Systematic generation of higher-order solitons and breathers of the Hirota equation on different backgrounds

Abstract

We investigate the systematic generation of higher-order solitons and breathers of the Hirota equation on different backgrounds. The Darboux transformation is used to construct proper initial conditions for dynamical generation of high-intensity solitons and breathers of different orders on a uniform background. We provide expressions for the Lax pair generating functions and the procedure for calculating higher-order solutions when Jacobi elliptic functions are the background seed solutions of the Hirota equation. We confirm that the peak height of each soliton or breather in the nonlinear Darboux superposition adds linearly, to form the intensity maximum of the final solution.

Appendix: The general Darboux transformation scheme

A higher-order soliton (breather) solution of the Nth order is a nonlinear superposition of N independent solitons (breathers), each determined by a complex eigenvalue $\lambda _j$, where $1 \le j \le N$ (the real part of the eigenvalue determines the angle between the localized solution and x-axis, while the imaginary part characterizes the periodic modulation frequency [27]). The Nth-order wave function given by the DT is:

$$\begin{aligned} {\psi _n} = {\psi _{n - 1}} + \frac{{2\left( {\lambda _n^* - {\lambda _n}} \right) {s_{n,1}}r_{n,1}^*}}{{{{\left| {{r_{n,1}}} \right| }^2} + {{\left| {{s_{n,1}}} \right| }^2}}}. \end{aligned}$$

(7.1)

In order to find $r_{n,1}$ and $s_{n,1}$, one has to analyze recursive relations between $r_{n,p}(x,t)$ and $s_{n,p}(x,t)$ functions of the Lax pair equations in the general form:

$$\begin{aligned} \begin{aligned} {r_{n,p}} =&\,\, [\left( {\lambda _{n - 1}^* - {\lambda _{n - 1}}} \right) s_{n - 1,1}^*{r_{n - 1,1}}{s_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - {\lambda _{n - 1}}} \right) {\left| {{r_{n - 1,1}}} \right| ^2}{r_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - \lambda _{n - 1}^*} \right) {\left| {{s_{n - 1,1}}} \right| ^2}{r_{n - 1,p + 1}}] \\&/\left( {{{\left| {{r_{n - 1,1}}} \right| }^2} + {{\left| {{s_{n - 1,1}}} \right| }^2}} \right) \\ {s_{n,p}} =&\,\, [\left( {\lambda _{n - 1}^* - {\lambda _{n - 1}}} \right) {s_{n - 1,1}}r_{n - 1,1}^*{r_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - {\lambda _{n - 1}}} \right) {\left| {{s_{n - 1,1}}} \right| ^2}{s_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - \lambda _{n - 1}^*} \right) {\left| {{r_{n - 1,1}}} \right| ^2}{s_{n - 1,p + 1}}] \\&/\left( {{{\left| {{r_{n - 1,1}}} \right| }^2} + {{\left| {{s_{n - 1,1}}} \right| }^2}} \right) . \\ \end{aligned} \end{aligned}$$

(7.2)

From the last equation, it can be deduced that all $r_{n,p}$ and $s_{n,p}$ can be calculated just from $r_{1,j}$ and $s_{1,j}$, with $1 \le j \le N$. The functions $r_{1,j}(x,t)$ and $s_{1,j}(x,t)$, forming the Lax pair $R = \left( {\begin{array}{*{20}{c}} r \\ s \\ \end{array}} \right) \equiv \left( {\begin{array}{*{20}{c}} {{r_{1,j}}} \\ {{s_{1,j}}} \\ \end{array}} \right) ,$ are determined by the eigenvalue $\lambda \equiv \lambda _j$, an embedded arbitrary center of the solution $\left( x_{0j},t_{0j}\right)$, and a system of linear differential equations:

$$\begin{aligned} \frac{{\partial R}}{{\partial t}} = U \cdot R,\quad \frac{{\partial R}}{{\partial x}} = V \cdot R. \end{aligned}$$

(7.3)

Particularly for the Hirota equation, matrices U and V are defined as ($\psi \equiv \psi _0$) [30]:

$$\begin{aligned} U= & {} i\left[ {\begin{array}{*{20}{c}} \lambda &{} {\psi {{(x,t)}^*}} \\ {\psi (x,t)} &{} { - \lambda } \\ \end{array}} \right] ,\nonumber \\ V= & {} \sum \limits _{k = 0}^3 {{\lambda ^k} \cdot i\left[ {\begin{array}{*{20}{c}} {{A_k}} &{} {B_k^*} \\ {{B_k}} &{} { - {A_k}} \\ \end{array}} \right] }, \end{aligned}$$

(7.4)

where the coefficients $A_k$ and $B_k$ are

$$\begin{aligned} {A_0}= & {} - \frac{1}{2}|\psi {|^2} - i\alpha \left( {\psi _t^*\psi - {\psi _t}{\psi ^*}} \right) ,\nonumber \\ {B_0}= & {} 2\alpha {\left| \psi \right| ^2}\psi + \frac{1}{2}i{\psi _t} + \alpha {\psi _{tt}}, \nonumber \\ {A_1}= & {} 2\alpha {\left| \psi \right| ^2},\quad {B_1} = \psi - 2i\alpha {\psi _t} \nonumber \\ {A_2}= & {} 1,\quad {B_2} = - 4\alpha \psi ,\quad {A_3} = - 4\alpha ,\quad {B_3} = 0.\nonumber \\ \end{aligned}$$

(7.5)