Purpose

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.

Design/methodology/approach

The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.

Findings

The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.

Research limitations/implications

The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.

Social implications

The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.

Originality/value

The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

中文翻译：

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新的可积 (2+1) 和 (3+1) 维浅水波动方程：多重孤子解和块解

目的

本研究旨在开发两个可积分的浅水波浪方程，它们具有更高的维度，分别具有常数和时间相关系数。作者推导出了多个孤子解和一类在空间各个方向都合理定域的集总解。

设计/方法/方法

作者使用简化的 Hirota 方法和块技术来确定多个孤子解和块解。作者表明，开发的 (2+1) 维和 (3+1) 维模型在 Painlené 意义上是完全可积的。

发现

该论文报告了属于浅水波介质的新的 Painlevé 可积扩展方程。

研究限制/影响

作者通过使用 Painlevé 分析解决了该模型的可积性特征。作者使用简化的 Hirota 方法报告了该方程的多个孤子解。

实际影响

得到的块解包括自由参数；一些参数与平移不变性有关，其他参数满足非零行列式条件。

社会影响

这项工作提出了用于构建新的可积方程和确定块解的有用算法。

原创性/价值

该论文展示了新开发的可积方程的原创工作，并展示了孤立波和块解的有用发现。