Abstract
We present a one-line closed-form expression for the three-parameter breather of the nonlinear Schrödinger equation. This provides an analytic proof of the time period doubling observed in experiments. The experimental check that some pulses generated in optical fibers are indeed such generalized breathers will be drastically simplified.
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Notes
The scaling invariance \((x,t,A)\to(kx,k^2t,kA)\) of the NLS equation reduces this number by one.
We never use the ambiguous term “periodic” for elliptic solutions, but always either “doubly periodic,” alias “elliptic” (example: Jacobi \(\operatorname{dn}\), Weierstrass \(\wp\)), or “quasi-doubly periodic,” alias “quasielliptic,” alias “elliptic of the second kind” in Hermite’s terminology [11, Vol. 1, p. 227; Vol. 2, p. 506] (example: the solution \(\mathrm{H}(t,a)\) of Lamé equation (10)).
To convert to the notation of Jacobi, see [21], § 18.9.11 and 18.10.8.
Under the addition of any of the two periods, a quasielliptic function is multiplied by a constant factor, the multiplier.
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Acknowledgments
The author is pleased to thank Micheline Musette for a critical reading of the manuscript.
Funding
This work was initiated at the Centre International de Rencontres Mathématiques, Marseille (grant 2311, year 2019), whose hospitality is gratefully acknowledged.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 209, pp. 46–58 https://doi.org/10.4213/tmf10095.
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Conte, R. Explicit breather solution of the nonlinear Schrödinger equation. Theor Math Phys 209, 1357–1366 (2021). https://doi.org/10.1134/S0040577921100032
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DOI: https://doi.org/10.1134/S0040577921100032