Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy

We present rogue wave solutions of the integrable nonlinear Schrödinger equation hierarchy with an infinite number of higher-order terms. The latter include higher-order dispersion and higher-order nonlinear terms. In particular, we derive the fundamental rogue wave solutions for all orders of the hierarchy, with exact expressions for velocities, phase, and “stretching factors” in the solutions. We also present several examples of exact solutions of second-order rogue waves, including rogue wave triplets.

Figure 1

Plot of relative factors ${\widehat{v}}_n\left(k\right)$ as a function of $k$ calculated using Eqs. (11), (12), and (14), for $n=1$ (uppermost curve, blue), $n=2$ (second curve from top, magenta), $n=3$ (third curve from top, olive), and $n=4$ (lowest curve, green).

Figure 2

Plot of the first-order rogue wave, defined by Eq. (8). The two coefficients in Eq. (3) are ${\alpha }_3=\frac{1}{2}$ and ${\alpha }_5=-\frac{1}{32}$ while all other ${\alpha }_j$'s zero. Here $k=1$.

Figure 3

Plot of the first-order rogue wave defined by Eq. (8). The coefficient ${\alpha }_3=1$. All other ${\alpha }_j$'s in Eq. (3) are zero. Here $k=\sqrt{2}$, giving ${\widehat{v}}_1=0$ as can be seen from Fig. 1. Hence, $v_0=0$. This choice leads to a symmetric intensity pattern.

Figure 4

Plot of the second-order real solution, ${\psi }_2$, given by Eq. (24). The coefficients of Eq. (1) are ${\alpha }_3=1$, ${\alpha }_5=1/2$, ${\alpha }_7=1/4$, ${\alpha }_9=1/8$, and ${\alpha }_{11}=1/16$.

Figure 5

Second-order complex solution, with $j=4$, i.e., ${\psi }_2^{\left(4\right)}$ from Eqs. (33, 34, 35, 36). Here ${\alpha }_8=1/60$ (and all other $\alpha$' s are zero).

Figure 6

Second-order triplet solution $|{\widehat{\psi }}_2^{\left(3\right)}|$, with $j=3$, which is defined by Eqs. (42, 43, 44, 45). Here ${\alpha }_6=0.03$ (and all other $\alpha$' s are zero), while additional parameters $\beta =-130$ and $\gamma =-180$.