Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy

Yue, Chao; Xia, Tiecheng

doi:10.1063/1.4891605

Cited by 11 publications

(8 citation statements)

References 32 publications

(30 reference statements)

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“…The generalized Kaup-Newell-type hierarchy of nonlinear evolution equations is explicitly related to Sharma-Tasso-Olver equation from [5]. Chao Yue in [6] provided theta function representation of algebro-geometric solutions and related crucial quantities for the complex Sharma-Tasso-Olver (CSTO) hierarchy. In [7] the simple symmetry reduction procedure is repeated by examining soliton fission and fusion to obtain the exact solutions for STO.…”

Section: Introductionmentioning

confidence: 99%

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

El-Danaf¹,

El-Sayed²,

Eissa³

et al. 2022

JAMP

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When analysing the thermal conductivity of magnetic fluids, the traditional Sharma-Tasso-Olver (STO) equation is crucial. The Sharma-Tasso-Olive equation's approximate solution is the primary goal of this work. The quintic B-spline collocation method is used for solving such nonlinear partial differential equations. The developed plan uses the collocation approach and finite difference method to solve the problem under consideration. The given problem is discretized in both time and space directions. Forward difference formula is used for temporal discretization. Collocation method is used for spatial discretization. Additionally, by using Von Neumann stability analysis, it is demonstrated that the devised scheme is stable and convergent with regard to time. Examining two analytical approaches to show the effectiveness and performance of our approximate solution.

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Section: Introductionmentioning

confidence: 99%

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

El-Danaf¹,

El-Sayed²,

Eissa³

et al. 2022

JAMP

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“…Over the recent decades, integrable equations related to 2 × 2 matrix spectral problems have been extensively researched. Several systematic methods have been developed to construct algebro-geometric solutions for integrable equations such as KdV, Kadomtsev-Petviashvili equation, modified KdV, sine-Gordon, Ablowitz-Kaup-Newell-Segur, the Camassa-Holm equations, and Ablowitz-Ladik lattice [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. But the study of algebrogeometric solutions of the whole hierarchy of 3 × 3 is still a challenging problem.…”

Section: Introductionmentioning

confidence: 99%

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Yue

Xia

2021

Advances in Mathematical Physics

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The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

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“…The bi-Hamiltonian formulation and the generalized Poisson bracket are presented in [17], the exact traveling solutions and symmetries of the STO equation were extensively studied in the literature [17][18][19]. Moreover, the multiple solitons, exact traveling solutions, kinks solutions, non-traveling wave solutions of the STO equation were obtained with different approaches [18][19][20][21][22]. In this paper, we concerned with the cSTO equation derived by Fan [23] u y + 1 2 (u xxx − 3i(u x u) x − 3u 2 u x ) = 0.…”

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confidence: 99%

“…µ 11 j (x, y; λ ) = µ22 j (x, y;λ ), µ 12 j (x, y; λ ) = µ 21 j (x, y;λ );(2) µ 11 j (x, y; −λ ) = µ 11 j (x, y; λ ), µ 12 j (x, y; −λ ) = −µ 12 j (x, y; λ ),µ 21 j (x, y; −λ ) = −µ 21 j (x, y; λ ), µ 22 j (x, y; −λ ) = µ 22 j (x, y; λ ). Proposition 2.4.3.…”

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confidence: 99%

A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line ^*

Zhang

Xia

2017

Commun. Theor. Phys.

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In this paper, we use the Fokas method to analyze the complex Sharma-Tasso-Olver(cSTO) equationAssuming that the solution u(x, y) of the cSTO equation is exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem(RHP) formulated in the complex λ −plane (the Fourier plane), which has a jump matrix with explicit (x, y)−dependence involving four scalar functions of λ , called spectral functions. The spectral functions {a(λ ), b(λ )} and {A(λ ), B(λ )} are obtained from the initial data u 0 (x) = u(x, 0) and the boundary data g 0 (y) = u(0, y), g 1 (y) = u x (0, y), g 2 (y) = u xx (0, y), respectively. The problem has the jump across {Imλ 6 = 0}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.Given initial and boundary values u 0 (x), g 0 (y), g 1 (y) and g 2 (y) such that there exist spectral functions satisfying the global relation, we show that the function u(x, y) defined by the above Riemann-Hilbert problem exists globally and solves the cSTO equation with the prescribed initial and boundary values. Key word: Riemann-Hilbert problem; the cSTO equation; initial-boundary value problem; jump matrix 2010 MR Subject Classification 35G31; 35Q15 IntroductionA unified method for analyzing boundary value problems with decaying initial data, extending ideas of the so-called inverse scattering transform(IST) method, was discovered in 1967 [1]. This method can be thought of as a nonlinear Fourier transform(FT) method. However, this nonlinear FT is not the same for every nonlinear evolution equation, but it is constructed from the x part of the Lax pair. Furthermore, neither the direct nonlinear FT of the initial data, nor the inverse nonlinear FT can be expressed in closed form: the former involves a linear Volterra integral equation and the latter involves a matrix Riemann-Hilbert problem(RHP). The y part is used only to determine the evolution of the direct nonlinear FT [2,3]. However, in many laboratory and field situations, the wave motion is initiated by what corresponds to the imposition of boundary conditions rather than initial conditions. This naturally leads to the formulation of an initial-boundary value (IBV) problem instead of a pure initial value problem.In 1997, a new unified approach based on the Riemann-Hilbert factorization problem to solve IVB problems for linear and nonlinear integrable PDEs was presented by Fokas [4], we call that Fokas method. The Fokas

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Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy

Cited by 11 publications

References 32 publications

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line ^*

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Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy

Cited by 11 publications

References 32 publications

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line *

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A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line ^*