In this paper, we investigate a five-component Gross–Pitaevskii equation, which is demonstrated to describe the dynamics of an

Bose–Einstein condensates (BECs) of the alkali-metal-atom gases have attracted certain attention in both experimental [^{87}Rb and ^{23}Na, respectively, and the corresponding ground-state structure and dynamical properties of those two cases are distinct [

The dynamics of nonlinear phenomena can be analyzed by means of the corresponding nonlinear evolution equations. There has been considerable work carried out on the control problem of nonlinear systems, such as those in chaotic and stochastic systems [

In this paper, we will consider a five-component Gross–Pitaevskii (GP) equation for the dynamics of an

Equation (

In this paper, we will concentrate on the soliton types and collisions in the same and different states in an

In order to understand the dynamics of equation (

To begin with, equation (

In order to obtain one-soliton solutions for equation (

It is noted that equation (

When

Here,

Intensity plots of the one soliton in the ferromagnetic state for the components

Intensity plots of the one soliton in the cyclic state for the components

For the case

Intensity plots of the one soliton in the polar state for the components

In this paper, we will concentrate on the two-soliton solutions and analyze the collisions between two solitons. Employing the following expansions,

We obtain the two-soliton solutions for equation (

Following the classification on the one-soliton solutions, we naturally consider the collisions between the same and different types of solitons, which are determined by

At first, we will consider the collision between two solitons in the ferromagnetic state. Under the condition

Before the collision

Soliton

Soliton

After the collision

Soliton

Soliton

with

The superscripts

Collision between two solitons in the ferromagnetic state for the components

We will analyze the collisions between soliton in the ferromagnetic state and one- and two-peak solitons in the polar state. Here, we choose

Before the collision.

Soliton

where

Soliton

where

After the collision.

Soliton

where

Soliton

where

Collision between one-peak solitons in the ferromagnetic and polar states for the components

Collision between one-peak soliton in the ferromagnetic state and two-peak soliton in the polar state for the components

Collision between one-peak solitons in the polar state for the components

Collision between two-peak solitons in the polar state for the components

Collision between one- and two-peak solitons in the polar state for the components

In this section, we will consider the collisions between the solitons in the polar state (

Before the collision.

Soliton

where

Soliton

where

After the collision.

Soliton

where

Soliton

where

In this paper, we have investigated the solitons and their collisions for the five-component GP equations, i.e., equation (

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China under Grant nos. 11501526 and 11871232, Key Projects of Science and Technology Research of the Henan Education Department, (No. 17A110035), and Doctoral Research Fund of Zhengzhou University of Light Industry.