Lump Interactions with Plane Solitons

Stepanyants, Yury; Zakharov, Dmitry; Захаров, В. Е.

doi:10.1007/s11141-022-10169-0

Cited by 23 publications

(9 citation statements)

References 15 publications

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“…The lumps and periodic lump chains can be absorbed and emitted by the line solitons. [28] A broad class of lump chains of the KPI equation were obtained by using a reduced version of τ-function in the form of Grammian. [29] The lump chain of the KP equation is already known, which is also called the breather and periodic soliton.…”

Section: Introductionmentioning

confidence: 99%

“…The breathers of the BKP equation and some patterns had been investigated by Yuan et al [43] An analytical method related to dominant domains was developed to analyze the interaction dynamical behaviors of the hybrid solutions. [44] Inspired by works, [28,29] the lump chains and interaction dynamics of the BKP equation will be investigated systematically.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

Zhao¹,

He²,

Wazwaz³

2023

Chinese Phys. B

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A large member of lump chain solutions of the (2+1)-dimensional BKP equation are constructed by means of the τ-function in form of Grammian. The lump chains are formed by periodic arrangement of individual lumps and travel with distinct group and velocities. An analytical method related dominant regions of polygon is developed to analyze the interaction dynamics of the multiple lump chains. The degenerate structures of parallel, superimposed and molecular lump chains are presented. The interaction solutions between lump chains and kink-solitons are investigated, where the kink-solitons lie on the boundaries of dominant region determined by the constant term in the τ-function. Furthermore, the hybrid solutions consisting of lump chains and individual lumps controlled by the parameter with high rank and depth are investigated. The analytical method presented in this paper can be further extended to other integrable systems to explore complex wave structures.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

Zhao¹,

He²,

Wazwaz³

2023

Chinese Phys. B

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“…The resonant interaction between the breathers and line solitons in Mel'nikov equation, KP-I equations and (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation as well as the resonant interaction between breathers in two-dimensional multi-component longwave-short-wave resonant interaction system were studied in Refs. [38][39][40][41]. In this paper, we investigate the resonant interaction among the line solitons, breathers and lumps for the (2 + 1)-dimensional elliptic Toda equation.…”

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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“…Lumps are 2D soliton solutions that are weakly localized and decay algebraically in all directions. [38][39][40][41][42][43][44][45][46] They are a type of rational function solutions that are frequently observed in high-dimensional systems. The investigation and examination of lump solutions in high-dimensional systems is an interesting area of research.…”

Section: Introductionmentioning

confidence: 99%

“…The investigation and examination of lump solutions in high-dimensional systems is an interesting area of research. [43][44][45][46][47] The primary target of this study is to obtain lump solutions from the breather and b-positon solutions of Eq. ( 3) when ε = 1.…”

Section: Introductionmentioning

confidence: 99%

From breather solutions to lump solutions: A construction method for the Zakharov equation

Yuan 袁,

Ghanbari,

Zhang 张

et al. 2023

Chinese Phys. B

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Periodic solutions of the Zakharov equation are investigated in this paper. Through performing the limit operation $\lambda_{2l-1}\to\lambda_1$ on the eigenvalues of the Lax pair obtained from the $n$-fold Darboux transformation, an order-$n$ breather-positon solution is first obtained from a plane wave seed. It is then proven that an order-$n$ lump solution can be further constructed through taking the limit $\lambda_1\to\lambda_0$ on the breather-positon solution, because the unique eigenvalue $\lambda_0$ associated with the Lax pair eigenfunction $\Psi(\lambda_0)=0$ corresponds to the limit of the infinite-periodic solutions. A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.

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Lump Interactions with Plane Solitons

Cited by 23 publications

References 15 publications

Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

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

From breather solutions to lump solutions: A construction method for the Zakharov equation

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