Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation

In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter ${\gamma }_1$, denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by ${\gamma }_1$ are discussed in detail.

Figure 1

The dynamical evolution of the first-order breather $|q^{\left[1\right]}{|}^2$ on the ($x,t$) plane. When the value of ${\gamma }_1$ increases, the number of peaks on same interval of time also increases.

Figure 2

The dynamical evolution of the first-order rogue wave $|q_{\mathrm{limit}}^{\left[1\right]}{|}^2$ on the ($x,t$) plane. For larger values of ${\gamma }_1$, it is clear that the compression effects in $t$ direction are quite high.

Figure 3

The dynamical evolution of the second-order rogue wave $|q_{\mathrm{rw}}^{\left[2\right]}{|}^2$ on the ($x,t$) plane. Comparing (a) and (b) with (c) and (d) indicates effective high compression in the $t$ direction.

Figure 4

The dynamical evolution of the second-order rogue wave $|q_{\mathrm{rw}\mathrm{trig}}^{\left[2\right]}{|}^2$ on the ($x,t$) plane. It is shown from (b), (c), and (d) that rogue wave compression increases as the value of ${\gamma }_1$ increases.

Figure 5

The dynamical evolution of the third-order rogue wave $|q_{\mathrm{rw}}^{\left[3\right]}{|}^2$ on the ($x,t$) plane. By comparison with (a) and (b), (c) and (d) are highly compressed.

Figure 6

The dynamical evolution of the third-order rogue wave $|q_{\mathrm{rw}\mathrm{trig}}^{\left[3\right]}{|}^2$ on the ($x,t$) plane. It is shown from (b), (c), and (d) that rogue wave is compressed more while increasing the value of ${\gamma }_1$.

Figure 7

The dynamical evolution of the third-order rogue wave $|q_{\mathrm{rwcirc}}^{\left[3\right]}{|}^2$ on the ($x,t$) plane. It is shown from (b), (c), and (d) that rogue wave is changing its shape and also compression increases by increasing the value of ${\gamma }_1$.