Traveling Wave Solutions Research Articles

Solitary wave solutions for the (1+1)-dimensional nonlinear Kakutani–Matsuuchi model of internal gravity waves

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.

Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions

Abstract The fractional-order differential equations (FO-DEs) faithfully capture both physical and biological phenomena making them useful for describing nature. This work presents the stable and more effective closed-form traveling-wave solutions for the well-known nonlinear space–time fractional-order Burgers equation and Lonngren-wave equation with additional terms using the exp ( − Φ ( ξ ) ) (-\Phi (\xi )) expansion method. The main advantage of this method over other methods is that it provides more accuracy of the FO-DEs with less computational work. The fractional-order derivative operator is the Caputo sense. The transformation is used to reduce the space–time fractional differential equations (FDEs) into a standard ordinary differential equation. By putting the suggested strategy into practice, the new closed-form traveling-wave solutions for various values of parameters were obtained. The generated 3D graphical soliton wave solutions demonstrate the superiority and simplicity of the suggested method for the nonlinear space–time FDEs.

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

Multi-wave complexiton, multi-wave, interaction-wave and the travelling wave solutions to the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for the incompressible fluid

Multi-wave complexiton, multi-wave, interaction-wave and the travelling wave solutions to the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for the incompressible fluid

Optical solitons perturbation and traveling wave solutions in magneto-optic waveguides with the generalized stochastic Schrödinger–Hirota equation

Optical solitons perturbation and traveling wave solutions in magneto-optic waveguides with the generalized stochastic Schrödinger–Hirota equation

Soliton dynamics and travelling wave solutions for high-order nonlinear Schrödinger equation in birefringent fibers using improved modified extended tanh function method

Soliton dynamics and travelling wave solutions for high-order nonlinear Schrödinger equation in birefringent fibers using improved modified extended tanh function method

New traveling wave solutions for paraxial wave equation via two integrating techniques

New traveling wave solutions for paraxial wave equation via two integrating techniques

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

Researchers are interested in the (1+1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations as they allow for the study of unidirectional wave propagation in shallow waters with a flat seabed, as well as nonlinear media exhibiting dispersion systems and superconductivity. This work has effectively developed exact wave solutions to the stated models, which may have significant consequences for characterising the nonlinear dynamical behaviour related to the phenomena. The extended -expansion technique is employed to procure a diverse array of progressive wave solutions characterized by hyperbolic, trigonometric, and rational functions. The solutions are shown as 3D profiles with a variety of shapes, including kink, singular kink, periodic, singular periodic, etc. The physical significance of the solutions is discussed by these plots, and the approach used in this study is considered efficient and capable of finding analytical solutions for the nonlinear models. Dhaka Univ. J. Sci. 72(1): 7-13, 2024 (January)

Lump, periodic, travelling, semi-analytical solutions and stability analysis for the Ito integro-differential equation arising in shallow water waves

Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the (G′/G2) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

Bifurcation of finger-like structures in traveling waves of epithelial tissues spreading

We consider a continuous active polar fluid model for the spreading of epithelial monolayers introduced by R. Alert, C. Blanch-Mercader, and J. Casademunt, 2019. The corresponding free boundary problem possesses flat front traveling wave solutions. Linear stability of these solutions under periodic perturbations is considered. It is shown that the solutions are stable for short-wave perturbations while exhibiting long-wave instability under certain conditions on the model parameters (if the traction force is sufficiently strong). Then, considering the prescribed period as the bifurcation parameter, we establish the emergence of nontrivial traveling wave solutions with a finger-like periodic structure (pattern). We also construct asymptotic expansions of the solutions in the vicinity of the bifurcation point and study their stability. We show that, depending on the value of the contractility coefficient, the bifurcation can be a subcritical or a supercritical pitchfork.

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

Analytic shock‐fronted solutions to a reaction–diffusion equation with negative diffusivity

AbstractReaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field according to diffusion and net local changes. Usually, the diffusivity is positive for all values of , which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity that is negative for . We use a nonclassical symmetry to construct analytic receding time‐dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, and we convert them to single‐valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan‐like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the and constant solutions, and prove for certain and that receding traveling waves are spectrally stable. In addition, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well‐known equal‐area rule, but for nonsymmetric diffusivity it results in a different shock position.

Unveiling novel dynamics in Q-deformed Sinh-Gordon model: exploring explicit wave solutions and stability analysis

The new generalized Q-deformed Sinh-Gordon equation is investigated in this article. Numerous physical interpretations and applications exist for the said model in several scientific fields. Therefore, for analyzing the underlying physical system, it is imperative to look at the traveling wave behavior displayed by this equation. To complete our thrust three innovative integration architectures including the modified sub-equation method, the G′/(bG′+G+a) -expansion method and G′G2 -expansion method have been utilized to create precise closed form traveling wave solutions. These methods provide solutions in the form of hyperbolic function solutions, trigonometric function solutions and rational solutions. The obtained solutions are discussed with 3D surface plots and 2D line plots to visualize and support the theoretical results. Furthermore, stability for the model is also discussed. This paper discusses a major development as well as gradual improvements to previous works on the current model. Moreover, we discuss the stability analysis of the studied equation and the outcomes are visually depicted in graphical form. To authenticate the solutions provided in this manuscript, each solution undergoes a meticulous validation process.

