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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.

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.

The purpose of this paper is to reveal the dynamical behavior of higher dimensional nonlinear wave by searching for the multi-wave solutions to the (3+1)-D Jimbo-Miwa equation.

The authors apply bilinear form and extended homoclinic test approach to the (3+1)-D Jimbo-Miwa equation.

In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breathertype of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breather-type of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving higher dimensional nonlinear evolution equations in mathematical physics.

The research manifests that the structures of the solution to higher dimensional nonlinear equations are diversified and complicated.

The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

The purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.

The newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.

The developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.

The paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.

The work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.

This study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.

The paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.

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 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.

The proposed model has been handled by using the Hirota’s method. Other techniques were used to obtain traveling wave solutions.

The examined extension of the Benjamin–Ono model features interesting results in propagation of waves and fluid flow.

The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple soliton solutions.

This work is entirely new and provides new findings, where although the new model gives multiple soliton solutions, it is nonintegrable.

The work develops two complete sets of multiple soliton solutions, the first set is real solitons, whereas the second set is complex solitons.

This study aims to develop a new (3 + 1)-dimensional Painlevé-integrable extended Vakhnenko–Parkes equation. The author formally derives multiple soliton solutions for this developed model.

The study used the simplified Hirota’s method for deriving multiple soliton solutions.

The study finds that the developed (3 + 1)-dimensional Vakhnenko–Parkes model exhibits complete integrability in analogy with the standard Vakhnenko–Parkes equation.

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

The work reports extension of the (1 + 1)-dimensional standard equation to a (3 + 1)-dimensional model.

The work presents useful algorithms for constructing new integrable equations and for handling these equations.

The paper presents an original work with newly developed integrable equation and shows useful findings.

The buckling behaviour of engineering materials has been researched extensively since the 1890s and more recently, thin shell theory has generalised the analysis to include complicated boundary conditions. However, the approximations and assumptions which form the basis of engineering models make them inappropriate for textile materials. Very small stresses on textile materials cause extremely large strains so that the deformations are highly nonlinear. In this paper, we develop a nonlinear mathematical method. In the final section, the nonlinear differential equations used are generalised into a nonlinear evolution equation which is completely integrable and thus solved analytically obtaining dynamical solution for three‐dimensional fabric drape. These analytical solutions are applicable under all conditions and are not subject to computational difficulties associated with finding numerical solutions for highly nonlinear problems. The use of this analytical approach to fabric mechanics and dynamics provides us with a very powerful tool to formulate and solve many long‐standing problems in fabric and clothing technology.

The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.

The authors investigate Painlevé integrability of the constructed linear structure.

The Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.

The research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.

The authors also furnish a variety of solitonic solutions and complex solutions as well.

The work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.

The paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

The newly developed Sakovich equation has been handled by using the Lie symmetries via using the Lie group method.

The developed extended Sakovich model exhibit symmetries and invariant solutions.

The present study is to address the two main motivations: the study of symmetry analysis and the study of soliton solutions of the extended Sakovich equation.

The work introduces symmetry analysis to the Painlevé-integrable extended Sakovich equation.

The work presents useful symmetry algorithms for handling new integrable equations.

The paper presents an original work with symmetry analysis and shows useful findings.