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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 study aims to propose an extended (3 + 1)-dimensional integrable Kadomtsev–Petviashvili equation characterized by adding three new linear terms.

This study formally uses Painlevé test to confirm the integrability of the new system.

The Painlevé analysis shows that the compatibility condition for integrability does not die away by adding three new linear terms with distinct coefficients.

This study uses the Hirota's bilinear method to explore multiple soliton solutions where phase shifts and phase variable are explored.

This study also furnishes a class of lump solutions (LSs), which are rationally localized in all directions in space, using distinct values of the parameters via using the positive quadratic function method.

This study also shows the power of the simplified Hirota’s method in handling integrable equations.

This paper introduces an original work with newly developed Painlevé integrable model and shows new useful findings.

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 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 construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena.

The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions.

As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections.

The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure.

By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections.

To the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.

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.

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 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.

The proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist.

The examined two-wave model features interesting results in propagation of waves and fluid flow.

The paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations.

A two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived.

The work shows the distinct features of the standard equation and the newly developed equation.

The work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.