Introduction

The purpose of this paper is to study, a (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system from Lie group analysis point of view. This system is studied in the literature for the first time. The authors found this system to be interesting since it is non-decouplable and possesses higher generalised symmetries.

Objectives

We look for the closed-form solutions and conservation laws of the system.

Methods

Optimal system of one-dimensional subalgebras for the system was obtained and then used to perform symmetry reductions and construct group invariant solutions. Power series solutions for the system were also obtained. The system has no variational principle and as such, we employed the multiplier method and used a homotopy integral formula to derive the conserved quantities.

Results

Group invariant solutions and power series solutions were constructed and three conserved vectors for the system were derived.

Conclusion

The paper studies the (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system for the first time and constructs its exact solutions and conservation laws.

中文翻译：

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（1 +1）维广义耦合mKdV型系统的守恒量，最优系统和显式解

介绍

本文的目的是从李群分析的角度研究（1 +1）维广义耦合改进Korteweg-de Vries型系统。该系统是文献首次研究。作者发现该系统很有趣，因为它是不可分解的，并且具有较高的广义对称性。

目标

我们寻找系统的封闭式解决方案和守恒定律。

方法

获得该系统的一维子代数的最优系统，然后将其用于执行对称约简并构造组不变解。还获得了该系统的电源系列解决方案。该系统没有变分原理，因此，我们采用了乘法器方法，并使用了同伦积分公式来导出守恒量。

结果

构造了组不变解和幂级数解，并推导了该系统的三个守恒向量。

结论

本文首次研究了（1 +1）维广义耦合改进的Korteweg-de Vries型系统，并构造了其精确解和守恒律。