Doubly periodic solutions and breathers of the Hirota equation: recurrence, cascading mechanism and spectral analysis

Abstract

The Hirota equation is an extension of the nonlinear Schrödinger equation by incorporating third-order dispersion. Doubly periodic solutions for the Hirota equation are established in terms of theta and elliptic functions. Cubic dispersion preserves numerical robustness, as slightly disturbed exact solutions as initial states still evolve to the analytical configurations. In contrast with the nonlinear Schrödinger case, the wavenumber of the envelope must satisfy an algebraic equation related to the magnitude of the quadratic / cubic terms and the period of the wave patterns. The long wave limits of these doubly periodic patterns will yield the widely studied Kuznetsov-Ma and Akhmediev breathers. A cascading mechanism for the Hirota equation is studied, which will elucidate the first formation of breather modes. Higher harmonics exponentially small initially will grow at a larger rate than the fundamental mode. Eventually the high frequency modes reach roughly the same magnitude at one moment in time (or spatial location), signifying the first occurrence of breather. Breathers then decay and modulation instability emerges again for sufficiently small amplitude. The cycle is repeated and constitutes a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence. These analytical doubly periodic solutions will permit the prediction of the period of recurrence. These results can be applied in hydrodynamic and optical contexts where third or higher-order dispersion is present.

Appendices

Appendix A

Theta functions are defined by Fourier series with exponentially decaying coefficients. In mathematical analysis, the Jacobi elliptic functions, sn, cn, dn can be expressed as ratio of theta functions. In classical mathematical analysis, theta functions are usually expressed as Fourier series with a real parameter q (τ purely imaginary):

$$ \theta_{1} \left( {x,{\uptau }} \right) = 2\sum\limits_{n = 0}^{\infty } {\left( { - 1} \right)^{n} } q^{{\left( {n + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} }} \sin \left( {2n + 1} \right)x $$

$$ = - \sum\limits_{m = - \infty }^{\infty } {\exp \left[ {{\uppi }i{\uptau }\left( {m + \frac{1}{2}} \right)^{2} + 2i\left( {m + \frac{1}{2}} \right)\left( {x + \frac{{\uppi }}{2}} \right)} \right]} , $$

(A1)

$$ \theta_{2} \left( {x,{\uptau }} \right) = 2\sum\limits_{n = 0}^{\infty } {q^{{\left( {n + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} }} } \cos \left( {2n + 1} \right)x $$

$$ = \sum\limits_{m = - \infty }^{\infty } {\exp \left[ {{\uppi }i{\uptau }\left( {m + \frac{1}{2}} \right)^{2} + 2i\left( {m + \frac{1}{2}} \right)x} \right]} , $$

(A2)

$$ \theta_{3} \left( {x,{\uptau }} \right) = 1 + 2\sum\limits_{n = 1}^{\infty } {q^{{n^{2} }} } \cos 2nx $$

$$ = \sum\limits_{m = - \infty }^{\infty } {\exp \left( {{\uppi }i{\uptau }m^{2} + 2imx} \right)} , $$

(A3)

$$ \theta_{4} \left( {x,{\uptau }} \right) = 1 + 2\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n} q^{{n^{2} }} } \cos 2nx $$

$$ = \sum\limits_{m = - \infty }^{\infty } {\exp \left[ {{\uppi }i{\uptau }m^{2} + 2im\left( {x + \frac{{\uppi }}{2}} \right)} \right]} , $$

(A4)

$$ 0 < q < 1, \, q = \exp \left( {{\uppi }i{\uptau }} \right), \, q = \exp \left( { - \frac{{{\uppi }K^{\prime}}}{K}} \right). $$

(A5)

K, K' are the complete elliptic integrals of the first kind with parameters k and (1 – k²)^1/2. In terms of the patterns in the summations, there is an extra minus sign in front of the function θ₁. In terms of basic mathematical properties, θ₁ is odd while the other three are even. The zeros or roots of θ₁, θ₂, θ₃, θ₄, are located at square grids in the complex plane at Mπ + Nπτ, (M + 1/2)π + Nπτ, (M + 1/2)π + (N + 1/2)πτ, Mπ + (N + 1/2)πτ respectively, with M, N integers. We can readily show by direct calculations that θ₁ and θ₂, as well as θ₃ and θ₄, are related by a phase shift of π/2.

