Discrete periodic problem for the modified nonlinear Korteweg-de Vries equation

Bogolyubov, N. N.; Prikarpatskii, A. K.; Самойленко, В. Г.

doi:10.1007/bf01027608

Cited by 17 publications

(15 citation statements)

References 11 publications

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“…To conclude, I want to point out the main differences between the approach of this paper and ones used earlier [4,5,2,3,6]. In the IST-based methods the hyperelliptic curves appear in the analysis of the spectral data of the scattering problem (2.1), while the dependence on the coordinates is derived from the system (2.2).…”

Section: Quasiperiodic Solutionsmentioning

confidence: 92%

Finite-genus solutions for the Ablowitz-Ladik hierarchy

Vekslerchik¹

1999

J. Phys. A: Math. Gen.

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The question of constructing the finite genus quasiperiodic solutions for the Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between the ALH and the Fay's identity for the θ-functions. It is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential-difference equations coinciding with the equations of the ALH, and that the original Fay's identity can be rewritten in a form similar to the functional equations representing the ALH which have been derived in the previous works of the author. This provides an algorithm for obtaining some class of quasiperiodic solutions for the ALH, which can be viewed as an alternative to the inverse scattering transform or the algebro-geometrical approach.

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Section: Quasiperiodic Solutionsmentioning

confidence: 92%

Finite-genus solutions for the Ablowitz-Ladik hierarchy

Vekslerchik¹

1999

J. Phys. A: Math. Gen.

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“…The authors of Refs. [17][18][19][20] discuss quasiperiodic solutions for the AL equations, which include the discrete nonlinear Schrodinger (NLS) equation and the discrete mod ified Korteweg-de Vries (mKdV) equation. Seeking and the *yufajun888@ 163.com investigation of the exact solutions are crucial tasks in the study of nonlinear DDEs.…”

Section: Introductionmentioning

confidence: 99%

Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation

2015

Phys. Rev. E

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We present the nonautonomous discrete bright soliton solutions and their interactions in the discrete Ablowitz-Ladik (DAL) equation with variable coefficients, which possesses complicated wave propagation in time and differs from the usual bright soliton waves. The differential-difference similarity transformation allows us to relate the discrete bright soliton solutions of the inhomogeneous DAL equation to the solutions of the homogeneous DAL equation. Propagation and interaction behaviors of the nonautonomous discrete solitons are analyzed through the one- and two-soliton solutions. We study the discrete snaking behaviors, parabolic behaviors, and interaction behaviors of the discrete solitons. In addition, the interaction management with free functions and dynamic behaviors of these solutions is investigated analytically, which have certain applications in electrical and optical systems.

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“…In Reference [22], the discrete Ablowitz-Ladik hierarchy [1,23] and their Hamiltonian structure were presented; it was studied how the discrete Ablowitz-Ladik hierarchy and their restricted flows are related to the AKNS hierarchy and to their restricted flows in the continuous limit with the aid of the corresponding spectral problem and its adjoint one. The authors of References [24][25][26][27][28][29][30][31] discuss quasi-periodic solutions for the discrete Ablowitz-Ladik equations, which include the discrete nonlinear Schrödinger equation and the discrete mKdV equation, through different ways. However, the decomposition of the discrete Ablowitz-Ladik hierarchy is still an open problem.…”

Section: Introductionmentioning

confidence: 99%

Decomposition of the Discrete Ablowitz–Ladik Hierarchy

2007

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The nonlinearization approach of Lax pairs is extended to the discrete Ablowitz-Ladik hierarchy. A new symplectic map and a class of new finite-dimensional Hamiltonian systems are derived, which are further proved to be completely integrable in the Liouville sense. An algorithm to solve the discrete Ablowitz-Ladik hierarchy is proposed. Based on the theory of algebraic curves, the straightening out of various flows is exactly given through the Abel-Jacobi coordinates. As an application, explicit quasi-periodic solutions for the discrete Ablowitz-Ladik hierarchy are obtained resorting to the Riemann theta functions.

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Discrete periodic problem for the modified nonlinear Korteweg-de Vries equation

Cited by 17 publications

References 11 publications

Finite-genus solutions for the Ablowitz-Ladik hierarchy

Finite-genus solutions for the Ablowitz-Ladik hierarchy

Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation

Decomposition of the Discrete Ablowitz–Ladik Hierarchy

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