Lump solutions of the 2D Toda equation

Sun, Yan; Ma, Wen‐Xiu; Yu, Jun

doi:10.1002/mma.6370

Cited by 16 publications

(5 citation statements)

References 28 publications

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“…There should exist more interesting problems which need to be discussed. We expect to investigate integrable properties 35,36 and high‐order lump solutions 37–44 to the presented Hirota bilinear equations in our future works.…”

Section: Discussionmentioning

confidence: 99%

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Cheng

2021

Math Methods in App Sciences

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A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)-dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)-dimensional generalized Hirota-Satsuma-Ito (HSI) equation, the special fourth-order (2+1)-dimensional nonlinear equation, and the fifth-order Korteweg-de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions.

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Section: Discussionmentioning

confidence: 99%

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Cheng

2021

Math Methods in App Sciences

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“…Similarly, if we let α 1 , α 2 ⟶ 0 and β 1 , β 2 ⟶ 0 in 2breather (21), we obtain the following 2-lump solution for (3):…”

Section: Advances In Mathematical Physicsmentioning

confidence: 99%

“…Nakamura derived exact solutions for the ð2 + 1Þ-dimensional cylindrical Toda equation and the ð3 + 1Þ-dimensional elliptic Toda equation in [19,20]. In [21], three classes of lump solutions for (2) were constructed through symbolic computation.…”

Section: Introductionmentioning

confidence: 99%

$M$ -Breather, Lumps, and Soliton Molecules for the $(2 + 1)$ -Dimen

Jia

et al. 2021

Advances in Mathematical Physics

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The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

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“…Utiliz-ing the symbolic computation, three classes of lump solutions for the (2 + 1)-dimensional elliptic Toda equation were constructed in Ref. [15]. More recently, N-soliton, M-breather and lump solutions of the (2 + 1)-dimensional elliptic Toda equation have been investigated by the Bäcklund transformation and nonlinear superposition formula, [16] which provides the possibility to obtain more hybrid solutions.…”

Section: Introductionmentioning

confidence: 99%

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the (2+1)-dimensional elliptic Toda equation

2023

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The (2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semi-discrete Kadomtsev-Petviashvili I equation. This paper focuses on investigating the resonant interactions between two breathers, a breather/lump and line solitons as well as lump molecules for the (2+1)-dimensional elliptic Toda equation. Based on the N-soliton solution, we obtain the hybrid solutions consisting of line solitons, breathers and lumps. Through the asymptotic analysis of these hybrid solutions, we derive the phase shifts of the breather, lump and line solitons before and after the interaction between a breather/lump and line solitons. By making the phase shifts infinite, we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons. Through the asymptotic analysis of these resonant solutions, we demonstrate the resonant interactions exhibit the fusion, fission, time-localized breather and rogue lump phenomena. Utilizing the velocity resonance method, we obtain lump-soliton, lump-breather, lump-soliton-breather and lump-breather-breather molecules. The above works have not been reported in the (2+1)-dimensional discrete nonlinear wave equations.

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Lump solutions of the 2D Toda equation

Cited by 16 publications

References 28 publications

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

$M$ -Breather, Lumps, and Soliton Molecules for the $(2 + 1)$ -Dimen

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the (2+1)-dimensional elliptic Toda equation

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Lump solutions of the 2D Toda equation

Cited by 16 publications

References 28 publications

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimen

Resonant interactions among two-dimensional nonlinear localized waves and lump molecules for the (2+1)-dimensional elliptic Toda equation

Contact Info

Product

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$M$ -Breather, Lumps, and Soliton Molecules for the $(2 + 1)$ -Dimen