Beam deflection coupled systems of fractional differential equations: existence of solutions, Ulam–Hyers stability and travelling waves

In this paper, we study a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo fractional derivatives. Under flexible/fixed end-conditions, two main theorems on the existence and uniqueness of solutions are proved by using two fixed point theorems. Some examples are discussed to illustrate the applications of the existence and uniqueness of solution results. Another main result on the Ulam–Hyers stability of solutions for the introduced system is also discussed. Some examples of stability are discussed. New travelling wave solutions are obtained for another conformable coupled system of beam type that has a connection with the first considered system. A conclusion follows at the end.

Lyapunov-Schmidt reduction in the study of bifurcation of periodic travelling wave solutions of a perturbed (1 + 1)−dimensional dispersive long wave equation

In this paper, the Lyapunov-Schmidt reduction is used to investigate the bifurcation of periodic travelling wave solutions of a perturbed (1+1)−dimensional dispersive long wave equation. We demonstrate that the bifurcation equation corresponding to the original problem is supplied by a nonlinear system of two cubic algebraic equations. As the bifurcation parameters change, this system has only one, three, or five regular real solutions. The linear approximation of the solutions to the main problem has been discovered.

Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation

Abstract The two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation has been used to model a variety of physical systems, including soliton propagation in asymmetric media, nonlinear waves in optical fibers, quantum field theory, and condensed matter physics. The equation is able to capture the complex dynamics of these systems and has been shown to be a powerful tool for studying them. This article discusses the two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation both analytically and numerically using Kudryashov’s approach and the finite difference method. In addition, the stability analysis and local truncation error of the equation are discussed. A number of illustrations are also included to show the various solitons propagation patterns. The proposed equation has opened up new possibilities for modeling asymmetric physical systems.

Abundant optical soliton solutions for the stochastic fractional fokas system using bifurcation analysis

In this study, the stochastic fractional Fokas system (SFFS) with M-truncated derivatives is considered. A certain wave transformation is applied to convert this system to a one-dimensional conservative Hamiltonian system. Based on the qualitative theory of dynamical systems, the bifurcation and phase portrait are examined. Utilizing the conserved quantity, we construct some new traveling wave solutions for the SFFS. Due to the fact that the Fokas system is used to explain nonlinear pulse transmission in mono-mode optical fibers, the given solutions may be applied to analyze an extensive variety of crucial physical phenomena. To clarify the effects of the M-truncated derivative and Wiener process, the dynamic behaviors of the various obtained solutions are depicted with 3-D and 2-D curves.

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Investigation of the dynamical structures of double-chain deoxyribonucleic acid model in biological sciences

The present research investigates the double-chain deoxyribonucleic acid model, which is important for the transfer and retention of genetic material in biological domains. This model is composed of two lengthy uniformly elastic filaments, that stand in for a pair of polynucleotide chains of the deoxyribonucleic acid molecule joined by hydrogen bonds among the bottom combination, demonstrating the hydrogen bonds formed within the chain’s base pairs. The modified extended Fan sub equation method effectively used to explain the exact travelling wave solutions for the double-chain deoxyribonucleic acid model. Compared to the earlier, now in use methods, the previously described modified extended Fan sub equation method provide more innovative, comprehensive solutions and are relatively straightforward to implement. This method transforms a non-linear partial differential equation into an ODE by using a travelling wave transformation. Additionally, the study yields both single and mixed non-degenerate Jacobi elliptic function type solutions. The complexiton, kink wave, dark or anti-bell, V, anti-Z and singular wave shapes soliton solutions are a few of the creative solutions that have been constructed utilizing modified extended Fan sub equation method that can offer details on the transversal and longitudinal moves inside the DNA helix by freely chosen parameters. Solitons propagate at a consistent rate and retain their original shape. They are widely used in nonlinear models and can be found everywhere in nature. To help in understanding the physical significance of the double-chain deoxyribonucleic acid model, several solutions are shown with graphics in the form of contour, 2D and 3D graphs using computer software Mathematica 13.2. All of the requisite constraint factors that are required for the completed solutions to exist appear to be met. Therefore, our method of strengthening symbolic computations offers a powerful and effective mathematical tool for resolving various moderate nonlinear wave problems. The findings demonstrate the system’s potentially very rich precise wave forms with biological significance. The fundamentals of double-chain deoxyribonucleic acid model diffusion and processing are demonstrated by this work, which marks a substantial development in our knowledge of double-chain deoxyribonucleic acid model movements.

Analytical investigations of propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide by three computational ideas

This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail.