Narrow Results

In this paper, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop–loop and loop–antiloop solutions have been explored and illustrated.

In this paper, the (2+1)-dimensional fifth-order KdV equation is analytically investigated. By using Hirota’s bilinear method combined with perturbation expansion, the high-order breather solutions of the fifth-order KdV equation are generated. Then, the high-order lump solutions are also derived from the soliton solutions by a long-wave limit method and some suitable parameter constraints. Furthermore, we extend this method to obtain hybrid solutions by taking long-wave limit for partial soliton solutions. Finally, the dynamic behavior of these solutions is presented in the figures.

In this work, we have considered the beta-fractional derivative form of the Vakhnenko–Parkes equation (VPE) to study its novel analytical solutions. To achieve the required new distinct traveling wave solutions, the new sub-equation method is used for the space–time fractional VPE. The dynamics of these solutions are analyzed under different parametric and fractional conditions by the graphical view. The produced wave patterns include combined bright–dark, w-shape, u-shape, v-shape, bright, propagation of two solitons, combined dark–bright, bell shape, a dark, periodic wave, breather, v-shape with two bright spots and combined anti-peakon–bright solutions. In recognizing the physical significance of the obtained wave solutions at each fraction of an interval, these solutions are highly commendable. In addition, to show the stability of the considered model, the stability analysis of the governing model is discussed by the linear stability analysis method which states the behavior of the model in the corresponding medium. These solutions are useful to study the proliferation of high-frequency waves in the relaxing medium.

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

In this paper, we present a coupled sine-Gordon (CSG) system, which describes the dynamics of two deoxyribonucleic acid (DNA) twisting strands in open configuration. We investigate the associated linear eigenvalue problem of the CSG system and obtain generalized formulae for multiple soliton solutions by employing the Darboux transformation (DT). Explicit expressions of one- and two-kink and line soliton solutions, single breather and first-order degenerate solutions are obtained by using particular solutions to the associated linear eigenvalue problem. We also present the dynamics of the different solutions.

The periodic solutions for the $(2+1)$-dimensional complex modified Korteweg–de Vries (cmKdV) equation are obtained by using the Darboux transformation (DT). Starting with a periodic seed, we get two types of periodic solutions that are respectively, induced by different properties of the parameter $l_2$, namely, the periodic line wave solutions for $l_2=\alpha \in ℝ$ and the breather solutions for $l_2=\alpha {\lambda }_1$. Especially, for each type, two different kinds of solutions can be obtained via choosing different constrains, namely, $c-2\gamma \ge 0$ or $c-2\gamma <0$. The detailed dynamical characteristics of these solutions are also analyzed, including period, velocity, extreme values, amputations and the rules of temporal evolutions.

In this work, by introducing a kind of parameters constraint to ensure that the well known formula of $n$-soliton solutions works well not only for integrable systems, but also for non-integrable systems. Then for the $(4+1)$-dimensional Fokas equation, based on this constraint and the modified $n$-soliton solutions, we further construct its different types of higher-order wave solutions, especially new types of interaction solutions.

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple.

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and $N$-soliton ($N$ is a nonnegative integer) are given. Some examples, including lump-$N$-exponential type, lump-$N$-logarithmic type, higher-order lump-type $N$-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-$N$-solitons, etc., are analyzed and simulated.

In this paper, based on the Hirota bilinear method, the high-order breathers and interaction solutions between solitons and breathers of the (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation are investigated. The lump and semi-rational solutions are obtained by applying the long wave limit of the $N$-soliton solution. Two types of semi-rational solutions are derived by choosing specific parameters, which are the mixture of the lump solution and solitons, and the mixture of the lump solution and breathers. Furthermore, the time evolution diagram illustrate the dynamic behavior of these solutions.

Starting from the bilinear form of the positive Gardner–Kadomtsev–Petviashvili equation, breather-type and N-breather solutions are obtained with the help of the extended homoclinic test approach. Breather-type, one-breather, two-breather and three-breather solutions are illustrated through 3D plots, density plots and contour plots and some appropriate parameter values have been chosen to better demonstrate their time evolution.

In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved $cosh(\mathrm{\Gamma }(\varpi ))-\mathrm{\text{sech}}(\mathrm{\Gamma }(\varpi ))$ function scheme and the combined $cos(\mathrm{\Gamma }(\varpi ))-sec(\mathrm{\Gamma }(\varpi ))$ function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.