Exact solutions and linear stability analysis for two-dimensional Ablowitz–Ladik equation

Construction of new exact rational form non-travelling wave solutions to the (2 + 1)-dimensional generalized Broer–Kaup system

A NEW GENERAL ALGEBRAIC METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW TRAVELING SOLUTION FOR THE (1 +1)-DIMENSIONAL DISPERSIVE LONG WAVE EQUATION

In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F-expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W-shaped, M-shape, V-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

Abundant soliton solutions for the Kundu–Eckhaus equation via tan(ϕ(ξ))-expansion method

The investigations of integrability, exact solutions and dynamics of nonlinear partial differential equations (PDEs) are vital issues in nonlinear mathematical physics. In this paper, we derive and solve a new Lax integrable nonisospectral integral-differential system. To be specific, we first generalize an eigenvalue problem and its adjoint equation by equipping it with a new time-varying spectral parameter. Based on the generalized eigenvalue problem and the adjoint equation, we then derive a new Lax integrable nonisospectral integral-differential system. Furthermore, we obtain exact solutions and their reduced forms of the derived system by extending the famous non-linear Fourier analysis method–inverse scattering transform (IST). Finally, with graphical assistance we simulate a pair of reduced solutions, the dynamical evolutions of which show that the amplitudes of solutions vary with time.

The \((\frac{G'}{G})\)-expansion method is utilized for a reliable treatment of space–time fractional biological population model. The method has been applied in the sense of the Jumarie’s modified Riemann–Liouville derivative. Three classes of exact traveling wave solutions, hyperbolic, trigonometric and rational solutions of the associated equation are characterized with some free parameters. A generalized fractional complex transform is applied to convert the fractional equations to ordinary differential equations which subsequently resulted in number of exact solutions. It should be mentioned that the \((\frac{G'}{G})\)-expansion method is very effective and convenient for solving nonlinear partial differential equations of fractional order whose balancing number is a negative integer.

In this paper, we derive a family of rational localized wave solutions with any order in the Bogoyavlenskii–Kadomtsev–Petviashvili-I equation in terms of the Kadomtsev–Petviashvili reduction method. These rational localized waves are expressed by the Grammian determinants, and the entries of the determinant are presented by means of the Schur polynomials, which provide convenience in discussing the dynamics of localized wave solutions. According to the parity of the element indexes in the higher-order determinants, we mainly discuss two different types of higher-order rational localized wave solutions. Tuning the free parameters of the higher-order rational localized waves, lump-type localized waves of various polygon patterns, such as triangle, quadrangle, and pentagon, are obtained. It is shown that when one of these free parameters in the higher-order rational localized waves becomes sufficiently large, the localized wave solutions given by the odd indexes element are made up of the first-order fundamental rational localized wave. However, the higher-order rational localized wave solutions given by the even indexes element can exhibit not only the polygon wave patterns given by the first-order fundamental rational localized wave but also other novel hybrid wave patterns. These hybrid wave patterns consist of the first-order fundamental rational localized wave and other higher-order fundamental rational localized wave. These results will help us to better understand the wave patterns and control of nonlinear localized waves in fluid dynamics.

In this paper, we obtained the exact solutions of the [Formula: see text]-dimensional nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) and the seventh-order Sawada–Kotera–Ito (S-K Ito) equations with the help of the [Formula: see text]-expansion method specially the [Formula: see text]-expansion and [Formula: see text]-expansion methods. Soliton solutions found for the given equations are in the form of hyperbolic, trigonometric and rational solutions. All obtained solutions were checked. 3D and 2D graphs of some solutions were given and discussed.

This work investigates the perturbed nonlinear Schrödinger equation using the modified (G′G2)-expansion method. The obtained results are generalized and classified into classes of trigonometric, hyperbolic, and rational solutions. The kinematics of soliton and kink profiles are very helpful to understand the propagation of electromagnetic waves inside nonlinear optical fibers. The proposed modified method is unique, straightforward, concise, and effective in the sense that it gives more traveling wave solutions. The findings of this study can strengthen a system's nonlinear dynamic behavior and show how practical the methodology used to attempt to replicate has been. Wolfram Mathematica 11 is used for mathematical simplification and MATLAB is used for graphical simulation.

