Asymptotic Analysis of Multilump Solutions of the Kadomtsev–Petviashvili-I Equation

We construct a broad class of solutions of the Kadomtsev-Petviashvili (KP)-I equation by using a reduced version of the Grammian form of the 𝜏function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

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“…The most general Grammian form of the multilump solutions of KP‐I was considered in Ref. 52. 3. Lump chains .…”

Section: The Grammian Form Of the τ‐Functionmentioning

confidence: 99%

Lump chains in the KP‐I equation

et al. 2021

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“…Analytical expressions of higher-order lumps have been derived by a wide variety of methods before [10][11][12][13][14][15][16][17][18][19][20][21]. Gorshkov, et al [10] reported a second-order lump solution that describes the interaction and anomalous scattering of two lumps.…”

Section: Introductionmentioning

confidence: 99%

“…Gaillard [19] studied a special class of higher-order lump solutions and reported lump patterns such as triangles and pentagons at t = 0 when some internal parameters in such solutions get large. Chang [20] studied the large-time asymptotics of higher-order lumps and showed that, for some special solutions, all lumps are located on a vertical line in the (x, y) plane at large negative time but rotate to a horizontal line at large positive time. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Yang,

Yang

2021

Preprint

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Pattern formation in higher-order lumps of the Kadomtsev-Petviashvili I equation at large time is analytically studied. For a broad class of these higher-order lumps, we show that two types of solution patterns appear at large time. The first type of patterns comprise fundamental lumps arranged in triangular shapes, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. As time evolves from large negative to large positive, this triangular pattern reverses itself along the x-direction. The second type of patterns comprise fundamental lumps arranged in non-triangular shapes in the outer region, which are described analytically by nonzeroroot structures of the Wronskian-Hermit polynomials, together with possible fundamental lumps arranged in triangular shapes in the inner region, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. When time evolves from large negative to large positive, the non-triangular pattern in the outer region switches its x and y directions, while the triangular pattern in the inner region, if it arises, reverses its direction along the x-axis. Our predicted patterns at large time are compared to true solutions, and excellent agreement is observed.

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“…(d) Several authors [13,10] have considered a more general choice for φ n such as φ n = D n k exp(iθ) with θ(k) same as before but D k := f (k)∂ k where f (k) is analytic. In such cases, it is possible to consider D k = ∂ z in terms of a (local) uniformizing variable z(k) with z (k) = 1/f (k) and an appropriate branch of its inverse k(z) to express θ(k(z)).…”

Section: Remarksmentioning

confidence: 99%

Dynamics of KPI lumps

Chakravarty,

Zowada

2021

Preprint

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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 xyplane 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| → ∞.

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Asymptotic Analysis of Multilump Solutions of the Kadomtsev–Petviashvili-I Equation

Cited by 26 publications

References 24 publications

Lump chains in the KP‐I equation

Lump chains in the KP‐I equation

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

Dynamics of KPI lumps

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