Search

Under investigation is a generalized variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources. Our main job is divided into four parts: (i) lump wave solution, (ii) interaction solutions between lump and solitary wave, (iii) breather wave solution and (iv) interaction solutions between lump and periodic wave. Furthermore, the interaction phenomenon of waves is shown in some 3D- and contour plots.

This study investigates the integrable Mikhailov–Novikov–Wang (MNW) equation in (1+1)-dimensional space–time, utilizing rigorous analytical methodologies including the Unified (UF) and Khater II (Khat.II) methods, alongside numerical solutions. Situated within the established mathematical framework of physics, the MNWequation holds significant relevance in elucidating various physical phenomena, encompassing solitons, nonlinear wave propagation, quantum mechanics, and field theories, contingent upon the specific context and parameters selected for analysis. The primary objective of this study is to conduct a comprehensive analysis of the MNWequation and propose effective resolution techniques. This is achieved through the amalgamation of diverse analytical and numerical methodologies, facilitating an in-depth exploration of the equation’s characteristics. Noteworthy findings indicate that the UF,Khat.IImethods, in conjunction with He’s variational iteration method, yield robust and highly accurate solutions. The significance of these results lies in their potential to address complex nonlinear equations, providing researchers and practitioners with versatile tools for analysis and interpretation. This research contributes a novel perspective by skillfully integrating these analytical methodologies, thereby advancing the field of mathematical physics and nonlinear differential equations. It is imperative to note that this study is purely theoretical in nature and does not involve specific subjects or participants. The approach adopted is grounded in meticulous numerical experimentation and analytical scrutiny, ensuring rigor and reliability in the findings presented.

AbstractIn this research, by applying the improved -expansion method, we have found the travelling and solitary wave solutions of the fractional-order biological population model, time fractional Burgers equation, the Drinfel’d–Sokolov–Wilson equation and the system of shallow water wave equations. The advantage of this method is providing a new and more general travelling wave solutions for many non-linear evolution equations, it supply three different kind of solutions in the form (the hyperbolic functions, the trigonometric functions and the rational functions). This method included the extended -expansion method when and the -expansion method when Ntakes only positive value and zero. All of these merits help us in survey of the physical meaning of each models mentioned above for investigating stability of these models. Rapprochement between our results and the previous renowned outcome presented.

This study gives more attention to soliton wave solutions of the nanoparticle hybrid system through two recent (generalized rational (GRat) and Kudryashov (GKud)) schemes. Because of the hybrid model’s nanoscale size, the nanoparticles in these new solitonic liquid wave structures exhibit unique electrical and optical characteristics crucial to several applications. The constructed results are numerically simulated in polar, contour, two, and three-dimensional graphs to demonstrate the dynamic behavior of an inviscid, incompressible, and non-rotating flow of fluid of constant depth h. Moreover, the obtained solutions’ accuracy is also checked by implementing two precise numerical (quintic-B-spline (QBS) and septic-B-spline (SBS)) schemes. The matching between computational and approximate solutions is explained through some distinct graphs in two-dimensional and distribution graphs. The novelty of our results is also elucidated by comparing our results with those constructed previously.