This collaborative proposal concerns the investigation of nonlinear, dispersive wave phenomena described by the Kadomtsev-Petviashvili (KP) equation and its physical applications, particularly in two-dimensional shallow water waves. The KP equation admits a particular solution called line-soliton, which is a steady propagating wave with high amplitude, like a beach wave. Broadly speaking, the project has two main goals that are interrelated, namely, (i) to study combinatorial and geometric aspects of the solution space of the KP solitons, and (ii) to develop an asymptotic theory where the KP equation as the leading order equation for real applications in two-dimensional wave phenomena. Detailed analytical and numerical studies of the interactions, stability and initial value problem of the KP solitons will be carried out in this project. The theoretical results will be carefully compared with experimental measurements.
Preliminary work suggests that some of these newly discovered solutions of the KP equation may have important physical applications such as the Mach reflection of an oblique incidence wave onto a vertical wall and in the generation of large amplitude "rogue" waves in shallow water near a beach. An objective of this research is to apply the results of this project in order to investigate possible mechanisms generating waves of extremely high elevations frequently observed in open seas and along coastlines, for example, tsunamis. Understanding the nature and dynamics of such extreme waves, and ultimately predicting such wave phenomena in oceans near highly populated coastal areas are significant and urgent tasks. The proposed research activities will involve several undergraduate and graduate students who will gain first-hand research experience in applied mathematics. Since the theory is shared by various other physical systems, it is anticipated that the results from the proposed work would provide insights into areas such as light waves in nonlinear optics, and spin waves in magnetic thin films.
At any beach with flat or nearly flat bottom, we often observe interesting web-like patterns generated by obliquely interacting solitary waves. The main purpose of the project supported by this award is to develop mathematical analyses for those patterns (non-stationary in general) and to investigate possible applications of the analyses to physical systems, based on the Kadomtsev-Petviashvili (KP) equation. The KP equation was proposed in 1970 by two Russian physicists Kadomtsev and Petviashvili, and it provides a model equation to describe two-dimensional surface waves on shallow water. The KP equation is also known to admit special classes of exact solutions in the form of solitary waves, referred to as the KP soliton solutions, or simply the KP solitons. However, these solutions have never been characterized in terms of their regularity and the interaction patterns. Over the past several years, the PI, together with S. Chakravarty (collaborative research partner), has investigated a number of problems related to the KP soliton solutions. They have studied the structure and interaction properties of the KP solitons as well as their applications to shallow water waves. In particular, they found a complete classification of the solution space of the non-singular solutions. Their studies have not only revealed a large variety of solutions that were totally overlooked in the past, but also found that some of these KP soliton solutions have important applications to describe solitary waves in shallow water. In this collaborative research project, the PI has continued his study on the KP equation and explored several issues which are mathematically significant as well as physically relevant. The project was mainly focused on the following two categories: (1) Study of shallow water wave interactions using the KP equation. (2) Classification of the web-like patterns generated by the KP solitons. In the category (1), the main objective was the Mach reflection phenomena which describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. The PI has been also collaborating with Harry Yeh at Oregon State University who has been conducting water-tank experiments of the Mach reflection phenomena. The results of this activity show that the resonant phenomena are well described by the KP equation with higher order corrections, and the wave patterns observed in the experiment fit quite well with some of the KP soliton solutions found in the previous works by the PI and his collaborators. These results also provide useful information for nonlinear wave phenomena in various other important physical systems including as plasma, optics, and ferromagnetics. In the category (2), the PI found that several mathematical tools recently developed in the fields of mathematics such as algebraic combinatorics and representation theory are quite useful for solving the classification problem. In particular, the PI used the ideas from cluster algebra in the combinatorics and total positivity in the representation theory. The classification results were also useful for solving an inverse problem of constructing KP solitons from the wave pattern observed, for example, in a surface of shallow water. The results from this project help advance the understanding of two-dimensional solitary waves for nonlinear wave equation of physical significance, and elucidate the deep connection that exists between integrable systems of partial differential equations and other fields of mathematics such as geometric as well as algebraic combinatorics and representation theory. The results were disseminated through archival journal articles and conference presentations. The PI also delivered lectures based on the investigation for the project at several national and international organizations. In particular, the PI was selected to give a series of ten lectures in â€˜â€™NSF/CBMS conference of Mathematical Sciencesâ€™â€™ at UTPA, Texas, May 20-24, 2013. The title of the series was â€˜â€™Solitons in two-dimensional water waves and applications to tsunamiâ€™â€™, and based on the KP solitons, the PI proposed a possible description of the dynamics of tsunami observed in the 2011 Tohoku earthquake.