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

Solitons on a finite 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 (1987)]. Having more parameters to vary, this significantly wider family has the 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 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 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.

Figure 1

An example of the B-type double-periodic solution. (a) False color plot of intensity ${\left|\psi (z,t)\right|}^2$. Two longitudinal periods are shown. (b) Evolution of the spectrum. (c) Intensity profile ${\left|\psi \left(t\right)\right|}^2$ at $z=0$ (dashed red curve, minimum of modulation) and at $z=L/2=3.78$ (solid blue curve, maximal modulation). (d) Evolution of the first three Fourier components in logarithmic scale ($20{log}_{10}\left|{\widehat{\psi }}_k\left(z\right)\right|$). Parameters are as follows: ${\alpha }_1=0.3$, ${\alpha }_2=0.4$, ${\alpha }_3=1$, giving the temporal period $T=3.9$ (modulation frequency $\omega =2\pi /T=1.61$) and the spatial period $L=7.55$.

Figure 2

Major characteristics of the B-type doubly periodic solutions on the plane (${\alpha }_1,{\alpha }_2$) when ${\alpha }_3=1$. (a) False color plot of frequency $\omega =2\pi /T$. (b) False color plot of wave number $q=2\pi /L$. The thin curves in (a) and (b) show the lines of equal frequency or wave number. (c) Maximal and minimal amplitudes of $\left|\psi \right(t=0,z\left)\right|$. (d) Absolute value of the CW component in $\psi (x,t)$ at $z=0$. Special cases in (a) as follows: PS, Peregrine solution; CW, the family of continuous wave solutions; DN, the family of DNoidal wave solutions; AB, the family of Akhmediev breathers; KM, the family of Kuznetsov-Ma solitons; BS, bright soliton on zero background; DP, the two-parameter family of doubly periodic solutions.

Figure 3

An example of the A-type doubly periodic solution. (a) False color plot of intensity ${\left|\psi (t,z)\right|}^2$. One complete period of evolution is shown. (b) Evolution of the spectrum. (c) The intensity profile ${\left|\psi \left(t\right)\right|}^2$ at $z=-L/4$ (dashed red curve, the minimal modulation) and at $z=0$ (solid blue curve, the maximal pulse compression). (d) Evolution of the first three Fourier components in logarithmic scale ($20{log}_{10}\left|{\widehat{\psi }}_k\left(z\right)\right|$). Parameters are as follows: ${\alpha }_3=1$, $\rho =0.355$, $\eta =0.073$. The resulting temporal period $T=3.9$ corresponds to the modulation frequency $\omega =2\pi /T=1.61$. The spatial period $L=13.6$.

Figure 4

Major characteristics of A-type doubly periodic solutions vs $\rho$ and $\eta$. False color plots of (a) frequency $\omega =2\pi /T$ and (b) wave number $q=2\pi /L$. (c) Maximal (${\psi }_{\max }$) and minimal (${\psi }_{\min }$) values of $\left|\psi \right(t=0,z\left)\right|$. (d) Difference between the maximal and minimal values of $\left|\psi \right(t=0,z\left)\right|$. Notations in (a) as follows: PS, Peregrine solution; CW, continuous wave; AB, Akhmediev breathers; KM, Kuznetsov-Ma solitons; BS, bright soliton. Here parameter ${\alpha }_3=1$.

Figure 5

False color plots of amplification of the first Fourier harmonics $|{\widehat{\psi }}_1|$ in decibels for (a) the B-type and (b) the A-type doubly periodic solutions. The superimposed level curves are for the constant frequencies $\omega$. The color map is saturated at 40 dB for readability. (c) Amplification curves on selected straight lines ${\alpha }_1={\alpha }_2-\mathrm{\Delta }a$ in the plot (a). Here $\mathrm{\Delta }a<{\alpha }_2<1$. (d) Amplification curves at selected lines of constant $\eta$ in the plot (b). Here $-3<\rho <1$. Vertical dashed lines in (c) and (d) show the MI threshold for a CW of unit amplitude. In all cases, ${\alpha }_3=1$.

Figure 6

Evolution of the A-type doubly periodic solution illustrating the amplification outside the conventional MI band. (a) False color plot of intensity ${\left|\psi (z,t)\right|}^2$. (b) Evolution of the first three Fourier components. (c) Intensity profiles ${\left|\psi \left(t\right)\right|}^2$ at $z=-L/4$ (dashed red curve, minimum of modulation) and at $z=0$ (solid blue curve, maximum pulse compression). (d) Input (solid red) and output (dashed blue) spectra. The input spectrum has a pair of small secondary (third-order) sidebands. Parameters of the solution are as follows: $\rho =0$, $\eta =1$, ${\alpha }_3=1$. In this case, the temporal period $T=2.74$ (modulation frequency $\omega =2.287>2$) and the spatial period $L=2.75$.

Figure 7

False color plots of minimum amplitude of the first Fourier harmonics $|{\widehat{\psi }}_1|$ in the plane of free parameters of the family of solutions. (a) B-type solutions (minimum sidebands obtained at $z=0$; parameters ${\alpha }_1,{\alpha }_2$). (b) A-type solutions (minimum sidebands obtained at $z=-L/4$; parameters $\rho ,\eta$). Here ${\alpha }_3=1$.