Dynamics of KPI lumps

Abstract

A family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the xy-plane are altered after collision. Thus they differ from the standard multi-peaked KPI simple n-lump solutions whose peak heights as well as peak trajectories remain unchanged after interaction.The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomena is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long time asymptotics of the solutions that the peak trajectories separate as O(√|t|) as |t| → ∞, and all the peak heights approach the same constant value corresponding to that of the simple 1-lump solution. Consequently, a multi-peaked n-lump solution evolves to a superposition of n 1-lump solutions asymptotically as |t| →∞.

Classification of KPI Lumps

“…This particular 2-lump solution and another solution which corresponds to n = 1, m 1 = 2, were found earlier in [20,45,3]. The latter solution was also studied more recently in [10], where the structure of the solution was analyzed in details. The analysis for the solutions corresponding to τ 2 in (2.18) is similar, so we do not include it here.…”

“…In this section, we will show that the above features also hold for an arbitrary positive integer N . Further evidence of such behavior exhibited by a special family of multi-lump solutions was presented in a recent paper [10] by the authors.…”

“…These roots describe the approximate peak locations of N -lump solutions of KPI when the associated partition λ has only one part, i.e., n = 1 and λ = (N ). These were recently studied in [10].…”

Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

“…[13,14] A large family of rational multi-lump solutions of the KPI equation were obtained by means of the Gramian method by Chakravarty and Zowada. They proved that there exists a oneto-one correspondence between the multi-lump solutions and partitions of a positive integer N. [15,16] A reduced version of the Grammian form solution was employed to present a direct method to study the multi-lump molecules of the KPI equation. [17] A new type of lump molecules appearing in a period of time of the KPI equation was obtained.…”

Asymptotic dynamics of higher-order lumps in the Davey-Stewartson II equation

“…The other class of lumps on the zero background of the DS II equation are also studied by using Wronskian determinant [16] and inverse scattering method [17,18], which behave highly nontrivially upon interaction (a head-on collision results in a orthogonal scattering). Such dynamical phenomenon are also of relevance to other 2 + 1-dimensional integrable systems, such as the KP I equation [19][20][21][22][23], 2 + 1-dimensional NLS equation [24,25], 2 + 1-dimensional asymmetric Nizhnik-Novikov-Veselov system [26] and the 2 + 1-dimensional chiral equation [27,28] where these lump peaks scatter with any desired angle. A natural question arises whether there exist novel lumps of DS II equation which scatter with nonorthogonal angle after collision.…”