Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

Feng, Bao Feng; Inoguchi, Jun ichi; Kajiwara, Kenji; Maruno, Ken Ichi; Ohta, Yuichi

doi:10.1088/1751-8113/44/39/395201

Cited by 43 publications

(40 citation statements)

References 79 publications

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“…We comment here that λ 1 is the reciprocal of the wave number p 1 in [31]. As discussed in [31], if λ Furthermore, by using the N -fold DT, we could drive the N -soliton solution through the formula (28):…”

Section: A Single Soliton Solution and N -Soliton Solutionmentioning

confidence: 99%

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

Ling

Feng

Zhu

2016

Physica D: Nonlinear Phenomena

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139

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In the present paper, we are concerned with the general localized solutions for the complex short pulse equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N -bright soliton solution in a compact determinant form, then the N -breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first-and second-order rogue wave solutions are given explicitly and illustrated by graphs. The asymptotic analysis is performed rigourously for both the N -soliton and the Nbreather solutions. All three forms of the localized solutions admit either smoothed-, cusped-or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation is different from the one to the nonlinear Schrödinger (NLS) equation, which could be a cusponed-or a looped one.

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Section: A Single Soliton Solution and N -Soliton Solutionmentioning

confidence: 99%

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

Ling

Feng

Zhu

2016

Physica D: Nonlinear Phenomena

Self Cite

139

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“…In [59], we have shown that both the sine-Gordon equation and the SP equation can be derived from the motion of plane curves. The reciprocal link between them can be interpreted as Euler-Lagrangian transformation for the motion of plane curves.…”

Section: The Link Of the CD And The Sp Equations To The Motion Of Spamentioning

confidence: 99%

“…Finally, although integrable discretizations of the real and complex CD equations, as well as the real and CSP equations were recently constructed by several authors [33,59,[62][63][64], how to understand and reconstruct them within the framework of geometry remains a topic to be explored in the future.…”

Section: Comments and Further Topicsmentioning

confidence: 99%

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

Shen

Feng²,

Ohta

2016

Stud Appl Math

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In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two-component extended Kadomtsew-Petviashvili (KP) hierarchy. As a by-product, the N -soliton solutions in the form of determinants for these equations are provided.

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“…In the last, the integrable discretizations for the CSP and the coupled CSP equations remain a topic to be explored in the future. Following a series work done by the authors regarding integrable discretizations of the real SP equation and its multicomponent generalizations [7,[53][54][55][56], we will construct the integrable discretizations of the focusing and defocusing CSP equation algebraically and clarify their geometric meaning within a general framework set in [57,58].…”

Section: Discussionmentioning

confidence: 99%

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

2016

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In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space R 2,1 , then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the defocusing CSP equation from the single-component extended Kadomtsev-Petviashvili (KP) hierarchy by the reduction method. As a by-product, the N -dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.

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Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

Cited by 43 publications

References 79 publications

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

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