KP solitons in shallow water
By Yuji Kodama. 2010
155 citations recorded by the source corpus.
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev–Petviashvili (KP) equation. The KP equation describes weakly dispersive and small amplitude wave propagation in a quasi-two-dimensional framework. Recently, a large variety of exact soliton solutions of the KP equation has been found and classified. These solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and are called line-solitons. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain classes of initial waves asymptotically approach to these exact solutions of the KP equation. We discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extraordinary fourfold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included in the quasi-two-dimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.
Subfields: Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/4d7fe3d5b5284c4be47d8a90d23fd782c88beaba
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
In Modeling and Simulation
In Statistical and Nonlinear Physics
Citations in the corpus
- KP solitons and total positivity for the Grassmannian (2011, methodology): “The KP equation can be also used to describe two-dimensional shallow water wave phenomena (see for example [15]).”