Modified Richards equation and its exact solutions for soil water dynamics on eroding hillslopes

Exact travelling wave solutions for the modified Novikov equation ∗

which was discovered in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity. By using the perturbation symmetry approach [7], Novikov found the first few symmetries and a scalar Lax pair for Eq. (1), then proved that it is integrable [9]. Hone and Wang [5] gave a matrix Lax pair for the Novikov equation and found its infinitely many conserved quantities, as well as a...

Exact Travelling Wave Solutions for a Modified Zakharov–Kuznetsov Equation

The modied Zakharov–Kuznetsov (mZK) equation, ut + uux + uxxx + uxyy = 0, (1) represents an anisotropic two-dimensional generalization of the Korteweg–de Vries equation and can be derived in a magnetized plasma for small amplitude Alfvén waves at a critical angle to the undisturbed magnetic field, and has been studied by many authors because of its importance [1–5]. However, Eq. (1) possesses m...

Burgers’ Equation and Layered Media: Exact Solutions and Applications to Soil-water Flow

Exact solutions are presented for Burgers’ equation in a fmite layer connected to an underlying semi-infinite medium of different conductivity and diffusivity. A constant-flux boundary condition is assumed at the surface. This has direct application to steady rainfall on layered field soils. At large times, a travelling wave profile develops in the deep layer and the concentration in the upper ...

Painlevé Analysis and Exact Solutions of a Modified Boussinesq Equation

We consider a modified Boussinesq type equation. The Painlevé test of the WTC method is performed for this equation and it shows that the equation has weak Painlevé property. Some exact solutions are constructed.

Exact and numerical solutions for nonlinear differential equation of Jeffrey-Hamel flow