Nondegeneracy, Morse Index and Orbital Stability of the KP-I Lump Solution

Liu, Yong; Wei, Juncheng

doi:10.1007/s00205-019-01413-5

Cited by 14 publications

(16 citation statements)

References 71 publications

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“…Recall that the lump, the soliton-like solution to KP-I with algebraic decay in space, does not decay in time. It was very recently proved to be orbitally stable in the space E 1 pR 2 q in another breakthrough by Liu and Wei [36]. Consequently, and in view of the results that we will describe below, the asymptotic stability of this object seems a fundamental open problem in the field.…”

Section: Introduction and Main Resultsmentioning

confidence: 88%

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Long time asymptotics of large data in the Kadomtsev-Petviashvili models

Mendez,

Muñoz,

Poblete

et al. 2021

Preprint

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We consider the Kadomtsev-Petviashvili equations posed on R 2 . For both equations, we provide sequential in time asymptotic descriptions of solutions, of arbitrarily large data, inside regions not containing lumps or line solitons, and under minimal regularity assumptions. The proof involves the introduction of two new virial identities adapted to the KP dynamics, showing decay in large regions of space, especially in the KP-I case, where no monotonicity property was previously known. Our results do not require the use of the integrability of KP and are adaptable to well-posed perturbations of KP.

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Section: Introduction and Main Resultsmentioning

confidence: 88%

“…Important advances in the understanding of this problem were obtained by Liu and Wei in the aforementioned work [36]. By using linear Bäcklund transformation techniques, they proved the orbital stability of the lump in the space E 1 pR 2 q.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Long time asymptotics of large data in the Kadomtsev-Petviashvili models

Mendez,

Muñoz,

Poblete

et al. 2021

Preprint

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“…it was proved in [30] that q 0 is nondegenerate and has Morse index one. (We note that the Morse index of the lump solution (1.5) has been shown numerically to be one ( [17]). )…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

From KP-I lump solution to travelling waves of Gross-Pitaevskii equation

Liu¹,

Wang²,

Wei³

et al. 2021

Preprint

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“…This system can be further simplified as follows. Multiply Equations (46) and (47) by − 1 and − 2 , respectively, and add the resulting equations. Then, substitute 2 from Equation 48, and…”

Section: Cartesian Casementioning

confidence: 99%

“…In Ref. 46, Bäcklund transformations are used to prove that the KP‐I lump is nondegenerate, with Morse index 1 and, as a consequence, it is orbitally stable.…”

Section: Approximate Soliton Solutionsmentioning

confidence: 99%

Multiscale expansions avector solitons of a two‐dimensional nonlocal nonlinear Schrödinger system

et al. 2020

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One-and two-dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a Kadomtsev-Petviashvili equation and its cylindrical counterpart, Johnson's equation. This way, we derive approximate soliton solutions of the nonlocal NLS system, which have the form of: (a) dark or antidark soliton stripes and (b) dark lumps in the Cartesian geometry, as well as (c) ring dark or antidark solitons in the cylindrical geometry. The type of the soliton, namely dark or antidark, is determined by the degree of nonlocality: dark (antidark) soliton states are formed for weaker (stronger) nonlocality. We perform numerical simulations and show that the derived soliton solutions do exist and propagate undistorted in the original nonlocal NLS system.

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Nondegeneracy, Morse Index and Orbital Stability of the KP-I Lump Solution

Abstract: Using Bäcklund transformation the spectral property of the KP-I lump solution is completely analyzed. It is proved that the lump solution is nondegenerate and has Morse index one. As a consequence we show that it is orbitally stable.

Cited by 14 publications

References 71 publications

Long time asymptotics of large data in the Kadomtsev-Petviashvili models

Long time asymptotics of large data in the Kadomtsev-Petviashvili models

From KP-I lump solution to travelling waves of Gross-Pitaevskii equation

Multiscale expansions avector solitons of a two‐dimensional nonlocal nonlinear Schrödinger system

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