Gauge theory of Bäcklund transformations. II

Sattinger, D. H.; Zurkowski, V. D.

doi:10.1016/0167-2789(87)90227-2

Cited by 53 publications

(25 citation statements)

References 11 publications

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“…Here G, G and w are m × m matrix functions (m > 0). Clearly, if u satisfies some initial system u x = Gu, then wu satisfies the transformed one u x = G u. Darboux matrix or gauge transformation is of great interest in this theory (see [15,41,43,47,52,79,81] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.

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Section: Introductionmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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“…In § 3 we refine results of [1], [4], and [10]. Though this is of very technical character, it gives the basis for the ensuing investigations.…”

Section: Introductionmentioning

confidence: 92%

“…Though this is of very technical character, it gives the basis for the ensuing investigations. In particular, we show how the "meromorphic splitting" of [1] can be reinterpreted (using [10]) as a Riemann-Hilbert splitting. Also, (3.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert Factorizations and Inverse Scattering for the AKNS-Equation with $L^1$-Potentials I

Dorfmeister

Szmigielski

1993

Publ. Res. Inst. Math. Sci.

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A detailed group theoretic approach is developed for the 2X2 AKNS system with potentials supported on the half-line. This approach uses consistently Riemann-Hilbert splittings interpreted as factorizations in certain Banach Lie groups. In particular, it is shown how meromorphic splittings introduced by Beals and Coifman can be replaced with regular Riemann-Hilbert splittings leading to the group theoretic notion of a scattering data.

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“…Thus, determining stability becomes simply a question of examining the behavior in time of f (x, t) and g(x, t). The Bäcklund-gauge transformation (Sattinger and Zurkowski, 1987) allows one to transform both the "seed" solution u(x, t) and its eigenfunctions while preserving spatial periodicity, as follows: let φ := α + φ + + α − φ − , α ± ∈ C, where φ + and φ − are linearly independent solutions of the Z-S system at (u, λ j ), with λ j one of the complex λ d j . Construct the following gauge matrix:…”

Section: Squared Eigenfunctions and Linear Stabilitymentioning

confidence: 99%

Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

Calini

Schober

2014

Nat. Hazards Earth Syst. Sci.

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Abstract. In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves.

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Gauge theory of Bäcklund transformations. II

Cited by 53 publications

References 11 publications

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Riemann–Hilbert Factorizations and Inverse Scattering for the AKNS-Equation with $L^1$-Potentials I

Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

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