Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations were constructed for a given curve y = f(x) whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) 27, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator D, ident...

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and...

A new theory for the emergence of dispersion in shallow-water hydrodynamics in two horizontal-space dimensions is presented. Starting with the key properties of uniform flow in open channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related prec...

The purpose of this article is to explore different types solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which termed as KP-Gardner equation, extensively used model strong nonlinear internal waves in ( 1 + 2 )-dimensions on stratified ocean shelf. This evoluti...

Pattern formation in higher-order lumps of the Kadomtsev–Petviashvili I equation at large time is analytically studied. For a broad class these lumps, we show that two types solution patterns appear time. The first type comprises fundamental arranged triangular shapes, which are described by root structures Yablonskii–Vorob’ev polynomials. As evolves from negative to positive, this pattern reve...

A substantial part of the energy of wake waves from high-speed ships sailing in shallow water is concentrated in nonlinear components which at times have a solitonic nature. Recent results of investigations into solitonic wave interactions within the framework of the Kadomtsev-Petviashvili equation and their implications for rogue wave theory are reviewed. A surface elevation four times as high...

We present a novel approach to the Kadomtsev-Petviashvili (KP) hierarchy and its modified counterpart, the mKP hierarchy based on factorizations of formal pseudo-differential operators and a matrix-valued Lax operator for the mKP hierarchy. As a result of this framework we obtain new Bäcklund transformations for the KP hierarchy and the possibility of transferring classes of KP solutions into t...

Under investigation is a generalized (3+1)-dimensional B-type Kadomtsev-Petviashvili equation with variable coefficients in fluid dynamics. Based on the Hirota's bilinear form and positive quadratic function, abundant lump solutions are obtained. The interaction between other solitons also presented. Their dynamical behaviors graphically shown different choices of free parameters.

Interaction of two long-crested shallow water waves is analysed in the framework of the two-soliton solution of the Kadomtsev-Petviashvili equation. The wave system is decomposed into the incoming waves and the interaction soliton that represents the particularly high wave hump in the crossing area of the waves. Shown is that extreme surface elevations up to four times exceeding the amplitude o...

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τ -function. We investigate the potential using the non-local ∂̄ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally...