Related papers: On the integrable elliptic cylindrical Kadomtsev-P…
The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by…
The Kadomtsev-Petviashvili (KP) nonlinear wave equation is the three dimensional generalization of the Korteveg-de Vries (KdV) equation. We show how to obtain the cylindrical KP (cKP) and cartesian KP in relativistic fluid dynamics. The…
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…
The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $\beta>1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric…
The propagation of nonlinear and dispersive waves in various materials can be described by the well-known Kadomtsev-Petviashvili (KP) equation, which is a (2+1)-dimensional partial differential equation. In this paper, we show that the KP…
Full dispersive models of water waves, such as the Whitham equation and the full dispersion Kadomtsev-Petviashvili (KP) equation, are interesting from both the physical and mathematical points of view. This paper studies analogous full…
A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are…
By considering the inhomogeneities of media, a generalized variable-coefficient Kadomtsev-Petviashvili (vc-KP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. In this…
The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and…
The KP-I equation \[ (u_t-2uu_x+\tfrac{1}{2}(\beta-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0 \] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number $\beta>1/3$). This equation admits ---…
The coupled Kadomtsev--Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo and Miwa [J. Phys. Soc. Jpn., 52:766--771, 1983], is reinvestigated within the direct linearisation framework, which provides us with more…
In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The…
The aim of this article is to derive surface wave models in the presence of surface tension and viscosity. Using the Navier-Stokes equations with a free surface, flat bottom and surface tension, we derive the viscous 2D Boussinesq system…
We study transverse stability and instability of one-dimensional small-amplitude periodic traveling waves of a generalized Kadomtsev-Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or…
A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation based on an elliptic Cauchy kernel. This yields a (3+1)-dimensional lattice system with one of the…
In this paper, we study the three-dimensional gravity-capillary water wave problem involving an irrotational, perfect fluid with gravity and surface tension. We focus on steady waves propagating uniformly in one direction. Assuming constant…
We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.
We classify the Lie point symmetries for the 2+1 nonlinear generalized Kadomtsev-Petviashvili equation by determine all the possible f(u) functional forms where the latter depends. For each case the one-dimensional optimal system is…
We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution…
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation,…