Dynamics of kink, antikink, bright, generalized Jacobi elliptic function solutions of matter-wave condensates with time-dependent two- and three-body interactions

By using the F-expansion method associated with four auxiliary equations, i.e., the Bernoulli equation, the Riccati equation, the Lenard equation, and the hyperbolic equation, we present exact explicit solutions describing the dynamics of matter-wave condensates with time-varying two- and three-body nonlinearities. Condensates are trapped in a harmonic potential and they exchange atoms with the thermal cloud. These solutions include the generalized Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. In addition, we have also found rational function solutions. Solutions constructed here have many free parameters that can be used to manipulate and control some important features of the condensate, such as the position, width, velocity, acceleration, and homogeneous phase. The stability of the solutions is confirmed by their long-time numerical behavior.

Figure 1

Spatiotemporal evolution of the wave function of Eq. (21) for $j=2$. The parameters selected are $\alpha =-0.005,k_0=1,{\chi }_0=\frac{1}{12},a_1=1,c=0.5,{\eta }_0=0,g\left(t\right)=-1$, and $\gamma =-0.005$. (a) $k=0.01$, density profile of an antikink soliton with initial speed $0.01$. (b) $k=-0.5$, effect of the linear frequency shift on the direction of the moving antikink soliton. The initial speed of the soliton is equal to $\left|k\right|$, and the direction of the soliton depends on the sign of $k$.

Figure 2

Propagation of the matter wave condensate of the solution given by Eq. (34). (a) Density profile of a growing periodic solution for $n=5,m=7$, and $\varepsilon =+1$. (b) Density profile of a moving kink soliton for $n=5,m=3$, and $\varepsilon =-1$. In both cases, the other parameters are the same as in Fig. 1, except ${\chi }_0=0.1,{\gamma }_0=0.005$.

Figure 3

(a) Sample of a bright soliton solution derived from solution (42) for $n=1,m=1$ with $k_0=1,{\chi }_0=-0.1,b_2=2$, and $b_6=-2$. (b) Same parameters as in panel (a) except $k_0=0.1$. The parameter $k_0$ affects both the width and the amplitude of the solution. The other parameters as the same as in Fig. 1.

Figure 4

Spatiotemporal evolution of density profiles of generalized Jacobi elliptic solutions of Eq. (1). (a) $c(\xi ,k_1,k_2)$, (b) $d_1(\xi ,k_1,k_2)$. Parameters are $\alpha =-0.005,k_1=0.6,k_2=0.2,{\chi }_0=-k_1^2{k}_2^2,\gamma =0.001,k=0.1$, and $g=1$.

Figure 5

Evolution of the density of a dark soliton of Eq. (1) obtained using the Bernoulli equation, with the same parameters as in Fig. 1 except $c=-0.5,a=1,b=-1$, and $\chi \left(t\right)=0$.

Figure 6

(a) Evolution of the density of a periodic solution of Eq. (1) obtained using the Riccati equation, with same parameters as in Fig. 2 except $M=-0.4,\chi \left(t\right)=0$. (b) Evolution of the density of a plane-wave solution of Eq. (1) derived using the Riccati equation, with the same parameters as in Fig. 2, except $M=-0.4,\chi \left(t\right)=0$.

Figure 7

Evolution of the density of a bright solution of Eq. (1) obtained using the ordinary auxiliary equation, with the same parameters as in Fig. 2 except $M=-0.4,\chi \left(t\right)=0$.

Figure 8

Spatiotemporal propagation of the stable antikink-like soliton of Fig. 1. Parameters are the same as in Fig. 1 except $\gamma =0$.

Figure 9

(a)–(c) Comparison between analytical (solid line) and numerical (dotted line) solutions at particular times of the antikink-like soliton of Fig. 1. (d) Spatiotemporal evolution of the stable antikink-like soliton of Fig. 1. Parameters are the same as in Fig. 1.

Figure 10

(a)–(c) Parallel between analytical (solid line) and numerical (dotted line) solutions at different times of the kinklike soliton of Fig. 2. (d) Spatiotemporal propagation of the stable antikink-like soliton. Parameters are the same as in Fig. 2.

Figure 11

(a)–(c) Parallel between analytical (solid line) and numerical (dotted line) solutions at different times of the bright soliton of Fig. 3. (d) Spatiotemporal propagation of the stable bright soliton. Parameters are the same as in Fig. 3 except $\gamma =0.005$.

Figure 12

Spatiotemporal evolution of a stable kink solution for $\chi =0,c=0.5$. The solution is that of Fig. 5 with the same parameters except that of $c$.