Types Of Exact Solutions Research Articles

Dynamics of fractional solitonic profiles to multicomponent Gross–Pitaevskii system

The fractional multicomponent Gross-Pitaevskii system arising in the Bose-Einestein condensate is under consideration. The Gross-Pitaevskii equation plays a significant role in Bose-Einstein condensation and engineering, where it characterizes the dynamics of the condensate wave function. Superfluidity and superconductivity are two characteristics of the low-temperature phenomenon that are linked to the Bose–Einstein condensate, which is generated by a diluted atomic gas. The investigation of multi-component equations has garnered considerable attention because of their capacity to clarify intricate physical phenomena and reveal the dynamic configurations of localized wave solutions. A variety of solutions have been secured in various forms, including bright, dark, singular, and combo solitons, in addition to solutions of hyperbolic, periodic, and exponential functions. For the purpose of ensuring the solutions, recently developed integration tools called the modified Sardar subequation method and enhanced modified extended tanh-expansion method have been implemented. In nonlinear dispersive media, solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The proposed approaches are certainly the most direct, efficient, and valuable method for dealing with multiple nonlinear models that arise in applied physics and mathematics, with the purpose of generating various types of exact solutions. In addition, 3D, 2D, contour, and density plots have been utilized to visually represent the obtained results, facilitating a greater understanding of the physical effects of the derived solutions. The solutions attained are of great importance with regard to their applicability across a wide range of quantum systems.

The Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional variable coefficients integrable coupled Burgers equations

The variable coefficients integrable coupled Burgers equations (vcICBs) are researched by the Lie symmetry analysis method. The infinitesimal generators, commutative relations and adjoint relations are given. Optimal system of vcICBs are studied by Olver’s method. Based on the optimal system, the (2+1)-dimensional vcICB equations are reduced to (1+1)-dimensional equations. Some new exact solutions of vcICB equations are derived. Some kink solutions, breathe wave solutions and soliton solutions are got by utilising -expansion method, Riccati equation method, Tanh-expansion method and Painlevé analysis method. By altering different coefficient functions, many different types of exact solutions can be derived. The effect of the coefficient functions on the solution is analysed. By studying the evolution of the solution with time t, the dynamic behaviors of the solutions are analysed.

A method for solving the generalized Camassa-Choi problem with the Mittag-Leffler function and temporal local derivative

This paper focuses on a reduction technique to discover exact solutions for the generalized Camassa-Choi equation with temporal local M-derivative. The paper presents various types of exact solutions along with their corresponding first integrals. Furthermore, the interactions between the orders of α and β in the M-derivative are taken into account and depicted graphically for the derived solutions. Remarkably, the paper demonstrates that in certain situations, exact solutions can be obtained for any value of n, which holds significant mathematical intrigue. The authors note that Nucci's reduction technique has not previously been employed for differential equations with M-derivative, to the best of their knowledge.

Symmetry reduction and exact solutions of the (3+1)-dimensional nKdV-nCBS equation

In this paper, the exact solutions of the (3+1)-dimensional nKdV-nCBS equation are investigated by various approaches. Firstly, according to Lie group theory, the (3+1)-dimensional nKdV-nCBS equation is reduced to two (1+1)-dimensional partial differential equations. Secondly, three different types of exact solutions are obtained by the homoclinic test approach, including the singular kinked-type periodic solitary wave solutions, the kinked solitary wave solutions and the kinked-cross periodic wave solutions. The diagrams of the solutions for the specified parameters are given. Furthermore, taking one reduced equation as an example, the N-soliton solutions are derived. And under some constraints, higher-order soliton solution can degenerate into lower-order soliton solution, which is a specific degeneration phenomenon of this equation.

Employing the exp $(-\varphi(z))$ - Expansion Method to Find Analytical Solutions for a (2+1)-dimensional Combined KdV-mKdV Equation

In this paper, we obtain exact solutions of the (2+1)-dimensional combined KdV-mKdV equation by using a symbol calculation approach. First, we give some background on the equation. Second, the exp$(-\varphi(z))$-expansion method will be introduced to solve the equation. After, using the exp$(-\varphi(z))$-expansion method to solve the equation, we can get four types of exact solutions, which are hyperbolic, trigonometric, exponential, and rational function solutions. Finally, we can observe the characteristics of the exact solutions via computer simulation more easily.