The Jacobi elliptic functions can be expressed as ratios of theta functions with ‘stretched’ arguments or variables as (for simplicity, the dependence on the parameter τ in Eqs. A1-A4 has been dropped in Eqs. A6, A7:

$$ \begin{gathered} {\text{sn}}\left( u \right) = \frac{{\theta_{3} \left( 0 \right)\theta_{1} \left( z \right)}}{{\theta_{2} \left( 0 \right)\theta_{4} \left( z \right)}},{\text{ cn}}\left( u \right) = \frac{{\theta_{4} \left( 0 \right)\theta_{2} \left( z \right)}}{{\theta_{2} \left( 0 \right)\theta_{4} \left( z \right)}}, \, \hfill \\ {\text{dn}}\left( u \right) = \frac{{\theta_{4} \left( 0 \right)\theta_{3} \left( z \right)}}{{\theta_{3} \left( 0 \right)\theta_{4} \left( z \right)}}, \hfill \\ \end{gathered} $$

(A6)

$$ z = \frac{u}{{\theta_{3}^{2} \left( 0 \right)}}, \, k = \frac{{\theta_{2}^{2} \left( 0 \right)}}{{\theta_{3}^{2} \left( 0 \right)}}, \, k^{\prime} = \frac{{\theta_{4}^{2} \left( 0 \right)}}{{\theta_{3}^{2} \left( 0 \right)}}, \, k^{2} + \left( {k^{\prime}} \right)^{2} = 1. $$

(A7)

Theta functions possess many properties which make them ideal candidates for handling bilinear forms of evolution equations.

Sum and difference identities: Theta functions possess a huge varieties of identities:

$$ \theta_{3} \left( {x + y} \right)\theta_{3} \left( {x - y} \right)\theta_{2}^{2} \left( 0 \right) = \theta_{4}^{2} \left( x \right)\theta_{1}^{2} \left( y \right) + \theta_{3}^{2} \left( x \right)\theta_{2}^{2} \left( y \right), $$

(A8)

$$ \theta_{4} \left( {x + y} \right)\theta_{4} \left( {x - y} \right)\theta_{2}^{2} \left( 0 \right) = \theta_{4}^{2} \left( x \right)\theta_{2}^{2} \left( y \right) + \theta_{3}^{2} \left( x \right)\theta_{1}^{2} \left( y \right). $$

(A9)

By differentiating Eq. A8 with respect to y twice and setting y = 0,

$$ \begin{gathered} D_{x}^{2} \theta_{3} \left( x \right) \cdot \theta_{3} \left( x \right) = \frac{{2\theta_{2}^{\prime \prime } \left( 0 \right)\theta_{3}^{2} \left( x \right)}}{{\theta_{2} \left( 0 \right)}} + 2\theta_{3}^{2} \left( 0 \right)\theta_{4}^{2} \left( 0 \right)\theta_{4}^{2} \left( x \right), \hfill \\ D_{x}^{2} \theta_{4} \left( x \right) \cdot \theta_{4} \left( x \right) = 2\theta_{3}^{2} \left( 0 \right)\theta_{4}^{2} \left( 0 \right)\theta_{3}^{2} \left( x \right) + \frac{{2\theta_{2}^{\prime \prime } \left( 0 \right)\theta_{4}^{2} \left( x \right)}}{{\theta_{2} \left( 0 \right)}}, \hfill \\ \end{gathered} $$

i.e., the bilinear derivatives (‘D’ operator, defined below Eq. 4d) of theta functions can be expressed back in terms of the theta functions themselves through such ‘sum’ and ‘difference’ identities Eqs. A8, A9. Similar principles apply to the computations of

$$ D_{x} \theta_{m} \cdot \theta_{n} , \, D_{x}^{2} \theta_{m} \cdot \theta_{n} , $$

where m, n are integers.

Derivatives at specified values – By Taylor expansion of these sum and difference formulas and comparing powers, we relate the derivatives of theta functions to the functions themselves [20, 22], e.g.,

$$ \frac{{\theta_{4}^{\prime \prime } \left( 0 \right)}}{{\theta_{4} \left( 0 \right)}} - \frac{{\theta_{3}^{\prime \prime } \left( 0 \right)}}{{\theta_{3} \left( 0 \right)}} = \theta_{2}^{4} \left( 0 \right), \, \frac{{\theta_{4}^{\prime \prime } \left( 0 \right)}}{{\theta_{4} \left( 0 \right)}} - \frac{{\theta_{2}^{\prime \prime } \left( 0 \right)}}{{\theta_{2} \left( 0 \right)}} = \theta_{3}^{4} \left( 0 \right), $$

$$ \frac{{\theta_{3}^{\prime \prime } \left( 0 \right)}}{{\theta_{3} \left( 0 \right)}} - \frac{{\theta_{2}^{\prime \prime } \left( 0 \right)}}{{\theta_{2} \left( 0 \right)}} = \theta_{4}^{4} \left( 0 \right). $$

By differentiation using elementary calculus, we can simplify bilinear derivatives of products of exponential functions and the auxiliary variables, e.g.,

$$ \begin{gathered} D_{x} \left[ {\exp \left( {imx} \right)g \cdot \exp \left( {inx} \right)f} \right] \hfill \\ = \left[ {D_{x} g \cdot f + i\left( {m - n} \right)g\;f} \right]\exp \left[ {i\left( {m + n} \right)x} \right], \hfill \\ \end{gathered} $$

$$ \begin{aligned} & D_{x}^{2} \left[ {\exp \left( {imx} \right)g \cdot \exp \left( {inx} \right)f} \right] \hfill \\& = \left[ {D_{x}^{2} g \cdot f + 2i\left( {m - n} \right)D_{x} g \cdot f - \left( {m - n} \right)^{2} g\;f} \right] \hfill \\& \quad \times \exp \left[ {i\left( {m + n} \right)x} \right]. \hfill \\ \end{aligned} $$

These formulas will be especially relevant in the present wave packet type calculations, as the carrier wave envelope is expressed as a complex exponential function multiplied by the auxiliary functions governing the actual packet structure (Eq. 4).

Appendix B

The parameters of analytical breather are as follows:

$$ {\upzeta } = {\upzeta }_{R} + i{\upzeta }_{I} = \sqrt {{\upgamma }_{1}^{2} + 4{\upgamma }_{1} {\upeta } + 4\left( {{\upeta }^{2} + {\upchi }^{2} } \right)}, $$

(B1)

$$ \begin{gathered} \Theta_{11} = {\upgamma }_{1} x + \left( { - {{\lambda \gamma }}_{1}^{2} + {{\sigma \gamma }}_{1}^{3} + 2{{\lambda \chi }}^{2} - 6{{\sigma \gamma }}_{1} {\upchi }^{2} } \right)t + {\upzeta }x \hfill \\ \quad \quad + {\upzeta }t\left[ { - {\uplambda }\left( {{\upgamma }_{1} - 2{\upeta }} \right) + {\upsigma }\left( {{\upgamma }_{1}^{2} - 2{\upgamma }_{1} {\upeta } + 4{\upeta }^{2} - 2{\upchi }^{2} } \right)} \right] \hfill \\ \end{gathered} ,$$

(B2)

$$\begin{aligned} & \Theta_{12} = - {\upgamma }_{1} x + \left( {{{\lambda \gamma }}_{1}^{2} - {{\sigma \gamma }}_{1}^{3} - 2{{\lambda \chi }}^{2} + 6{{\sigma \gamma }}_{1} {\upchi }^{2} } \right)t + {\upzeta }x \\ & + {\upzeta }t\left[ { - {\uplambda }\left( {{\upgamma }_{1} - 2{\upeta }} \right) + {\upsigma }\left( {{\upgamma }_{1}^{2} - 2{\upgamma }_{1} {\upeta } + 4{\upeta }^{2} - 2{\upchi }^{2} } \right)} \right], \end{aligned}$$

(B3)

$$ \Theta_{2} = {\upzeta }\left\{ {x - t\left[ {{\uplambda }\left( {{\upgamma }_{1} - 2{\upeta }} \right) - {\upsigma }\left( {{\upgamma }_{1}^{2} - 2{\upgamma }_{1} {\upeta } + 4{\upeta }^{2} - 2{\upchi }^{2} } \right)} \right]} \right\}. $$

(B4)

Based on Eq. B1, ζ_R and ζ_I can be determined by

$$ \begin{gathered} {\upzeta }_{R}^{2} - {\upzeta }_{I}^{2} = {\upgamma }_{1}^{2} - 4{\upeta }_{I}^{2} + 4{\upgamma }_{1} {\upeta }_{R} + 4{\upeta }_{R}^{2} + 4{\upchi }^{2} , \, \hfill \\ 2{\upzeta }_{R} {\upzeta }_{I} = 4{\upgamma }_{1} {\upeta }_{I} + 8{\upeta }_{I} {\upeta }_{R} . \hfill \\ \end{gathered} $$

(B5)