Secondly combining group theory and singularity analysis we obtained some information about the similarity solutions of the fourth order generalized Boussinesq equation GBE4. It has been shown that this equation possess a three dimensional Lie algebra of the point symmetries and the similarity reductions due to these symmetries have been obtained.

In this Letter, we study (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation by using the new generalized transformation in Homogeneous Balance Method (HBM). As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.  2002 Elsevi...

We review recent computational results for hexagon patterns in nonBoussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a verti...

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the 2D lattice, having 3D consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discr...

We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE’s for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Bo...

Two different linearized schemes are applied to a parametric finite-difference scheme concerning the numerical solution of the Boussinesq equation. At the first linearized scheme the nonlinear term of the equation is substituted by an appropriate value, while at the second scheme we use Taylor’s expansion. Both schemes are analyzed for local truncation error, stability and convergence. The resu...

where u (x, t) denotes the unknown function, f (s) is the given nonlinear function, u0 (x) and u1 (x) are the given initial value functions, k is a constant, the subscript t indicates the partial derivative with respect to t, n is the dimension of space variable x, and △ denotes the Laplace operator in R. The effects of small nonlinearity and dispersion are taken into consideration in the deriv...

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...