Assorted exact explicit solutions for the generalized Atangana’s fractional BBM–Burgers equation with the dissipative term

In this study, the generalized Atangana’s fractional BBM–Burgers equation (GBBM-B) with the dissipative term is investigated by utilizing the modified sub-equation method and the new G'/(bG' + G + a)-expansion method; with the aid of symbolic computations, many types of new exact explicit solutions including solitary wave solutions, trigonometric function periodic solutions, and the rational function solutions are obtained. Some 3D and 2D plots of these solutions are simulated, which show the novelty and visibility of the propagation behavior and dynamical structure of the corresponding equation. Moreover, with the selection of different values on the parameters and orders, we can deduce many types of exact solutions in special cases. We also discussed the changes and characteristics of these solutions, which can help us further understand the inner structure of this equation. The obtained solutions indicate that the approach is easy and effective for nonlinear models with high-order dispersion terms.

A reduction technique to solve the generalized nonlinear dispersive mK(m,n) equation with new local derivative

In this work, we consider the generalized nonlinear dispersive mK(m,n) equation with a recently defined local derivative in the temporal direction. Different types of exact solutions are extracted by Nucci’s reduction technique. Combinations of the exponential, trigonometric, hyperbolic, and logarithmic functions constitute the exact solutions especially of the soliton and Kink-type soliton solutions. The influence of the derivative order α, for the obtained results, is graphically investigated. In some cases, exact solutions are achieved for arbitrary values of n and m, which can be interesting from the mathematical point of view. We provided 2-D and 3-D figures to illustrate the reported solutions. Computational results indicate that the reduction technique is superior to some other methods used in the literature to solve the same equations. To the best of the author’s knowledge, this method is not applied for differential equations with the recently hyperbolic local derivative.

New Solutions of Nonlinear Dispersive Equation in Higher-Dimensional Space with Three Types of Local Derivatives

The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently.

Applications of rogue wave, breathers, multiwave and interaction solutions to long water-wave equation

In this paper, we study different analytical solutions for long water-wave equation (LWWE) like lump, breathers, rogue waves, periodic cross-kink, periodic wave, lump with one and two-kink, multiwave and periodic-cross lump. We compute the essential relationship between interaction between lump periodic and kink wave, periodic cross-kink and other types of exact solutions by studying the unknown nonlinear evolution equations (NLEEs) for long water wave. We show the graphical representation for our retrieve analytical solutions to observe the behavior of effective waves.

Multi lump and interaction solutions for Atangana conformable Boussinesq-like equation

Atangana conformable Boussinesq-like equations (NLACBEs) plays an important role in shape-memory alloys, nonlinear string and coupled electrical circuits. In this paper, we study the different analytical solutions of various types of NLACBEs like multi-wave, homoclinic breather approach, rogue waves, M-shaped rational solutions, lump with one kink, lump interaction with kink and rogue wave, periodic-cross lump, lump with two kink, periodic cross-kink and periodic solutions. We compute the essential relationship between lump interaction with kink and rogue wave, periodic cross-kink and other types of exact solutions by studying the unknown nonlinear confirmable differential equations for the corresponding parameters. We show the graphical representation for our retrieve analytical solutions to observe the behavior of effective waves.

New Solitary Wave Solutions of the Fisher Equation

In this paper, we use Riccati equation to find new solitary wave solutions of Fisher equation, which describes the process of interaction between diffusion and reaction. It is of great importance to comprehend the equation to solve the problems in chemical kinetics and population dynamics. We resolve the Ricatti equation through diverse function transformation and many types of exact solutions are obtained. Then it is used as an auxiliary equation to solve Fisher equation. In the process, we select different coefficients in the Racatti equation, as a result, abundant solitary wave solutions are obtained, some of which haven’t been found in other documents yet. Moreover, these solutions we got in this paper will be favorable for understanding the Fisher equation.

Exact solutions and aysmptotic state analysis of a modified Toda lattice system with a perturbation parameter

Under consideration is a modified Toda lattice system with a perturbation parameter, which may describe the particle motion in a lattice. With the aid of symbolic computation Maple, the discrete generalized [Formula: see text]-fold Darboux transformation (DT) of this system is constructed for the first time. Different types of exact solutions are derived by applying the resulting DT through choosing different [Formula: see text]. Specifically, standard soliton solutions, rational solutions and their mixed solutions are given via the [Formula: see text]-fold DT, [Formula: see text]-fold DT and [Formula: see text]-fold DT, respectively. Limit states of various exact solutions are analyzed via the asymptotic analysis technique. Compared with the known results, we find that the asymptotic states of mixed solutions of standard soliton and rational solutions are consistent with the asymptotic analysis results of solitons and rational solutions alone. Soliton interaction and propagation phenomena are shown graphically. Numerical simulations are used to explore relevant soliton dynamical behaviors. These results and properties might be helpful for understanding lattice dynamics.

