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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 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.

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.

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 find solutions for the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation and to research the quality of B-type Kadomtsev-Petviashvili equation.

The authors apply the extended three-wave approach and the homoclinic test technique to solve the B-type Kadomtsev-Petviashvili equation.

The authors obtain breather type of cross-kink solutions, doubly breather type of kink solitary solutions and the breather type of kink wave solutions for B-type Kadomtsev-Petviashvili equation.

As nonlinear evolution equations are characterized by rich dynamical behaviors, the authors have just found some of them and others are still to be found.

These results may help us to investigate the local structure and the interaction of waves in high-dimensional models.

The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.

By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.

The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.

The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear equations, derived from connection between the Hirota’s transformation and Bell polynomials.

Based on the bilinear form of new reduced form of the (3 + 1)-dimensional generalized BKP equation, lump solutions with sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are discovered. Also, extended homoclinic approach is applied to considered equation for finding solitary wave solutions.

A class of the bright-dark lump waves are fabricated for studying different attributes of (3 + 1)-dimensional generalized BKP equation and some new exact solutions including kinky periodic solitary wave solutions and line breathers periodic are also obtained by Following the extended homoclinic approach.

The paper presents that the implemented methods have emerged as a promising and robust mathematical tool to manage (3 + 1)-dimensional generalized BKP equation by using the Hirota’s bilinear equation.

By considering important characteristics of lump and solitary wave solutions, one can understand the shapes, amplitudes and velocities of solitons after the collision with another soliton.

The analysis of these higher-dimensional nonlinear wave equations is not only of fundamental interest but also has important practical implications in many areas of mathematical physics and ocean engineering.

To the best of the authors’ knowledge, the acquired solutions given in various cases have not been reported for new reduced form of the (3 + 1)-dimensional generalized BKP equation in the literature. These obtained solutions are advantageous for researchers to know objective laws and grab the indispensable features of the development of the mathematical physics.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.

The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.

The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.

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

The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.

The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.

The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

This paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.

The authors formally use 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.

The Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level, but integrability characteristics is justified through the Lax sense.

Multiple-soliton solutions are explored using the Hirota's bilinear method. The authors also furnish a class of lump solutions using distinct values of the parameters via the positive quadratic function method.

The authors also retrieve a bunch of other solutions of distinct structures such as solitonic, periodic solutions and ratio of trigonometric functions solutions.

This work formally furnishes algorithms for extending integrable equations and for the determination of lump solutions.

To the best of the authors’ knowledge, this paper introduces an original work with newly developed Lax-integrable equation and shows new useful findings.