We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrödinger (NLS) equation, the λφ4 model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual pr...

A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of one, or two variables. The conditions are determined under which the equation allows an infinite dimensional symmetry algebra. This algebra can involve up to three arbitrary functions of time. It depends on precisely...

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial diierential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of diierential equations. A general existence and uniqueness...

A bilinear estimate in terms of Bourgain spaces associated with a linearised Kadomtsev-Petviashvili-type equation on the three-dimensional torus is shown. As a consequence, time localized linear and bilinear space time estimates for this equation are obtained. Applications to the local and global well-posedness of dispersion generalised KP-II equations are discussed. Especially it is proved tha...

Using the method of positions of the complex singularities, we identify a class of new, exact solutions of the Flierl-Petviashvili equation. The solutions are periodic and have the geometry of the zonal flow. We examine the physical properties and find that the solution can reproduce data from experimental observations and numerical simulations.

The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained.