Nonintegrable semidiscrete Hirota equation: Gauge-equivalent structures and dynamical properties

In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete ${\mathrm{Hirota}}^{-}$ equation and the nonintegrable semidiscrete ${\mathrm{Hirota}}^{+}$ equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete ${\mathrm{Hirota}}^{-}$ equation and the nonintegrable semidiscrete ${\mathrm{Hirota}}^{+}$ equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.

Figure 1

Evolution of orbits of the map $M$ with increase of the parameter $\tilde{\gamma }$ for $F=-1.5,J=0.1,b=1.5$ and $(x_1,z_1)=(0.15,-0.1)$. (a) For nonintegrable semidiscrete ${\mathrm{Hirota}}^{-}$: black dot, $\tilde{\gamma }=0$; red circle, $\tilde{\gamma }=0.1$; green square, $\tilde{\gamma }=0.2$; purple cross,$\tilde{\gamma }=0.3$; blue plus,$\tilde{\gamma }=0.4$; magenta triangle, $\tilde{\gamma }=0.5$. (b) For nonintegrable semidiscrete ${\mathrm{NLS}}^{-}$: black dot, $\tilde{\gamma }=0$; red circle, $\tilde{\gamma }=0.1$; green square, $\tilde{\gamma }=0.2$; purple cross, $\tilde{\gamma }=0.21$; blue plus, $\tilde{\gamma }=0.22$; magenta triangle, $\tilde{\gamma }=0.23$. (c) The special orbit for nonintegrable stationary discrete ${\mathrm{Hirota}}^{-}$ as $\tilde{\gamma }=0.51$.

Figure 2

Evolution of orbits of the map $M$ with increase of the parameter $b$ for $F=-0.5,J=0.1,\tilde{\gamma }=0.1$ and $(x_1,z_1)=(2,0)$. (a) For nonintegrable semidiscrete ${\mathrm{Hirota}}^{-}$: cyan dot, $b=0.5$; black circle, $b=1$; red square, $b=1.5$; magenta cross, $b=2.5$; green plus, $b=5$; blue triangle, $b=10$. (b) For nonintegrable semidiscrete ${\mathrm{NLS}}^{-}$: black circle, $b=1$; red square, $b=1.5$; magenta cross, $b=2.5$; green plus, $b=5$; blue triangle, $b=10$. (c) The special orbit for nonintegrable stationary discrete ${\mathrm{Hirota}}^{-}$ at $b=1.1$.

Figure 3

Orbits of nonintegrable stationary discrete ${\mathrm{Hirota}}^{-}$ evolve into integrable ones with the increase of $a$ due to the fact that the integrable stationary discrete complex mKdV becomes significant with the increase of the parameter $a$. Here the parameters are $F=1.5,\tilde{\gamma }=0.3,J=0.3,b=1.5$, and $(x_1,z_1)=(j,0)(j=1,2,3,\cdots ,10)$.

Figure 4

Orbits of the map $M$ for the following parameters: (a) regular orbit with initial value $(x_1,z_1)=(2,1)\left(\tilde{\gamma }⩽JF/b\right)$: $F=1.5,\tilde{\gamma }=0.02,J=0.1,a=b=0.5$; (b) irregular orbit: $F=0.5,\tilde{\gamma }=0.3,J=0.3,a=0.2,b=1.0$.

Figure 5

Evolution of orbits of the map $M$ with the increase of the parameter $\tilde{\gamma }$ when $F=0.5,J=0.1,a=0.5,b=1.5$, and $(x_1,z_1)=(3,-2.5)$: red triangle, $\tilde{\gamma }=0$; green diamond, $\tilde{\gamma }=0.1$; purple dot, $\tilde{\gamma }=0.15$; black circle, $\tilde{\gamma }=0.2$; cyan square, $\tilde{\gamma }=0.21$; magenta plus, $\tilde{\gamma }=0.25$; blue cross, $\tilde{\gamma }=0.27$.

Figure 6

Evolution of orbits of the map $M$ with increase of the parameter $b$ with initial value $(x_1,z_1)=(3,-2.5)$, but when $b=8$, all the points distribute in two trajectories. $F=0.5,\tilde{\gamma }=0.2,J=0.1,a=1.5$. Black circle, $b=0.5$; purple dot, $b=2$; green cross, $b=5$; red plus, $b=8$; blue square, $b=15$.

Figure 7

The nonintegrable case evolves to the integrable one with the increase of $a$ for the nonintegrable discrete ${\mathrm{Hirota}}^{+}$ when $F=1.0,\tilde{\gamma }=0.2,J=0.25,b=1$.

Figure 8

(a): A broad chaotic sea for nonintegrable stationary discrete ${\mathrm{Hirota}}^{+}$ has been developed with parameters $F=0.5,\tilde{\gamma }=0.2,J=0.1$, and $b=1$. [(c) and (d)] The broad chaotic sea disappeared with the increase of the parameter $a$.

Figure 9

Evolution of bounded regime with the increase of parameter $a$ at $(x_1,z_1)=(j,0)(-1800⩽j⩽-100)$ with an interval of 100, and $j$ takes the integral. Here $F=-1.7,\tilde{\gamma }=0.5,J=1.1$, and $b=1$. The larger $a$ the smoother the boundary of orbits becomes. Panel (e) is the local enlargement of (b).

Figure 10

Another spatial feature of the nonintegrable semidiscrete ${\mathrm{Hirota}}^{+}$ equation when $F=-1.5,\tilde{\gamma }=0.3,J=1,a=0.5$, and $b=1.5$. Three orbits are surrounded by lots of curves regularly. Black orbits and green ones are formed with initial value $(x_1,z_1)=(j,0)(1⩽j⩽10)$ and initial value $(x_1,z_1)=(0,j)(0⩽j⩽6)$ with an interval of 0.5, respectively.