Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves

A large member of lump chain solutions of the(2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili(BKP)equation are constructed by means of theτ-function in the form of Grammian.The lump chains are formed by periodic arrangement of individual lumps and travel with distinct group and velocities.An analytical method related dominant regions of polygon is developed to analyze the interaction dynamics of the multiple lump chains.The degenerate structures of parallel,superimposed,and molecular lump chains are presented.The interaction solutions between lump chains and kink-solitons are investigated,where the kink-solitons lie on the boundaries of dominant region determined by the constant term in theτ-function.Furthermore,the hybrid solutions consisting of lump chains and individual lumps controlled by the parameter with high rank and depth are investigated.The analytical method presented in this paper can be further extended to other integrable systems to explore complex wave structures.

New Painlevé Integrable(3+1)-Dimensional Combined pKP–BKP Equation:Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable(3+1)-dimensional combined potential Kadomtsev–Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation(pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new(3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new(3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined p KP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

A New Integrable (2+1)-Dimensional Generalized Breaking Soliton Equation:N-Soliton Solutions and Traveling Wave Solutions

In this work,we study a new(2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters.We derive multiple soliton solutions,traveling wave solutions,and periodic solutions as well.We use the simplified Hirotas method and a variety of ansatze to achieve our goal.

The simplified Hirota’s method for studying three extended higher-order KdV-type equations

In this work we study three extended higher-order KdV-type equations.The Lax-type equation,the Sawada-Kotera-type equation and the CDG-type equation are derived from the extended KdV equation.We use the simplified Hirota’s direct method to derive multiple soliton solutions for each equation.We show that each model gives multiple soliton solutions,where the structures of the obtained solutions differ from the solutions of the canonical form of these equations.

Hypergeometric Series Solution to a Class of Second-Order Boundary Value Problems via Laplace Transform with Applications to Nanofluids

Very recently, it was observed that the temperature of nanofluids is finally governed by second-order ordinary differential equations with variable coefficients of exponential orders. Such coefficients were then transformed to polynomials type by using new independent variables. In this paper, a class of second-order ordinary differential equations with variable coefficients of polynomials type has been solved analytically. The analytical solution is expressed in terms of a hypergeometric function with generalized parameters. Moreover, applications of the present results have been applied on some selected nanofluids problems in the literature. The exact solutions in the literature were derived as special cases of our generalized analytical solution.

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work,we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics,scientific fields,and ocean engineering.This equation will be reduced to the Korteweg-de Vries equation via using the perturbation analysis.We derive the corresponding vectors,symmetry reduction and explicit solutions for this equation.We readily obtain Bäcklund transformation associated with truncated Painlevéexpansion.We also examine the related conservation laws of this equation via using the multiplier method.Moreover,we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.

A new integrable nonlocal modified KdV equation:Abundant solutions with distinct physical structures

In this work we study a new integrable nonlocal modified Korteweg-de Vries equation(mKdV)which arises from a reduction of the AKNS scattering problem.We use a variety of distinct techniques to determine abundant solutions with distinct physical structures.We show that this nonlocal equation possesses a family of traveling solitary wave solutions that include solitons,kinks,periodic and singular solutions.