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The purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov’s methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation.

The article’s methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions.

As for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation.

The originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field.

This paper aims to explore new variable separation solutions for a new generalized (3 + 1)-dimensional breaking soliton equation, construct novel nonlinear excitations and discuss their dynamical behaviors that may exist in many realms such as fluid dynamics, optics and telecommunication.

By means of the multilinear variable separation approach, variable separation solutions for the new generalized (3 + 1)-dimensional breaking soliton equation are derived with arbitrary low dimensional functions with respect to {y, z, t}. The asymptotic analysis is presented to represent generally the evolutions of rogue waves.

Fixing several types of explicit expressions of the arbitrary function in the potential field U, various novel nonlinear wave excitations are fabricated, such as hybrid waves of kinks and line solitons with different structures and other interesting characteristics, as well as interacting waves between rogue waves, kinks, line solitons with translation and rotation.

The paper presents that a variable separation solution with an arbitrary function of three independent variables has great potential to describe localized waves.

The roles of parameters in the chosen functions are ascertained in this study, according to which, one can understand the amplitude, shape, background and other characteristics of the localized waves.

The work provides novel localized waves that might be used to explain some nonlinear phenomena in fluids, plasma, optics and so on.

The study proposes a new generalized (3 + 1)-dimensional breaking soliton equation and derives its nonlinear variable separation solutions. It is demonstrated that a variable separation solution with an arbitrary function of three independent variables provides a treasure-house of nonlinear waves.

The purpose of this study is to produce families of exact soliton solutions (2+1)-dimensional Korteweg-de Vries (KdV) equation, that describes shallow water waves, using an ansätze approach.

This article aims to introduce a recently developed ansätze for creating soliton and travelling wave solutions to nonlinear nonintegrable partial differential equations, especially those with physical significance.

A recently developed ansätze solution was used to successfully construct soliton solutions to the (2 + 1)-dimensional KdV equation. This straightforward method is an alternative to the Painleve test analysis, yielding similar results. The strategy demonstrated the existence of a single soliton solution, also known as a localized wave or bright soliton, as well as singular solutions or kink solitons.

The ansätze solution used to construct soliton solutions to the (2 + 1)-dimensional KdV equation is novel. New soliton solutions were also obtained.

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets.

The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method.

The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures.

This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations.

The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.

The results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.

These results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.

The results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.

These results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

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.

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

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.

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

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.

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.

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

This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth.

Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties.

N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied.

Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions.

The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution.

The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution.

These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

This paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.

Hirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.

The fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.

This paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.