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations, we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem. The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair. By Lax pair, an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found. The solutions with specific properties are distinct from those of the local Hirota equation. In order to further describe the properties and the dynamic features of the solutions explicitly, several kinds of graphs are depicted.

The reduction of nonlinear response in near-surface layers by adjusting the electric field amplitude

In this work, we use a special nonlinear dependence of dielectric permittivity to study theoretically the effect of a decrease in the nonlinear response in near-surface layers of a medium, which occurs with an increase in the amplitude of the electric field. We propose a model of nonlinearity in which the Kerr-type nonlinearity abruptly disappears with an increase in the field, and the dielectric permittivity becomes constant and independent of the field. Increasing the electric field leads to the formation of a local zone (optical domain) near the surface with linear optical properties where the dielectric permittivity becomes independent of the electric field. We formulate a nonlinear equation with stepwise dependence of the dielectric permittivity on electric field, and obtain its two types of exact solutions corresponding to the surface waves in media with positive (self-focusing) and negative (defocusing) nonlinear responses. We calculate and analyze the total power flows of thesurface waves of both types. We discuss in detail the features of the obtained solutions in comparison with previously published results. It is shown that the choice of a crystal with an appropriate nonlinear response makes it possible to increase or decrease the field intensity near the crystal surface with practically the same thickness of the near-surface layer with altered optical properties.

Lie symmetry analysis and dynamical structures of soliton solutions for the (2 + 1)-dimensional modified CBS equation

In this present article, the new [Formula: see text]-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, commutation table, invariant surface condition, Lie symmetry reductions, infinitesimal generators and explicit solutions are constructed. As we all know, an optimal system contains constructively important information about the various types of exact solutions and it also offers clear understandings into the exact solutions and its features. The symmetry reductions of [Formula: see text]-dimensional mCBS equation is derived from an optimal system of one-dimensional subalgebra of the Lie invariance algebra. Then, the mCBS equation can further be reduced into a number of nonlinear ODEs. The generated explicit solutions have different wave structures of solitons and they are analyzed graphically and physically in order to exhibit their dynamical behavior through 3D, 2D-shapes and respective contour plots. All the produced solutions are definitely new and totally different from the earlier study of the Manukure and Zhou (Int. J. Mod. Phys. B 33, (2019)). Some of these solutions are demonstrated by the means of solitary wave profiles like traveling wave, multi-solitons, doubly solitons, parabolic waves and singular soliton. The calculations show that this Lie symmetry method is highly powerful, productive and useful to study analytically other nonlinear evolution equations in acoustics physics, plasma physics, fluid dynamics, mathematical biology, mathematical physics and many other related fields of physical sciences.

Using of fast expansions in the construction of two-dimensional exact solutions of the Poisson equation

Using the fast expansion method, we obtained several exact solutions of the boundary value problem for the Poisson equation in a rectangular domain. We have given graphs of exact solutions corresponding to different boundary conditions and different types of the Poisson equation free term. We showed the influence of the type of boundary conditions and the right-hand side on the type of exact solution. We obtained a solution to the problem of membrane deflections under the action of variable load. We have given graphs of the stress components, from the analysis of which it follows that the greatest stress is in the middle of the rectangular membrane long sides.

A generalization of the Landau-Lifschitz equation: breathers and rogue waves

A generalization of the Landau-Lifschitz equation with uniaxial anisotropy is proposed, which can also reduce to the derivative nonlinear Schrödinger equation under an infinitesimal parameter. Based on the gauge transformation between Lax pairs, an N-fold generalized Darboux transformation is constructed for the generalization of the Landau-Lifschitz equation with uniaxial anisotropy. As applications of the N-fold generalized Darboux transformations, several types of exact solutions of the generalization of the Landau-Lifschitz equation with uniaxial anisotropy are obtained, including soliton solutions, Akhmediev breather solutions and rogue-wave solutions.

Exact solution of perturbed nonlinear Schrödinger equation using ($$G^\prime /$$G, 1/G)-expansion method

By constructing auxiliary equations and combining the expansion method of ( $$G^{\prime }/G,1/G$$ ), we study a class of nonlinear Schrodinger equation with perturbation terms which describes the propagation of the waves in optical metamaterials. More types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

Non-traveling lump solutions and mixed lump–kink solutions to (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

New exact solutions of generalized convection-reaction-diffusion equation

In this paper, we explain how to construct a complete classification of invariant subspaces for the generalized nonlinear convection-reaction-diffusion equation. Also, we have explicitly shown that the convection-reaction-diffusion equation admits more than one invariant subspaces in different dimensions which in turn helps to derive more than one different types of exact solution.