Mach-Type Soliton in the Novikov-Veselov Equation

Chang, Jen-Hsu

doi:10.3842/sigma.2014.111

Cited by 6 publications

(31 citation statements)

References 30 publications

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“…Please see the figure 1. One obtains two kink fronts (from left to right): • For y << 0: one has one kink front [1,2]-front and the line soliton [2, 3]-soliton.…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…From (15), the kink bounded by [2, 3]-front and [1,3]-front has the height: k 3 2 (1 − tanh θ 3 2 ), and the kink bounded by [2, 3]-front and [1,2]-front has the height:…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…2 ). The wave vectors and velocities of [1,3]-front and [1,2]-front are given by (21) for j = 3, 2. These three fronts satisfy the resonant conditions (25).…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…Next, we compute the amplitude of the intersection part of [1,2]-soliton and [3,4]-soliton . It is determined by the linear system…”

Section: O-type Solitonmentioning

confidence: 99%

“…Recently, the soliton interaction in integrable models has attracted much attention, especially the resonant theory in the KP-(II) theory [1,8,9,10,11,12,13] (references therein) and Novikov-Veselov (NV) equation [2,3]. The key point of soliton interaction in the KP-(II) equation is the τ -function structure, i.e., the Wronskian formula.…”

Section: Introductionmentioning

confidence: 99%

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Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation

Chang

2018

Applicable Analysis

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We study soliton interaction in the Modified Kadomtsev-Petviashvili-(II) equation (MKP-(II)) using the totally non-negative Grassmannian. One constructs the multi-kink soliton of MKP equation using the τ -function and the Binet-Cauchy formula, and then investigates the interaction between kink solitons and line solitons. Especially, Y-type kink-soliton resonance, O-type kink soliton and P-type kink soliton of X-shape are investigated. Their amplitudes of interaction are computed after choosing appropriate phases.

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“…Please see the figure 1. One obtains two kink fronts (from left to right): • For y << 0: one has one kink front [1,2]-front and the line soliton [2, 3]-soliton.…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…From (15), the kink bounded by [2, 3]-front and [1,3]-front has the height: k 3 2 (1 − tanh θ 3 2 ), and the kink bounded by [2, 3]-front and [1,2]-front has the height:…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…2 ). The wave vectors and velocities of [1,3]-front and [1,2]-front are given by (21) for j = 3, 2. These three fronts satisfy the resonant conditions (25).…”

Section: Basic Resonant Solutionsmentioning

confidence: 99%

“…Next, we compute the amplitude of the intersection part of [1,2]-soliton and [3,4]-soliton . It is determined by the linear system…”

Section: O-type Solitonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation

Chang

2018

Applicable Analysis

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Soliton molecules in Sharma–Tasso–Olver–Burgers equation

Yan

Sen-Yue

2020

Applied Mathematics Letters

115

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Soliton molecules have been experimentally discovered in optics and theoretically investigated for coupled systems. This paper is concerned with the formation of soliton molecules by the resonant mechanism for a noncoupled system, the Sharma-Tasso-Olver-Burgers (STOB) equation. In terms of introducing velocity resonance conditions, we derive the soliton (kink) molecules, half periodic kink (HPK) molecules and breathing soliton molecule of STOB equation. Meanwhile, the fission and fusion phenomena among kinks, kink molecules, HPKs and HPK molecules have been revealed. Moreover, we also discuss the central periodic kink solutions from the multiple solitary wave solutions.Solitons have been experimentally found in plasma physics and optics, as well as in nonlinear science area including Bose-Einstein condensation and DNA mechanical waves [15,16]. It is wellknown that solitons have an important application in a variety of areas, such as atmospheric and ocean dynamics [17], optical fibers [18]-[22], photonic crystals [23] and plasmas [24]. Later on the soliton molecules which are the soliton bound states have attracted considerable attention. Soliton molecules have been observed in optical systems [25]-[29] and analyzed in Bose-Einstein condensates [30]. Various theoretical proposals to form soliton molecules have been established [31]-[33]. It is known that soliton molecules in coupled systems have been well discussed [34]. Recently, the breathing soliton molecules have been experimentally observed in a mode-locked fiber laser [28]. The resonant theory of solitons is applicable to a variety of integrable systems, such as the KP-(II) equation [35, 36] and Novikov-Veselov equation [37] in which Wronskian representation of the τ -function have been employed. By restricting different resonant conditions on the solitons, it may derive many types resonant excitations. The resonant solutions have been generated by using linear superposition principle and the Bell polynomials for the bilinear equations [38]. By virtue of Hirotas direct method and the Bäcklund transformation, the researchers study the fission and fusion of the solitary wave solution of the Burgers and STO equations [14]. Furthermore, the interaction between the lump soliton and the pair of resonance stripe solitons form a rogue wave solution [39]. The rational solutions to the modified Korteweg-de Vries equation have been analyzed by imposing the wave number resonance constraints [40]. Lately, by means of introducing velocity resonant mechanism, Lou investigated some types of soliton molecules in fluid systems [41] and nonlinear optical systems [42] involving the third-fifth order Korteweg de-Vries (KdV) equation, the KdV-Sawada-Kotera (KdVSK) equation, the KdV-Kaup-Kupershmidt (KdVKK) equation, the Hirota system and the potential modified KdV-sG system which derive kink-kink molecule, kink-antikink molecule, kink-breather molecule and breatherbreather molecule etc. Although the structure and properties of the integrable systems have been widely investigated,...

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The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Yurova

Yurov

2020

Symmetry

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We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

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Mach-Type Soliton in the Novikov-Veselov Equation

Cited by 6 publications

References 30 publications

Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation

Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation

Soliton molecules in Sharma–Tasso–Olver–Burgers equation

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

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