Purpose

This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.

Design/methodology/approach

The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.

Findings

This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.

Research limitations/implications

The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.

Practical implications

This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.

Social implications

The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.

Originality/value

This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.

中文翻译：

---

扩展 (3 + 1) 维 Kairat-II 和 Kairat-X 方程：Painlevé 可积性、多重孤子解、集总解和呼吸波解

目的

本研究旨在研究两个新开发的 (3 + 1) 维 Kairat-II 和 Kairat-X 方程，它们说明了曲线微分几何和等价方面的关系。

设计/方法论/途径

Painlevé 分析证实了 Kairat-II 和 Kairat-X 方程的完全可积性。

发现

本研究探索了两个研究模型的多个孤子解决方案。此外，作者表明只有 Kairat-X 给出了块解和呼吸波解。

研究局限性/影响

Hirota 的双线性算法用于提供各种具有有用物理结构的孤子解。

实际影响

这项研究还为 Kairat-II 方程提供了多种周期解、扭结解和奇异解。此外，从 Kairat-X 模型中还获得了块解和呼吸波解。

社会影响

这项工作正式提供了用于研究新建系统的算法，这些系统检查等离子体物理、光通信、海洋以及曲线的微分几何等。

原创性/价值

本文提出了一项原创工作，其中提出了两个新开发的 Painlev\'{e} 可积模型，并具有深刻的发现。