Optical Solutions of the Date–Jimbo–Kashiwara–Miwa Equation via the Extended Direct Algebraic Method

In this study, the solutions of 2 + 1 -dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups

In this article, new exact solutions of 2 + 1 -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2 + 1 -dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.

Analytical and numerical treatment to the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation

In this work, we study the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation. We employ the extended tanh function method and the simple equation method to achieve analytical soliton solutions. Moreover, numerical treatment for this equation is introduced by the finite difference method. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs.

Dynamical structures of solitons and some new types of exact solutions for the (2+1)-dimensional DJKM equation using Lie symmetry analysis

This paper is devoted to obtaining some new types of exact solutions of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation utilizing the Lie symmetry method. All the Lie symmetries, infinitesimal generators, and possible geometric vector fields have been obtained by using the invariance condition of the group-theoretic method. Meanwhile, the Lie symmetry reductions and explicit exact solutions are obtained by a one-dimensional (1D) optimal system. All the obtained exact solutions are absolutely new and completely different from the earlier established results in the literature. Moreover, the dynamical behavior of obtained solitons like doubly solitons, dark solitons, kink wave, curved shaped multi-solitons, parabolic waves, solitary waves, and annihilation of elastic multi-soliton profiles is depicted graphically via interesting 3D-shapes. That will be widely used to provide many more attractive complex physical phenomena in the fields of plasma physics, statistical physics, fiber optics, fluid dynamics, condensed matter physics, and so on. Finally, we have verified all the achieved soliton solutions through symbolic computations with Mathematica.

Grammian solutions for a (2+1)-dimensional integrable coupled modified Date–Jimbo–Kashiwara–Miwa equation

In this paper, Grammian solutions to a (2[Formula: see text]+[Formula: see text]1)-dimensional modified Date–Jimbo–Kashiwara–Miwa (mDJKM) equation are presented by using Hirota bilinear method and perturbation expansion. Starting from the Grammian solutions, an integrable coupled mDJKM equation is then obtained and the corresponding Grammian solutions are first derived by utilizing the source generation procedure. Besides, we also construct and solve a coupled DJKM equation via source generation procedure. It is interesting that the coupled mDJKM system constitute a bilinear Bäcklund transformation for the coupled DJKM system. This means that the commutativity of source generation procedure and Bäcklund transformation is valid for the (2[Formula: see text]+[Formula: see text]1)-dimensional DJKM equation.