Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

Geng, Xianguo; Li, Ruomeng

doi:10.1016/j.aop.2015.06.017

Cited by 17 publications

(6 citation statements)

References 12 publications

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“…Choose a constant 1 = 11 + 2 . Let Ω( 1 ) be the solution to Lax pair (6) and (16) when = 1 , and let Ω ′ ( 1 ) = Ω (1) ( 1 ) be defined by (25). Choose…”

Section: F I G U R E 3 a First-order Rogue Wavementioning

confidence: 99%

“…Choose a constant 1 = 11 + 2 . Let Ω( 1 ) be the solution to Lax pair (6) and (16) when = 1 , and let Ω ′ ( 1 ) = Ω (1) ( 1 ), Ω ′′ ( 1 ) = Ω (2) ( 1 ) be defined by (25). Choose By using (20) and its derivatives with respect to , we can obtain ( 1 ), ( 1 ), ′ ( 1 ), ′ ( 1 ), ′′ ( 1 ), and ′′ ( 1 ).…”

Section: F I G U R E 3 a First-order Rogue Wavementioning

confidence: 99%

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On a vector long wave‐short wave‐type model

Geng

2019

Stud Appl Math

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A new vector long wave-short wave-type model is proposed by resorting to the zero-curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave-short wave-type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two-component long wave-short wavetype model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena. K E Y W O R D S breathers, Darboux transformations, long wave-short wave-type model, Riccati equations, rogue waves, solitons 164

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“…Choose a constant 1 = 11 + 2 . Let Ω( 1 ) be the solution to Lax pair (6) and (16) when = 1 , and let Ω ′ ( 1 ) = Ω (1) ( 1 ) be defined by (25). Choose…”

Section: F I G U R E 3 a First-order Rogue Wavementioning

confidence: 99%

Section: F I G U R E 3 a First-order Rogue Wavementioning

confidence: 99%

On a vector long wave‐short wave‐type model

Geng

2019

Stud Appl Math

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“…It should be pointed out that the two-component mYOLS Equation (10) is different from the long-wave-short-wave resonance Equation (1), and the two-component mYOLS Equation (10) is equivalent to the long-wave-short-wave model (2) under some transformation [21]. Then, with the help of Riccati equations for the Lax pair associated with the vmYOLS Equation (8) [22][23][24], Bäcklund transformation [25,26], Darboux transformation [27][28][29][30][31][32][33][34][35][36][37][38][39], and others [40][41][42][43][44][45][46][47][48][49][50][51]. Some interesting explicit solutions have been found, the most important among which are pure-soliton solutions, quasi-periodic solutions, and rogue waves solutions.…”

Section: Introductionmentioning

confidence: 99%

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Geng

2019

Mathematics

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A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

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“…where the solution is defined. Example 2: As the next example, we consider the time-fractional coupled Drinfeld--Sokolov-Satsuma-Hirota (DSSH) as [23]:…”

Section: Application Of the Methodsmentioning

confidence: 99%

Group invariant solution of some time fractional evolution equations

Mandal¹,

Bira²

2018

J. Appl. Math. Comput. Mech.

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In this paper, we consider some classes of a system of nonlinear fractional differential equations (FDEs) arising in some important physical phenomena. Using symmetry group of transformations, the given systems of fractional partial differential equations (FPDEs) are reduced to systems of fractional ordinary differential equations (FODEs). Further, using the group invariant condition, we solve the reduced systems of FODEs and exact solutions of the given equations are constructed. Finally, the physical significance of the solutions are investigated graphically based on the exact solutions in order to highlight the importance of the study.

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Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

Cited by 17 publications

References 12 publications

On a vector long wave‐short wave‐type model

On a vector long wave‐short wave‐type model

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Group invariant solution of some time fractional evolution equations

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