Shallow ocean soliton and localized waves in extended (2 + 1)-dimensional nonlinear evolution equations

In recent years, the derivation and solution of integrable nonlinear evolution equations (NLEEs) in one, two, or more dimensions have been the apex in the field of applied mathematics. As is well-known, NLEEs are capable of describing a wide variety of nonlinear physical phenomena in the fields of applied science and engineering. In this article, the integrability, shallow ocean wave soliton, Peregrine soliton, multi-soliton, lumps, breathers (such as Akhmediev, general, and Kuznetsov-Ma breather) solutions, and wave interactions for four extended (2+1)-dimensional NLEEs are investigated. The Painlevé test is used to analyze the integrability of these models, and then their bilinear forms are constructed using the simplified bilinear approach. Furthermore, various forms of the aforementioned solutions are constructed using the Hirota bilinear method and several ansatz functions. The shallow ocean waves for the X-, Y-, and H-types, as well as other complex interactions are replicated by the two, three, and four soliton solutions. These structures, along with a few other intriguing structures are represented by graphs in three-dimensional and contour plots. Some two dimensional plots are also presented. We hope these findings might well be useful in explaining the complex dynamic behaviors of shallow water waves. •Four extended (2+1)-dimensional nonlinear evolution equations are studied.•The Painlevé analysis is used to check the integrability of these models.•Shallow ocean wave soliton and localized waves are investigated.•Some plots are presented to clearly illustrate the dynamic features of these solutions.•The shallow ocean waves for the X-, Y-, and H-types, as well as other complex interactions are simulated.