Travelling-wave solutions of the Degasperis–Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodicwave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well. Hone and Wang showed a connection between the DPE and the Vakhnenko equation (VE). Comparing the solutions of the ...

In a recent paper [Abazari R. Application of ( ′ G )-expansion method to travelling wave solutions of three nonlinear evolution equation. Computers & Fluids 2010;39:1957–1963], the (G/G)-expansion method was used to find travelling-wave solutions to three nonlinear evolution equations that arise in the mathematical modelling of fluids. The author claimed that the method delivers more general fo...

A model describing non-equilibrium processes in relaxing media is considered. Restrictions arising from the symmetry principles and the second law of thermodynamics are stated. System of ODE describing a set of travelling wave solutions is obtained via group theory reduction. Bifurcation analysis of this system reveals the existence of periodic invariant solutions as well as limiting to them so...

In this paper, we establish the existence of shock solutions for a simplified version of the Othmer–Stevens chemotaxis model (SIAM J. Appl. Math. 1997; 57:1044–1081). The existence of these shock solutions was suggested by Levine and Sleeman (SIAM J. Appl. Math. 1997; 57:683–730). Here, we consider the general Riemann problem and derive the shock curves in parameterized forms. By studying the t...

In this paper, we construct exact travelling wave solutions for the generalized (2+1)dimensional ZK-MEW equation by using the solutions of an auxiliary ordinary differential equation given by Sirendaoreji [1]. It is shown that some solutions obtained in this study are new solutions which have not been reported yet.

A new procedure for finding exact travelling wave solutions to the modified CamassaHolm and Degasperis-Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.

The recent theoretical discovery of families of travelling wave solutions in pipe flow (Faisst & Eckhardt 2003; Wedin & Kerswell 2004; Hof et al. 2004) at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these ...

Travelling wave solutions for the FitzHugh-Nagumo equations have been proved to exist, by various authors, close to a certain singular limit of the equations. In this paper it is proved that these waves are stable relative to the full system of partial differential equations; that is, initial values near (in the sup norm) to the travelling wave lead to solutions that decay to some translate of ...

Travelling-wave solutions of the standard and compound form of Kortewegde Vries-Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli equations of nonlinearity 3/2 and 2 (Riccati), respectively. Introducing the initial conditions through an imaginary phase in the travelling coordinate, we obta...

The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a uid, or the interface between two homogeneous uids of di ering densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincaré theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of ...