Decay Estimates and Smoothness for Solutions of the Dispersion Managed Non-linear Schrödinger Equation

Hundertmark, Dirk; Lee, Young-Ran

doi:10.1007/s00220-008-0612-4

Cited by 38 publications

(45 citation statements)

References 31 publications

(43 reference statements)

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“…The proof of Theorem 1.1 is given in Section 3, see Theorem 3.2 and Corollary 3.3. Similar to our study of decay properties of dispersion managed solitons in [16], the main tool in the proof of our super-exponential decay Theorem 1.1 is the self-consistency bound from Proposition 3.1 on the tail distribution of weak solutions of the diffraction management equation (1.11). Our existence proof for diffraction management solitons is given in Section 4.…”

Section: Introductionmentioning

confidence: 88%

“…As in the continuous case, see [16], the key idea is not to focus on the solution f directly, but to study its tail distribution defined, for n ∈ N 0 , by…”

Section: Self-consistency Bound and Super-exponential Decaymentioning

confidence: 99%

“…Moreover, it was shown in [34], that the corresponding minimizer is decaying faster than polynomial, which again yields the orbital stability of solutions of (1.4) for d av = 0 and initial conditions close to a minimizer. In this paper we continue our study of regularity properties of the dispersion management technique initiated in [16] and study the decay properties of diffraction management solitons for vanishing average dispersion, i.e., weak solutions of (1.11). Our main result is a significant strengthening of the super-polynomial decay result for diffraction managed solitons in [34].…”

Section: Introductionmentioning

confidence: 98%

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Super-Exponential Decay of Diffraction Managed Solitons

Hundertmark

Lee

2011

Commun. Math. Phys.

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Abstract. This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the non-linear Schrödinger equation. In this part we study the diffraction managed discrete nonlinear Schrödinger equation, an equation which describes coupled waveguide arrays with periodic diffraction management geometries. We show that, for vanishing average diffraction, all solutions of the non-linear and non-local diffraction management equation decay super-exponentially. As a byproduct of our method, we also have a simple proof of existence of diffraction managed solitons in the case of vanishing average diffraction.

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Section: Introductionmentioning

confidence: 88%

“…As in the continuous case, see [16], the key idea is not to focus on the solution f directly, but to study its tail distribution defined, for n ∈ N 0 , by…”

Section: Self-consistency Bound and Super-exponential Decaymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Super-Exponential Decay of Diffraction Managed Solitons

Hundertmark

Lee

2011

Commun. Math. Phys.

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“…The GTE is autonomous and does not depend on ε. It has been intensively studied in analysis [18,23,33,37,41] and by means of numerical approximations [37,40], see also [36] for a review. Because of the averaging step in its derivation the GTE yields only an approximation to the DMNLS.…”

Section: Introductionmentioning

confidence: 99%

Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation

Jahnke

Mikl

2017

Numer. Math.

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The dispersion-managed nonlinear Schrödinger equation contains a rapidly changing discontinuous coefficient function. Approximating the solution numerically is a challenging task because typical solutions oscillate in time which imposes severe step-size restrictions for traditional methods. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes.Mathematics Subject Classification (2010): 65M12, 65M15, 65M70, 65Z05, 35B40, 35Q55

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“…In particular, he predicted that / and / decay exponentially at infinity. For dav = 0, the first rigorous x-space decay bounds were established in [12]:…”

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confidence: 99%