Cheng Xueping, Jin Guiming, Wang Jianan
School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China.
School of Information Engineering, Zhejiang Ocean University, Zhoushan 316022, China.
Chaos. 2024 Oct 1;34(10). doi: 10.1063/5.0223870.
Through the modified deformation algorithm related to conservation laws, the (1+1)-dimensional AKNS(-1) equations are extended to a (4+1)-dimensional AKNS(-1) system. When one, two, or three of the independent variables are removed, the (4+1)-dimensional AKNS(-1) system degenerates to some novel (3+1)-dimensional, (2+1)-dimensional, and (1+1)-dimensional AKNS(-1) systems, respectively. Under a simple dependent transformation, the (1+1)-dimensional AKNS(-1) equations turn into the classical sine-Gordon equation. Then using the same deformation procedure, the (1+1)-dimensional sine-Gordon equation is generalized to a (3+1)-dimensional version. By introducing the deformation operators to the Lax pairs of the original (1+1)-dimensional models, the Lax integrability of both the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation is proven. Finally, the traveling wave solutions of the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation are implicitly given and expressed by tanh function and incomplete elliptic integral, respectively. These results may enhance our understanding of the complex physical phenomena described by the nonlinear system discussed in this paper.
通过与守恒律相关的修正变形算法，将(1 + 1)维AKNS(-1)方程扩展为(4 + 1)维AKNS(-1)系统。当去掉一个、两个或三个自变量时，(4 + 1)维AKNS(-1)系统分别退化为一些新的(3 + 1)维、(2 + 1)维及(1 + 1)维AKNS(-1)系统。在一个简单的因变量变换下，(1 + 1)维AKNS(-1)方程变为经典的正弦 - 戈登方程。然后使用相同的变形过程，将(1 + 1)维正弦 - 戈登方程推广为(3 + 1)维形式。通过将变形算子引入到原(1 + 1)维模型的拉克斯对中，证明了(4 + 1)维AKNS(-1)系统和(3 + 1)维正弦 - 戈登方程的拉克斯可积性。最后，分别隐式给出了(4 + 1)维AKNS(-1)系统和(3 + 1)维正弦 - 戈登方程的行波解，它们分别由双曲正切函数和不完全椭圆积分表示。这些结果可能会增进我们对本文所讨论的非线性系统所描述的复杂物理现象的理解。
