... 08 2025 01 09 2025 08 09 2025 09 09 2025 © 2025 Emerald Publishing Limited 2025 Emerald Publishing Limited Licensed re-use rights only Kairat-II-X-extended equation Kairat-II-X-type equation Multiple soliton solutions Lump wave solutions The canonical forms...

... of curves and the concept of equivalence. Design/methodology/approach The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability. Findings This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion...

... 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...

... are used to furnish multiple soliton solutions. Practical implications The authors also furnish a variety of numerous periodic solutions, kink solutions and singular solutions. Social implications The work formally furnishes algorithms for investigating several physical systems, including plasma...

... to study integrable equations with multiple soliton solutions and a sufficient number of conservation laws or symmetries (Adem and Khalique, 2013 ; Adeyemo and Khalique, 2022 ; Alejo, 2012 ; Ismael et al., 2020). For investigating the integrability of nonlinear partial differential equations...

... ; Khuri, 2005 ; Khuri, 2008). Besides, it is Painlevé integrable; in the sense, it shows multiple soliton solutions (MSSs) and infinite conservation laws (Wazwaz, 2010 , 2016 , 2017a , 2017b , 2011b). The main objective of this part is to determine the dispersion relation and PhS of solitons...

... the simplified Hirota's method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space. Findings The Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level...

.../methodology/approach The newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria. Findings The authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct...

.../methodology/approach This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space. Findings This paper confirms the complete integrability of the newly developed (3 + 1...

...Abdul-Majid Wazwaz; Gui-Qiong Xu Purpose The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a time-dependent equation. Design/methodology/approach The newly developed time...

...Abdul-Majid Wazwaz Purpose The purpose of this paper is concerned with developing a (2 + 1)-dimensional Benjamin–Ono equation. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for this equation. Design/methodology/approach The proposed model...

...Abdul-Majid Wazwaz Purpose The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters. Design...