Completely resonant collision of lumps and line solitons in the Kadomtsev–Petviashvili I equation

Abstract

Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP‐I) equation are studied. The KP‐I equation describes the evolution of weakly nonlinear, weakly dispersive waves with slow transverse variations. Lumps can only exist for the KP equation when the signs of the transverse derivative and the weak dispersion in the propagation direction are different, that is, in the KP‐I regime. Collisions among lumps and solitons for “integrable” equations are normally elastic, that is, wave shapes are preserved except possibly for phase shifts. For resonant collisions, mathematically the phase shift will become indefinitely large. Physically a lump may be detached (or emitted) from a line soliton, survives for a brief transient period in time, and then merges with the next adjacent soliton. This special lump is thus localized in time as well as the two spatial dimensions, and can be termed a “rogue lump.” By employing a reduction method for the KP hierarchy in conjunction with the Hirota bilinear technique, a general L‐lump/(L+1)‐line‐soliton solution is obtained for the KP‐I equation for an arbitrary positive integer L. For L≥2, collisions among modes become increasingly complicated, with multiple lumps detaching from one single or different line solitons and then disappearing into the remaining solitons. In terms of applications, such “rogue lumps” represent a mode truly localized both in space and time, and will be valuable in modeling physical problems.

Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation

“…Ma [21] derived a fundamental lump solution which contains more free parameters; but that solution can be made equivalent to the original lump solution as reported in [8,9]. We note by passing that non-rational KP-I solutions in the form of a linear periodic chain of lumps, and those that describe the resonant collision between lumps and line solitons, have also been reported recently [22,23].…”

“…Based on this conjecture, W Λ (z) would have N W nonzero simple roots, where N W is given in Eq. (23). We have checked this conjecture on a number of examples of W Λ (z), and found it to always hold.…”

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

“…Explicit solutions of integrable equations for RWs help to understand these phenomena in the physical systems. Such solutions have been found for many integrable models, such as the NLS equation and its multicomponent version, 44–54 the Davey–Stewartson (DS), 55–57 and Ablowitz‐Ladik 58,59 equations 60–69 . The RW solutions can be regarded as the limit case of breathers, which are periodic in time or spatial coordinate.…”

“…Such solutions have been found for many integrable models, such as the NLS equation and its multicomponent version, [44][45][46][47][48][49][50][51][52][53][54] the Davey-Stewartson (DS), [55][56][57] and Ablowitz-Ladik 58,59 equations. [60][61][62][63][64][65][66][67][68][69] The RW solutions can be regarded as the limit case of breathers, which are periodic in time or spatial coordinate. The breathers can also provide a model for the observation of RWs in experiments.…”

“…Derivation of RW solutions to the 2-LSRI model. In this Section, we construct the rational RW solutions to the 2-LSRI model by starting with matrix elements m (n,k) s,j of tau functions τ n,k(65) in the following form:…”