In this work, first it is shown that the Hirota–Ramani equation, which governs the nonlinear propagation of coupled Langmuir and dust-acoustec wave in a multicomponent dusty plasma, possesses kinds of wave profile such as singular periodic profile, periodic profile, singular soliton profile, M-shape rational profile, bright soliton profile, kink profile in the form of the trigonometric, hyperbolic, and rational solutions. With the aid of symbolic computation, we select the Hirota–Ramani equation with a source to investigate the validity and advantage of the improved [Formula: see text]-expansion method and construct some frames of the 3D profiles and the contour profiles to the equation along with the 2D profiles of some solutions to understand the traveling wave dynamics. Following the selection of appropriate values for the associated parameters, more generalized solutions are provided, along with certain patterns in the solutions that are examined. This improved method is effective, concise, reliable, and can be applied for further futuristic applications.

Some new exact solutions of the generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation (KPE) are explored in this study. Firstly, the resonant multiple soltion solutions (RMSs) are discussed via employing the linear superposition principle and weight algorithm. Then, by introducing pairs of the conjugate parameters to the RMSs, the complexiton solutions including the non-singular complexiton and singular complexiton solutions are extracted. In addition, the complex multiple kink soliton solutions are also probed by employing the bilinear approach. Finally, we investigate the rational wave solutions via the test function method and symbolic computation. By choosing the appropriate parameters, the graph descriptions of the derived solutions are presented to show the dynamical properties. The outcomes of this work are desirous to bring some new perspective to the study of the complexiton, complex solutions and rational wave solutions to the other PDEs.

Application of tan(ϕ/2)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity

On some new travelling wave structures to the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli model

Following our earlier work [Phys. Rev. Lett. 84, 3570 (2000)] we present an exact linear stability analysis of one-site monochromatic breathers in a piecewise smooth discrete nonlinear Schrödinger-type model. Destabilization of the breather occurs by virtue of a growth rate becoming positive as a stability border is crossed, while above a critical spatial decay rate (lambda(c)) the breather is found to be intrinsically unstable. The model admits of other exact breather solutions, including multisite monochromatic breathers for which the profile variable (phi(n)) crosses a relevant threshold at more than one site. In particular, we consider exact two-site breather solutions with phase difference delta between the two sites above threshold, and present stability results for delta=pi (antiphase breather; the in-phase breather with delta=0 happens to be intrinsically unstable). We obtain a band of extended eigenmodes, together with a pair of localized symmetric modes and another pair of localized antisymmetric ones. The frequencies of the localized modes vary as the parameters characterizing the breather are made to vary, and destabilization occurs through the Krein collision of a quartet of growth rates, leading to temporal growth of a pair of symmetric eigenmodes of nonzero frequency. We clarify the limit N--> infinity (N is the gap length between the sites above threshold) when the two-site breather reduces to a pair of decoupled one-site breathers. The model offers the possibility of obtaining spatially random vortex-type breathers.

The Hirota–Maccari (HM) system is a fundamental model in wave propagation that has been widely utilized to investigate complex nonlinear phenomena in nonlinear optics, optical communications, and mathematical physics. The HM system sheds light on critical insights into soliton dynamics, wave interactions, and other nonlinear effects. Therefore, the main goal of this work is to compile new soliton structures to the HM system in hyperbolic, trigonometric, exponential, and rational forms, both single and combined version, by using three robust analytical approaches: the new extended rational sinh‐Gordon equation expansion method (ShGEEM), the ( G ′ / G , 1/ G )‐expansion method, and the new extended hyperbolic function method (EHFM). The solutions thus obtained are also tested for their validity and accuracy using Mathematica. In order to exhibit the physical properties of the soliton pulses, we depict 2D, 3D, and contour graphs by using suitable values of the parameters. These methods play a major role compared with other methods in the literature because new and more general solutions are obtained with additional free parameters. Remarkably, all the known solutions are special cases of our generalized solutions. An important feature of the proposed methods is their simplicity, robustness, and computational efficiency that can be widely extended to nonlinear partial differential equations in all scientific disciplines. The efficiency of this framework thus indicates at its application in addressing complicated problems in the future.