Study on Date–Jimbo–Kashiwara–Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions

In this article, we construct the exact dynamical wave solutions to the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative by using an efficient and well-established approach, namely: the two-variable G’/G, 1/G-expansion method. The solutions of the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative play a vital role in many scientific occurrences. The regular dynamical wave solutions of the abovementioned equation imply three different fundamental functions, which are the trigonometric function, the hyperbolic function, and the rational function. These solutions are classified graphically into three categories, such as singular periodic solitary, kink soliton, and anti-kink soliton wave solutions. Furthermore, the effect of the fractional parameter on these solutions is discussed through 2D plots.

Study on the (3+1)-dimensional KP equation with lump solution and its collision phenomena

The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied in this paper by constructing the Hirota bilinear form. The lump solution of the equation is obtained by bilinear form, and the conditions for the existence of the solution are obtained. The picture description of lump solution is further given. On the other hand, we also give the collision phenomena of lump solution, periodic wave solution and a single-kink soliton solution when the (3+1)-dimensional KP equation reduces to [Formula: see text] and [Formula: see text] by means of the Hirota method. The collision phenomenon is shown in the 3D plot description, the dynamic characteristics of the collision are also analyzed.

Painlevé analysis of Fokas–Lenells equation with fractal temporal evolution

This paper presents new optical solitons of a fractal Fokas–Lenells equation that models the dynamics of optical fibers. The Painlevé technique is employed to identify kink soliton solutions. The constraint conditions guarantee the existence of these solitons. The outcomes of this research give new solutions that are not achieved using some already defined algorithms. The derived method is efficient and its applications are promising for other nonlinear problems. The 3D graphical illustrations of obtained results are depicted for various values of the fractal parameter.

Kink-Soliton, Singular-Kink-Soliton and Singular-Periodic Solutions for a New Two-Mode Version of the Burger-Huxley Model: Applications in Nerve Fibers and Liquid Crystals

New two-mode version of Burger-Huxley equation is derived using Korsunsky's operators. The new model arises in the applications of nerve fibers and liquid crystals and describes the interaction of two symmetric waves moving simultaneously in the same direction. Kink-soliton, singular-kink-soliton and singular-periodic solutions are obtained to this model by means of the simplified bilinear method, polynomial-function method and the Kudryashov-expansion method. A comprehensive graphical analysis is conducted to show the physical aspects of this new type of nonlinear equations. Finally, all obtained solution are verified by direct substitution in the model.

Lump-soliton, lump-multisoliton and lump-periodic solutions of a generalized hyperelastic rod equation

In this paper, we will obtain lump-soliton solution for (1[Formula: see text]+[Formula: see text]1)-dimensional generalized hyperelastic rod equation, also known as generalized KdV equation by aid of Hirota bilinear method (HBM). We also obtain lump-multisoliton (which is an interaction of lump with one kink or two kink soliton) and lump-periodic solutions (which is formed by an interaction between lump and periodic waves). The dynamics of these solution are examined graphically by selecting significant parameters.