Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions

We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.

Figure 1

A first-order Akhmediev breather of the quintic equation with eigenvalue $\lambda =0.9i$. It is given by Eq. (4) with $\delta =\frac{1}{32}$. Here $x$ is the evolution variable and $t$ is the transverse one.

Figure 2

A first-order rogue-wave solution of the quintic equation for $\delta =\frac{1}{32}$.

Figure 3

(a) A first-order Kuznetsov-Ma breather of the quintic equation with eigenvalue $\lambda =1.2i$. It is given by Eq. (6) with $\delta =\frac{1}{64}$. Here $x$ is the evolution variable and $t$ is the transverse one.

Figure 4

A two-breather solution with eigenvalues ${\lambda }_1=0.65i$ and ${\lambda }_2=0.85i$, and $\delta =\frac{1}{32}$. The two breathers are shifted in $x$ due to translations $\Delta x_1=5$ and $\Delta x_2=-5$.

Figure 5

Two-breather solution with one being a Kuznetsov-Ma soliton with eigenvalue ${\lambda }_2=1.35i$ and the other one being an AB with ${\lambda }_1=0.85i$. The parameter $\delta =\frac{1}{64}$.

Figure 6

Contour plots of two superimposed ABs with $\delta =\frac{1}{64}$ and various frequency ratios: (a) ${\kappa }_1=\frac{1}{2}$ and ${\kappa }_2=\frac{1}{10}$; (b) ${\kappa }_1=\frac{1}{5}$ and ${\kappa }_2=\frac{1}{10}$; (c) ${\kappa }_1=\frac{1}{8}$ and ${\kappa }_2=\frac{1}{10}$; (d) ${\kappa }_1=\frac{1}{9}$ and ${\kappa }_2=\frac{1}{10}$.

Figure 7

A separated superposition of a rogue wave with an AB with eigenvalue ${\lambda }_1=0.65i$ and $\delta =\frac{1}{32}$.

Figure 8

Rogue wave on top of AB with eigenvalue ${\lambda }_1=0.65i$ and $\delta =\frac{1}{32}$.

Figure 9

A second-order quintic rogue wave with $\delta =\frac{1}{64}$ given by Eq. (7).

Figure 10

Degenerate two-breather solutions. (a) A two-AB solution with $\kappa =0.8$ and $\delta =\frac{1}{64}$. (b) A two-KM soliton solution with $\kappa =0.8i$ and $\delta =\frac{1}{256}$.

Figure 11

Quintic rogue wave triplet given by Eq. (8) with $x_d=-25$, $t_d=-50$ and $\delta =\frac{1}{64}$.

Figure 12

Two-breather collision. The two complex eigenvalues are ${\lambda }_1=0.05+0.9i$ and ${\lambda }_2=-0.05+0.9i$. The parameter $\delta =\frac{1}{64}$.

Figure 13

Locus of points on the plane of complex eigenvalues ${\lambda }_1$ that lead to parallel breathers. The second eigenvalue is fixed and equal to ${\lambda }_2=0.08+0.9i$ for solid (blue) lines and equal to ${\lambda }_2=0.08+1.5i$ for the dashed (red) lines. Parameter $\delta =\frac{1}{64}$. Solid red dots, blue rectangles, green triangles, and magenta stars on the curves are chosen to signify the examples in the figures below.

Figure 14

Two-breather superposition with the condition Eq. (11), marked by solid red dots in Fig. 13. Here ${\lambda }_2=0.08+0.9i$ and $\delta =\frac{1}{64}$. (a) ${\lambda }_1=-1.70693+0.8i$; (b) ${\lambda }_1=0.141238+0.8i$; (c) ${\lambda }_1=1.1181+0.8i$.

Figure 15

Two-breather superposition with the condition of Eq. (11), marked by green triangles in Fig. 13. Here, ${\lambda }_2=0.08+1.5i$ and $\delta =\frac{1}{64}$. (a) ${\lambda }_1=-1.71834+0.9i$; (b) ${\lambda }_1=0.990488+0.9i$.

Figure 16

Two-breather superposition with the condition of Eq. (11), marked by blue rectangles in Fig. 13. Here (a) ${\lambda }_1=1.22549+0.9i$, ${\lambda }_2=0.08+0.9i$; (b) ${\lambda }_1=0.08+0.9i$, ${\lambda }_2=0.08+0.9i$ (degenerate case). Parameter $\delta =\frac{1}{64}$.

Figure 17

Two-breather superposition with the condition Eq. (11), marked by magenta stars in Fig. 13. Here, ${\lambda }_2=0.08+1.5i$ and $\delta =\frac{1}{64}$. (a) ${\lambda }_1=-2.29479+1.5i$; (b) ${\lambda }_1=0.273178+1.5i$.