Purpose

This study aims to find the symmetries and conservation laws of a new Painlevé integrable Broer-Kaup (BK) system with variable coefficients. This system is an extension of dispersive long wave equations. As the system is generalized and new, it is essential to explore some of its possible aspects such as conservation laws, symmetries, Painleve integrability, etc.

Design/methodology/approach

This paper opted for an exploratory study of a new Painleve integrable BK system with variable coefficients. Some analytic solutions are obtained by Lie classical method. Then the conservation laws are derived by multiplier method.

Findings

This paper presents a complete set of point symmetries without any restrictions on choices of coefficients, which subsequently yield analytic solutions of the series and solitary waves. Next, the authors derive every admitted non-trivial conservation law that emerges from multipliers.

Research limitations/implications

The authors have found that the considered system is likely to be integrable. So some other aspects such as Lax pair integrability, solitonic behavior and Backlund transformation can be analyzed to check the complete integrability further.

Practical implications

The authors develop a time-dependent Painleve integrable long water wave system. The model represents more specific data than the constant system. The authors presented analytic solutions and conservation laws.

Originality/value

The new time-dependent Painleve integrable long water wave system features some interesting results on symmetries and conservation laws.

中文翻译：

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新的 Painlevé 可积 Broer-Kaup 系统：对称性分析、解析解和守恒定律

目的

本研究旨在寻找具有可变系数的新 Painlevé 可积 Broer-Kaup (BK) 系统的对称性和守恒定律。该系统是色散长波方程的扩展。由于该系统是广义的和新的，因此有必要探索其一些可能的方面，例如守恒定律、对称性、Painleve 可积性等。

设计/方法/方法

本文选择对具有可变系数的新 Painleve 可积 BK 系统进行探索性研究。一些解析解是用李经典方法得到的。然后用乘法法推导出守恒定律。

发现

本文提出了一套完整的点对称性，对系数的选择没有任何限制，随后产生级数和孤立波的解析解。接下来，作者推导出从乘数中出现的每一个公认的非平凡守恒定律。

研究限制/影响

作者发现所考虑的系统很可能是可集成的。因此，可以通过分析 Lax 对可积性、孤子行为和 Backlund 变换等其他方面来进一步检查完全可积性。

实际影响

作者开发了一种时间相关的 Painleve 可积长水波系统。该模型表示比常数系统更具体的数据。作者提出了解析解和守恒定律。

原创性/价值

新的时间相关 Painleve 可积长水波系统在对称性和守恒定律方面具有一些有趣的结果。