Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations

The ‘tanh-coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh-coth expansion. It is known that, for every tanh-function expansion solution, there is a corresponding coth-function expansion solution. It is shown that there is a tanh-coth expansion solution that is merely a disguised version of the coth solution. In many papers, such tanh-coth solutions are erroneously claimed to be ‘new’. However, other tanh-coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh-function method. Similar remarks apply to tan, cot and tan-cot expansion solutions.     

Bifurcations of travelling wave solutions for the generalized KP-BBM equation

By using the bifurcation theory of dynamical systems to the generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, compactons solution, non-smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Explicit peakon and solitary wave solutions for the modified Fornberg-Whitham equation

In this paper, the modified Fornberg-Whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis. In some parametric conditions, some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained.     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the (2 + 1)-dimensional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Under some parameter conditions, exact solitary wave solutions and kink wave solutions are obtained.     

A Note on Solitary Subgroups of Finite Groups

We say that a subgroup H of a finite group G is solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of G such that no other subgroup (respectively, normal subgroup) of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. We show some new results about lattice properties of these subgroups and their relation with classes of groups and present examples showing a negative answer to some questions about these subgroups.     

Solitary wave solutions of the generalized two-component Hunter–Saxton system

The existence of solitary wave solutions of the generalized two-component Hunter–Saxton system is determined. It is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions.     

Domain wall motion in ferromagnets modelled by a quintic complex Ginzburg-Landau equation

A quintic complex Ginzburg-Landau equation is derived from a Landau-Lifshitz-Gilbert equation and is used to describe the magnetization dynamics in a one-dimensional uni-axial ferromagnet. Trough the use of suitable approximations, we derive the magnetic solitary wave excitations solutions which have pulse-like shapes. Subsequent numerical simulations reveal domain wall propagation.     

Asymmetric Synthesis of Chiral Flavan-3-Ols

Flavan-3-ols are a series of natural products widely present in plants and show versatile biological activities. The structures of such compounds are characterized by owing two adjacent chiral centers and three rings. Their interesting structures and promising biological activities have driven increasing research developments toward the preparation of enantioenriched flavan-3-ols. This review summarizes the recent approaches for the asymmetric synthesis of chiral flavan-3-ols from two strategies in the construction of chiral centers. The key steps in the synthetic protocol involve Sharpless asymmetric dihydroxylation, Shi asymmetric epoxidation and Sharpless asymmetric epoxidation.     

A normalized solitary wave solution of the Maxwell-Dirac equations

We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.