Integrable pair-transition-coupled nonlinear Schrödinger equations

We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.

Figure 1

The dynamics of waterfall structure nonlinear waves: (a) for component $q_1$ and (b) for component $q_2$. It is seen that the background amplitudes before and after the breather emerges are unequal, in contrast to the breather reported before. This corresponds to a drastic nonlinear particle transition behavior. Parameters: $\theta =arccos\left(\frac{4}{5}\right), {\vartheta }_1=arcsin\left(\frac{3}{5}\right)$, and ${\kappa }_1=0$.

Figure 2

Fundamental rogue wave which possesses one hump and two valleys: (a) for component $q_1$ and (b) for component $q_2$. It is seen that the pattern is quite different from the eye-shaped rogue wave reported before. Parameters: $\theta =arccos\left(\frac{\sqrt{2}}{2}\right), x_1=t_1=0.$

Figure 3

The evolution of kink-dark waves: (a) for component $q_1$ and (b) for component $q_2$. It is seen that there are a kink and a dark soliton coexisting in the intensity distribution. Parameters: $c_1=0, \theta =\pi /6, {\vartheta }_2=\pi /2$.

Figure 4

The evolution of rogue wave and rogue wave pair in the coupled model: (a) for component $q_1$ and (b) for component $q_2$. Parameters: $\theta =arccos(3/5), x_1=0$, and $t_1=1$.

Figure 5

The numerical simulation of the rogue wave and rogue wave pair with small noise. The initial excitation condition is given by the exact ones at $t=-3$ in Fig. 4 by multiplying a factor $(1+0.01Random[-1,1\left]\right)$. It is seen that the rogue wave pair are robust against small noise.