Invertible Coupled KdV and Coupled Harry Dym Hierarchies

Błaszak, Maciej; Marciniak, Krzysztof

doi:10.1111/sapm.12008

Cited by 4 publications

(1 citation statement)

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“…Equation (1) is one of the most important soliton equations i.e., a soliton is deemed as solitary, travelling wave pulse solution of nonlinear partial differential equations (NPDEs) [2] e.g., the Korteweg-de Vries (KdV) equation (see Refs. [3] [4]). Moreover, according to Drazin and Johnson [5], a soliton can interact strongly with other solitons and retain its identity.…”

Section: Introductionmentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

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Section: Introductionmentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation

Chen

2018

Int. J. Geom. Methods Mod. Phys.

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In this paper, the backward and forward Neumann type systems are generalized to deduce the quasi-periodic solutions for a negative-order integrable system of 2-component KdV equation. The 2-component negative-order KdV (2-nKdV) equation is depicted as the zero-curvature representation of two spectral problems. It follows from a symmetric constraint that the 2-nKdV equation is reduced to a pair of backward and forward Neumann type systems, where the involutive solutions of Neumann type systems yield the finite parametric solutions of 2-nKdV equation. The negative-order Novikov equation is given to specify a finite-dimensional invariant subspace for the 2-nKdV flow. With a spectral curve given by the Lax matrix, the 2-nKdV flow is linearized on the Jacobi variety of a Riemann surface, which leads to the quasi-periodic solutions of 2-nKdV equation by using the Riemann-Jacobi inversion.

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On a Harry–Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

2022

Mathematics

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Resorting to the characteristic polynomial of Lax matrix for a Harry–Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker–Akhiezer function and meromorphic function are introduced. With the help of the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we obtain the quasi-periodic solutions for the entire Harry–Dym-type hierarchy.

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Invertible Coupled KdV and Coupled Harry Dym Hierarchies

Cited by 4 publications

References 10 publications

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation

On a Harry–Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

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