Asymptotic expansion of homoclinic structures in a symplectic mapping

Nakamura, Katsuhiro; Hamada, Masato

doi:10.1088/0305-4470/29/22/025

Cited by 11 publications

(9 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the commutative anti-self-dual Yang-Mills hierarchies, see e.g. [2,40,43,52,53]. The noncommutative anti-self-dual Yang-Mills hierarchy equations can be rewritten as the following form:…”

Section: Nc Asdym Hierarchy and Soliton Solutionsmentioning

confidence: 99%

Soliton Scattering in Noncommutative Spaces

Hamanaka¹,

Okabe²

2018

Theor Math Phys

View full text Add to dashboard Cite

We discuss exact multi-soliton solutions to integrable hierarchies on noncommutative space-times in diverse dimension. The solutions are represented by quasi-determinants in compact forms. We study soliton scattering processes in the asymptotic region where the configurations could be real-valued. We find that the asymptotic configurations in the soliton scatterings can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized lump of energy and each solitary wave-packet lump preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. As new results, we present multi-soliton solutions to noncommutative anti-self-dual Yang-Mills hierarchy and discuss 2-soliton scattering in detail.

show abstract

Section: Nc Asdym Hierarchy and Soliton Solutionsmentioning

confidence: 99%

Soliton Scattering in Noncommutative Spaces

Hamanaka¹,

Okabe²

2018

Theor Math Phys

View full text Add to dashboard Cite

show abstract

“…Existing beyond-all-orders investigations of relevance to (1.1) concern the limit → 0 of its steady state form with λ = 0 (the significance of λ arises from the constraint (2.4) below on F(φ)); see, for example, Lazutkin et al [21], Amick et al [1], Hakim & Mallick [13], Gelfreich et al [11], Nakamura & Hamada [22] and Delshams & Ramírez-Ros [7]. It would be inappropriate to review these studies here; the optimal truncation approach we shall adopt is somewhat different and is, we hope, transparent and widely applicable (cf.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

King

Chapman

2001

Eur. J. Appl. Math

View full text Add to dashboard Cite

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

show abstract

“…While numerical iterations of low-dimensional mappings easily provide these complicated structures, it is extremely difficult to derive them analytically. For nearly integrable systems, the difficulty may have been overcome by means of the asymptotic expansion beyond all orders supplemented with the Borel sum and the stokes phenomenon [2] [3] [4] [5]. However, this method can not be applied to the systems far from integrable such as a finitetime-discrete dynamical system with double-well potential (the double-well map) and the strange attractor in the dissipative Hénon map, which will be analyzed in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

Goto¹,

Nozaki²

2001

Progress of Theoretical Physics

View full text Add to dashboard Cite

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Hénon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion , a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Hénon map.

show abstract

Asymptotic expansion of homoclinic structures in a symplectic mapping

Cited by 11 publications

References 6 publications

Soliton Scattering in Noncommutative Spaces

Soliton Scattering in Noncommutative Spaces

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

Contact Info

Product

Resources

About