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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 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.

The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion KdV (KdV8) and many other related equations.

The newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria.

The authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct resonance branches, whereas the phase shift retains the KdV type of shifts for any member.

This paper presents an efficient algorithm for handling a hierarchy of integrable equations of diverse orders.

Multisoliton solutions are derived for each member of the hierarchy, and then generalized for any higher-order model.

This work presents useful algorithms for finding and studying integrable equations of a hierarchy of nonlinear equations. The developed models exhibit complete integrability, by investigating the compatibility conditions for each model.

This paper presents an original work with a variety of useful 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 study aims to explore novel solitary wave solutions of a new (3 + 1)-dimensional nonlocal Boussinesq equation that illustrates nonlinear water dynamics.

The authors use the Painlevé analysis to study its complete integrability in the Painlevé sense.

The Painlevé analysis demonstrates the compatibility condition for the model integrability with the addition of new extra terms.

The phase shifts, phase variables and Hirota’s bilinear algorithm are used to furnish multiple soliton solutions.

The authors also furnish a variety of numerous periodic solutions, kink solutions and singular solutions.

The work formally furnishes algorithms for investigating several physical systems, including plasma physics, optical communications and oceans and seas, among others.

This paper presents an original work using a newly developed Painlevé integrable model, as well as novel and insightful findings.