence of the above solutions are given and the dynamical behaviors of SWCNTs are well-known. [ABSTRACT FROM AUTHOR]

The turbulent flow in a reacting hydro turbine was simulated with large eddy simulation (LES) to investigate the spatial and temporal distributions of the turbulence in a 3D skew blade passage. The simulations show that the intrinsic features are quite complicated. LES can capture the details of the structures, and the results are generally agreeable with those by direct numerical simulation. [ABSTRACT FROM AUTHOR]