KdV mKdV Equation Research Articles

Dynamical behavior of soliton solutions to the space-time fractional combined KDV-MKDV equation through two robust techniques

Dynamical behavior of soliton solutions to the space-time fractional combined KDV-MKDV equation through two robust techniques

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Wave solutions to the combined KdV-mKdV equation via two methods with the Riccati equation

Wave solutions to the combined KdV-mKdV equation via two methods with the Riccati equation

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In this article, we investigated the solitary wave solutions of the KdV–mKdV equation using Hirota’s bilinear method. Closed-form analytical single and multiple solitary wave solutions were obtained. Through qualitative methods and the analysis of solitary waveforms, we discovered that in addition to sech-type solitary waves, the system also contains Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary waves. By employing the trial functions method, we obtained a single Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary wave and verified its existence and stability using the split-Step Fourier Transform method. Furthermore, we use the collision of two Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type single solitary waves to excite a stable Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type double solitary wave. Similarly, we excite a stable triple solitary wave with three Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type single solitary waves. This method can also be used to excite stable multiple solitary waves. It is shown that these solitary wave solutions enrich the dynamic behavior of the KdV–mKdV equation and provide methods for solving Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary waves, which hold significant theoretical value.

Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

Abstract The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

The study of coherent structures of combined KdV-mKdV equation through integration schemes and stability analysis

The study of coherent structures of combined KdV-mKdV equation through integration schemes and stability analysis

Brief Communication: A modified Korteweg–de Vries equation for Rossby–Khantadze waves in a sheared zonal flow of the ionospheric E layer

Abstract. The system of non-linear equations for electromagnetic Rossby–Khantadze waves in a weakly ionized conductive ionospheric E-layer plasma with sheared zonal flow is given. Use of multiple-scale analysis allows reduction of an obtained set of equations to a (1+1)D non-linear modified KdV (mKdV) equation with cubic non-linearity describing the propagation of solitary Rossby–Khantadze solitons.

Solutions of the KdV-MKdV equations arising in non-linear elastic rods under fractal dimension

A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately.

Study on the simplified MCH equation and the combined KdV–mKdV equations with solitary wave solutions

This paper is concerned with the traveling wave solutions and analytical treatment of the simplified MCH equation and the combined KdV–mKdV equations. Based on a polynomial about x and t, the rational solutions are investigated. The abundant soliton and periodic wave solutions are obtained by using a direct function. The hyperbolic-type and trigonometric-type solutions are given by utilizing the improved tan(ϕ/2)-expansion method. The dynamic properties of these derived results are shown in some three-dimensional, density and 2D plots. Traveling wave solutions are utilized to depict water waves in the mentioned equations, and that is used in physical science to model quantum field theory, dust-acoustic waves, ion acoustic waves, which is taken into account through the application of the improved tan(ϕ/2)-expansion technique. In addition, the recommended technique allowed us to produce some dynamical wave patterns of kink, kink single soliton, single soliton, compacton, periodic shape and other structures are developed, which are shown using three-dimensional, density and 2D plots to more clearly illustrate the physical layout. The method is one of the proficient and effective approaches which have swiftly developed in order to searching appropriate responses to partial differential equations with nonlinear sciences.

The combined KdV-mKdV equation: Bilinear approach and rational solutions with free multi-parameters

In this paper, we investigate the combined KdV-mKdV equation which serves as a valuable tool in the study of water waves, enabling researchers to understand and predict the behaviour of various wave phenomena, including solitary waves, wave breaking, turbulence, wave-structure interactions, and tsunamis. Using the bilinear approach, its rational solutions with free multi-parameters and N-soliton solutions in the determinant form are obtained. The novel idea in this paper is that we develop the bilinear approach allowing free multi-parameters in rational solutions. The dynamics of rational solutions are presented by selecting appropriate parameters. By the bilinear approach, different to the Darboux transformation method, there are no limiting techniques involved on the spectrum parameters to obtain rational solutions. The approach can be extended to obtain other types of solutions if outside the framework of polynomials.

Consequence of head‐on collision and double‐layer soliton with linear analysis in multi‐component dusty plasmas

AbstractThe unmagnetized multicomponent collisional dusty plasmas have been considered to investigate the consequence of head‐on collision of dust‐ion‐acoustic waves (DIAWs) with nonlinear characteristic analysis of DIAWs and collisionless plasmas have been considered for studying the production of double layer (DL). The plasma system consists of positively charged ion fluid, negatively charged cold dust fluid, thermal electrons, q‐nonextensive electrons, and immobile background neutral particles. The extended Poincaré‐Lighthill‐Kuo (ePLK) method has been used to derive two‐sided damped Korteweg–De Vries (dKdV) equations and the reductive perturbation method has been used to derive modified KdV (mKdV) equation and the standard Gardner equation (SGE). Linear dispersion relation is derived using linear perturbation technique. The nonlinear propagation of DIA Gardner solitons (DIAGSs) and double layers (DLs) as well as phase shifts have been investigated for the effects of the concerned parameters. It is found that the DIA soliton structures are significantly modified by the species densities, dust mass, and superextensive parameter. The rarefactive dip shape solitary Gardner solitons become produced for the effects of concerned parameters and the positive (negative) DLs become produced in the range of () for the effect of temperatures on the species. The superextensive and the subextensive parameters of electrons significantly affect the DL soliton structures.

