On the construction of the KP line-solitons and their interactions

Chakravarty, Sarbarish; Lewkow, Tim; Maruno, Ken Ichi

doi:10.1080/00036810903403343

Cited by 19 publications

(44 citation statements)

References 26 publications

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“…We already mentioned the two families associ- 15 See also [44] for a relation between integrable PDEs and polytopes.…”

Section: Summary Of Further Results and Conclusionmentioning

confidence: 99%

“…four phases, the regularity condition only allows the two 2-forms τ O = (e 1 + e 2 ) ∧ (e 3 + e 4 ) and τ P = (e 1 − e 4 ) ∧ (e 2 + e 3 ) , which belong to classes called "O-type" and "P-type" by some authors (see e.g. [15,19]). We will consider the O-type solution in detail in Example 4.3.…”

Section: On the General Class Of Kp Line Soliton Solutionsmentioning

confidence: 99%

“…More comprehensive studies of the structure of the rather complex networks emerging in this way have been undertaken quite recently [6][7][8][9][10][11][12][13][14][15][16][17][18] (see also the review [19] and the references cited therein). Whereas in these works a classification in terms of the asymptotic behavior at large negative and positive times, and large (positive or negative) values of the coordinate transverse to the main propagation direction, has been addressed, in the present work we proceed toward an understanding of the full evolution.…”

Section: Introductionmentioning

confidence: 99%

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KP line solitons and Tamari lattices

Dimakis¹,

Müller‐Hoissen²

2010

J. Phys. A: Math. Theor.

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The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).

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“…We already mentioned the two families associ- 15 See also [44] for a relation between integrable PDEs and polytopes.…”

Section: Summary Of Further Results and Conclusionmentioning

confidence: 99%

Section: On the General Class Of Kp Line Soliton Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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KP line solitons and Tamari lattices

Dimakis¹,

Müller‐Hoissen²

2010

J. Phys. A: Math. Theor.

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“…(46), we can find that the amplitudes of the two solitons s 1 and s 2 for the long wave component in the asymptotic analysis are 2m 2 1 and 2m 2 2 , respectively. According the classifications of the soliton solutions in the Kadomtsev-Petviashvili (KP) equation [43,44], the two bright solitons in v can be classified as O-type ('O' for original) and P-type ('P' for physical) soliton interactions.…”

Section: Two Dark-dark Solitonsmentioning

confidence: 99%

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

Chen

Zhang

2019

Mod. Phys. Lett. B

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Based on the KP hierarchy reduction method, the general multi-dark soliton solutions in Gram type determinant forms for the multi-component coupled Maccari system are constructed. Especially, the three-component coupled Maccari system comprised of two-component short waves and single-component long waves are discussed in detail. Besides, the dynamics of one and two darkdark solitons are analyzed. It is shown that the collisions of two dark-dark solitons are elastic by asymptotic analysis. Additionally, the two dark-dark solitons bound states are studied through two different cases (stationary and moving cases). The bound states can exist up to arbitrary order in the stationary case, however, only two-soliton bound state exists in the moving case. Besides, the oblique stationary bound state can be generated for all possible combinations of nonlinearity coefficients consisted of positive, negative and mixed cases. Nevertheless, the parallel stationary and the moving bound states are only possible when nonlinearity coefficients take opposite signs.

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“…Using the above expressions for a i , the transformed eigenfunction ψ[N] = gψ[0] can be expressed as the ratio of two Wronskian determinants, namely, can be read off from this relation using the form of L given in (8). .…”

Section: Darboux Transformationmentioning

confidence: 99%

Line-soliton solutions of the KP equation

Chakravarty¹,

Kodama²,

Xiu³

et al. 2010

AIP Conference Proceedings

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A review of recent developments in the study and classification of the line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation is provided. Such solution u(x, y,t) is defined by a point of the totally non-negative Grassmannian Gr(N, M), and for fixed t, decays exponentially except along N distinct directions for y 0 and M − N distinct directions for y 0. These solutions can be constructed algebraically as u = 2(ln τ) xx where the function τ is a wronskian of N independent solutions of f y = f xx and f t = f xxx . The KP equation admits a large variety of such multi-soliton solutions in contrast to its one-dimensional counterpart -the Kortweg de-Vries (KdV) equation. It is shown how these solutions can be classified using combinatorial methods.should admit solutions that provide good approximations for two soliton interaction with smaller angle. Thus, it seems strange that one does not have a reasonable solution in the parameter regimes where the KP equation is supposed to give a better approximation. Miles also found that at the critical angle the two line-solitons of the O-type solution interact resonantly, and a third wave (soliton) is created to make a "Y-shape" solution. Indeed, it turns out that such Y-shape resonant solutions are exact solutions of the KP equation (see also [25]). Miles then used the Y-shape solution to describe the Mach reflection observed in shallow water waves reflected by a rigid wall. However the recent papers [7,19] shows that the Mach reflection should be described by other resonant solution of the KP equation (described also in the present paper).After the discovery of the resonant phenomena in the KP equation, several numerical and experimental studies were performed to investigate resonant interactions in other physical two dimensional equations such as the ion-acoustic and shallow water wave equations (see for examples [15,16,26,11,22]). However, after these activities, no significant progress has been made in the study of the solution space or physical applications of the KP equation. It would appear that the general perception was that there were not many new and significant results left to be uncovered in the soliton solutions of the KP theory. During the past 5 years, we have studied the classification problem of the soliton solutions of the KP equation, and our investigation has revealed a large variety of soliton solutions which were totally overlooked in the past. The purpose of this paper is to give a survey of our recent works [4, 18, 3, 5, 6, 7, 8] on a classification theory of soliton solutions of the Kadomtsev-Petviashvili (KP) equation. Most of the materials in this paper except for the discussion of Darboux transformations in Section 3 were presented by one of the authors (YK) as a tutorial at the international workshop "Nonlinear and Modern Mathematical Physics" held in Beijing during July 15-21, 2009. The organization of the paper is as follows. In Section 2, we provide some background material by introducing the wronskian form of the KP τ-function. Th...

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On the construction of the KP line-solitons and their interactions

Cited by 19 publications

References 26 publications

KP line solitons and Tamari lattices

KP line solitons and Tamari lattices

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

Line-soliton solutions of the KP equation

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