# Parametric resonance and radiative decay of dispersion-managed solitons

## Summary (1 min read)

### 1. Introduction.

- The authors analysis starts with the standard NLS equation (1.3) for m = 0, such that the right-hand side of (1.3) is treated as a small perturbation.
- Methods of their analysis are similar to the soliton perturbation theory in [13] , but their calculations are more systematic.
- Rigorous analysis of decay rates in the linear Schrödinger equation was recently considered in [17, 18] , where the bound states were supported by a time-dependent periodic potential in [17] and by a time-independent potential in [18] .
- Section 4 describes a comparison between the analytical and numerical results.
- Appendices A and B describe technical details of the first-order solution in the perturbation series expansions.

### 4. Numerical simulations of DM solitons.

- When the initial value of µ is large, the asymptotic analysis predicts that the soliton decays exponentially according to the bounds in (3.18) .
- Thus, the accuracy of the analytical prediction needs to be examined.
- Further evolution of the DM soliton shows that the DM soliton character is lost after the fifth critical resonance at the average amplitude about 9.38.
- Figures 6(b) and (c ) indicate that when µ(0) is close to or above the lowest critical resonance value µ 1 = 2π, the pulse deviates further from the DM soliton than in the case of m = 0.1, and the pulse amplitude oscillates with a period further away from the unit period of the dispersion map.

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### Cites methods from "Parametric resonance and radiative ..."

...This phenomenon was modelled analytically and numerically for the strong dispersion management in [YK01] and for the weak dispersion management in [ PY04 ]....

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##### References

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### "Parametric resonance and radiative ..." refers background or methods in this paper

...We use Kaup’s method [ 19 ] to solve the inhomogeneous problem (2.14)–(2.15) with the spectral decomposition for a linearized NLS operator....

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...The linearized NLS operator L possesses a complete set of eigenfunctions [ 19 ] that consists of eigenfunctions associated with two branches of the continuous spectrum and eigenfunctions associated with the zero eigenvalue of the discrete spectrum....

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...The orthogonality conditions (A.10)–(A.11) are modified compared with the original definition in [ 19 ]....

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...It follows from (2.12) that the system (2.14)–(2.15) is supplemented with zero initial conditions: U (1) n (0 ,t ; µ0) = 0 for any |n |≥ 1. Solutions of the system (2.14)–(2.15) are constructed in Appendix A with the use of the spectral decomposition for a linearized NLS operator [ 19 , 20]....

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### "Parametric resonance and radiative ..." refers background in this paper

...For this dispersion map, the DM soliton is chirp-free at mod(z, 1) = 0 and mod(z, 1) = 1 2 (see [21], for instance)....

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