Comparative qualitative analysis and numerical solution of conformable fractional derivative generalized KdV-mKdV equation

Comparative qualitative analysis and numerical solution of conformable fractional derivative generalized KdV-mKdV equation

A survey of (2+1)-dimensional KdV–mKdV equation using nonlocal Caputo fractal–fractional operator

We analyze the nonlinear (2+1)-dimensional KdV–mKdV equation with Caputo fractal–fractional operator. Some theoretical features are demonstrated via fixed point results. The solution of the considered KdV–mKdV is studied by the composition of the J-transformation and decomposition method. For the validity and effectiveness of the considered method, two examples with suitable initial conditions are solved, where best agreements observed. The validity of the suggested approach is verified by convergence analysis and Picard stability. From the simulations of the obtained results, it is noted that fractional order and fractal dimension significantly affects the amplitude and shape of wave solutions.

Traveling wave behavior of new (2+1)-dimensional combined KdV–mKdV equation

The KdV and mKdV equations have many significant applications in the field of plasma physics and ocean dynamics. In this work, the traveling wave behavior of the newly proposed (2+1)-dimensional combined KdV–mKdV equation has been examined using two direct analytical approaches. The two variable G′G,1G-expansion method and generalized projective Riccati equation method are used to determine a variety of solitons and solitary waves for the governing model. The graphical observations are made for suitable choice of parametric values. The obtained results will provide useful insight in further explorations of the wave phenomena depicted by the considered equation.

Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures

<abstract><p>This article mainly uses two methods of solving the conservation laws of two partial differential equations and a system of equations. The first method is to construct the conservation law directly and the second method is to apply the Ibragimov method to solve the conservation laws of the target equation systems, which are constructed based on the symmetric rows of the target equation system. In this paper, we select two equations and an equation system, and we try to apply these two methods to the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation, and simply verify them. The combined KdV-MKdV equation describes the wave propagation of bound particles, sound waves and thermal pulses. The Klein-Gordon equation describes the nonlinear sine-KG equation that simulates the motion of the Josephson junction, the rigid pendulum connected to the stretched wire, and the dislocations in the crystal. And the coupled KdV equation has also attracted a lot of research due to its importance in theoretical physics and many scientific applications. In the last part of the article, we try to briefly analyze the Hamiltonian structures and adjoint symmetries of the target equations, and calculate their linear soliton solutions.</p></abstract>

Physics-informed neural network methods based on Miura transformations and discovery of new localized wave solutions

We put forth two physics-informed neural network (PINN) schemes based on Miura transformations. The novelty of this research is the incorporation of Miura transformation constraints into neural networks to solve nonlinear PDEs, which is an implementation method of unsupervised learning. The most noteworthy advantage of our method is that we can simply exploit the initial–boundary data of a solution of a certain nonlinear equation to obtain the data-driven solution of another evolution equation with the aid of Miura transformations and PINNs. In the process, the Miura transformation plays an indispensable role of a bridge between solutions of two separate equations. It is tailored to the inverse process of the Miura transformation and can overcome the difficulties in solving solutions based on the implicit expression. Moreover, two schemes are applied to perform abundant computational experiments to effectively reproduce dynamic behaviors of solutions for the well-known KdV equation and mKdV equation. Significantly, new data-driven solutions are successfully simulated and one of the most important results is the discovery of a new localized wave solution: kink-bell type solution of the defocusing mKdV equation and it has not been previously observed and reported to our knowledge. It provides a possibility for new types of numerical solutions by fully leveraging the many-to-one relationship between solutions before and after Miura transformations. Performance comparisons in different cases as well as advantages and disadvantages analysis of two schemes are also discussed. On the basis of the performance of two schemes and no free lunch theorem, they both have their own merits and thus more appropriate one should be chosen according to specific cases.

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.

BÄCKLUND TRANSFORMATION AND DIVERSE EXACT EXPLICIT SOLUTIONS OF THE FRACTAL COMBINED KdV–mKdV EQUATION

A fractal modification of the combined KdV–mKdV equation which plays a key role in various fields of physics is presented in this work for the first time. Aided by the fractal two-scale transform, the homogeneous balance method is employed to construct the fractal Bäcklund transformation. By means of the Bäcklund transformation, some new exact explicit solutions such as the algebraic solitary wave solution of rational function, single-soliton solution, double-soliton solutions, N-soliton solutions, singular traveling solutions and the periodic wave solutions of trigonometric function are obtained. Finally, some solutions are illustrated with different fractal orders in the form of the 3D plot, 3D density and 2D curves by assigning reasonable parameters with the help of Mathematica. The findings in this paper are expected to present some new insights into the fractal theory of the fractal PDEs.

Analysis of Kink Behaviour of KdV-mKdV Equation under Caputo Fractional Operator with Non-Singular Kernel

The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution’s existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation.

Invariant analysis, optimal system, power series solutions and conservation laws of Kersten-Krasil'shchik coupled KdV-mKdV equations

The primary goal of this research is to carry out the closed-form analytical solutions of the generalised (1+1)-dimensional Kersten-Krasil'shchik coupled KdV-mKdV equations, which are frequently used to represent shallow water waves, elastic media, electric circuits, electrodynamics, etc. By following the Lie symmetry analysis for this coupled system, optimal system of subalgebras of the secured symmetries has been constructed. Furthermore, the given system is reduced to the system of ordinary differential equations by utilizing an optimal system of subalgebra. Using power series solutions and their convergence analysis, explicit exact solutions of the reduced ordinary differential equations were obtained. Additionally, the conservation laws of the aforementioned coupled system are derived by applying the “new conservation theorem” proposed by Ibragimov.