Explicit breather solution of the nonlinear Schrödinger  equation

Conte, Robert

doi:10.1134/s0040577921100032

Cited by 2 publications

(6 citation statements)

References 24 publications

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“…Remarkably, the new results (as claimed) in [7] are singular solutions of the NLSE that have been omitted completely in [1] and [2] as they are unphysical. We should also note that although the major part of derivations in [7] have been copied from [1] and [2], no any errors were detected. This is in striking contradiction with the claims of the Authors of the Comment.…”

Section: On the Adequacy Of Ansatz (2)mentioning

confidence: 94%

“…[2] (see the end of Section 2). This alternative approach was used recently by Conte [7]. Mathematically, the two representations are equivalent.…”

Section: About the Representation Of Solutions: The Solutions Reporte...mentioning

confidence: 99%

“…Moreover, it does not report any novelty: the solutions in terms of Weierstrass functions have been first suggested in [2] and have been recently implemented in more details by Conte [7] (ref.…”

Section: On the Adequacy Of Ansatz (2)mentioning

confidence: 99%

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Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

et al. 2020

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Solitons on a finite a background, also called breathers, are solutions of the focusing nonlinear Schrödinger equation, which play a pivotal role in the description of rogue waves and modulation instability. The breather family includes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons (PS), which have been successfully exploited to describe several physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally derived by Akhmediev, Eleonskii and Kulagin [Theor. Math. Phys. 72, 809-818 (1987)]. Having more parameters to vary, this significantly wider family has a potential to describe many more physical effects of practical interest than its subsets mentioned above. The complexity of this class of solutions prevented researchers to study them deeply. In this paper, we overcome this difficulty and report several new effects that follow from more detailed analysis. Namely, we present the doubly periodic solutions and their Fourier expansions. In particular, we outline some striking properties of these solutions. Among the new effects, we mention (a) regular and shifted recurrence, (b) period doubling and (c) amplification of small periodic perturbations with frequencies outside the conventional modulation instability gain band.

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Section: On the Adequacy Of Ansatz (2)mentioning

confidence: 94%

“…[2] (see the end of Section 2). This alternative approach was used recently by Conte [7]. Mathematically, the two representations are equivalent.…”

Section: About the Representation Of Solutions: The Solutions Reporte...mentioning

confidence: 99%

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Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

et al. 2020

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“…(vi) In a recent article [15] new elliptic ansatz (4) -solutions of the CNLSE have been presented. The solutions Q(x, t) of Eq.…”

Section: Conclusion (A) Summarymentioning

confidence: 99%

“…The solutions Q(x, t) of Eq. ( 5) in [15] (with h(t) according to Eq. ( 7) in [15]), derived by using Eq.…”

Section: Conclusion (A) Summarymentioning

confidence: 99%

Non-existence of certain elliptic solutions of the Cubic-nonlinear Schrödinger Equation

Schürmann¹,

Serov²

2023

Preprint

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For a certain class of solutions of the cubic nonlinear Schrödinger equation we prove non-existence in the generic case. In the nongeneric case we present a two-parameter set of solutions, bounded or unbounded, depending on corresponding constraints. I. INTRODUCTIONAs is well known, elliptic (travelling-wave) solutions of the non-integrable complex Ginzburg-Landau equation 1) is integrable and coincides with the cubic nonlinear Schrödinger equation (CNLSE)where, in the "fiber optics notation", z denotes the distance along the fiber, and t the (retarded) time. Besides general solutions derived by direct methods (e.g., IST, Darboux), particular solutions of the CNLSE, suitable for specific physical applications, are interesting and may suffice. In this context, methods have been proposed for seeking elliptic solutions of Eq.( 1) [2(b)]. For the CNLSE particular elliptic travelling-wave solutions exist in the form (see, e.g., [3] and references therein)Thus, in correspondence to the non-existence of elliptic solutions of Eq.( 1), the question is obvious whether elliptic solutions of Eq.( 2), more general than those represented by (3), exist. In particular, if f (z)e −iλt in (3) is replaced by f (t, z) + id(z) (without assuming travelling-wave reduction z = x − ct), we are led to Ψ(t, z) = (f (t, z) + id(z))e iφ(z) , f, φ, d ∈ R, (4) as a possible ansatz, suitable to obtain elliptic solutions of the CNLSE (2).

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Explicit breather solution of the nonlinear Schrödinger equation

Cited by 2 publications

References 24 publications

Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

Non-existence of certain elliptic solutions of the Cubic-nonlinear Schrödinger Equation

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