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In this paper, Lie symmetry analysis method is applied to (3 + 1)-dimensional time fractional modified KdV–Zakharov–Kuznetsov equation. All Lie symmetries and the corresponding conserved vectors for the equation are obtained. The one-dimensional optimal system is utilized to reduce the aimed equation with Riemann–Liouville fractional derivative to a low dimensional fractional partial differential equation with Erdélyi–Kober fractional derivative. Then the power series solution of the reduced equation is given. Moreover, some other low dimensional reduced fractional differential equations with Riemann–Liouville fractional derivative are obtained and can be solved by different methods in the literatures herein. [ABSTRACT FROM AUTHOR]

The article investigates some new properties such as invariant transformations, conserved vectors, and closed-form series solutions of (2+1) dimensional nonlinear coupled Burger system, a model of the evolution of the scaled volume concentration or sedimentation of the two kinds of particles in fluid suspensions of colloids, under the effect of gravity, with fractional temporal evolution by the Lie symmetry method. It is worth noting that the series solutions of (2+1) or higher-dimensional system is not much explored. Moreover, nonlocal conservation laws, one of the important aspects of symmetries, are constructed by the generalized Noether theorem. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the study of Lie symmetries and conservation laws for fractional partial differential equations featuring mixed derivatives of fractional and integer orders. Firstly, we explore the Lie symmetries permitted by the (2+1)-dimensional time-fractional dissipative long-wave system. Subsequently, leveraging these symmetries, we reduce the original (2+1)-dimensional partial differential equations to (1+1)-dimensional counterparts. We then proceed to develop explicit power series solutions along with several exact solutions. Lastly, employing the concept of nonlinear self-adjointness in the fractional context, we establish the conservation properties of the system, resulting in the derivation of novel non-trivial conservation law. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied. [ABSTRACT FROM AUTHOR]

In this paper, the Lie symmetry analysis method is applied to the high dimensional fractional Zakharov-Kuznetsov equation. All Lie symmetries and the corresponding conserved vectors for the equation are obtained. The one-dimensional optimal system is utilized to reduce the aimed equation with Riemann-Liouville fractional derivative to a low dimensional fractional partial differential equation with Erdélyi-Kober fractional derivative. Then the power series solution of the reduced equation is given. Moreover, some other low-dimensional reduced fractional differential equations with Riemann-Liouville fractional derivatives are obtained and can be solved by different methods in the literatures herein. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a q-th order PDE that admits a Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order q-1. In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs. [ABSTRACT FROM AUTHOR]

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we present Lie symmetry analysis for time fractional Black-Scholes equation with time-dependent coefficients. The group classification is carried out by investigating the time-dependent coefficients σ (t) , r(t) and s(t). Then the obtained group generators are used to reduce the equation under study, some of the reduced equations are fractional ordinary equations with Erdélyi-Kober fractional derivative, and some exact solutions including power series solutions are constructed. [ABSTRACT FROM AUTHOR]

This paper investigates the concatenation model under the influence of power-law self-phase modulation through the Lie symmetry. We employ two integration schemes, namely the extended tanh approach and the F-expansion algorithm, to rigorously integrate the reduced ordinary differential equations governing the system. Through this methodological framework, we uncover a diverse array of soliton solutions and systematically classify them, shedding light on their intricate dynamics and characteristics. Our research unveils previously undiscovered soliton solutions, enriching the existing understanding of concatenation models. We introduce a comprehensive classification scheme for these solitons, providing valuable insights into their behavior and interactions. Numerical simulations validate the stability and persistence of the identified soliton solutions across various parameter regimes. Our findings contribute to the theoretical framework of nonlinear wave dynamics and hold potential for innovative applications in fields such as nonlinear optics and information processing. [ABSTRACT FROM AUTHOR]

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili Benjamin–Bona–Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being “out of phase” with each other. The kink, acting as a twist in the soliton’s value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation. [ABSTRACT FROM AUTHOR]

We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter γ = 0 and γ = 1 , respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer's sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to γ = 0 and γ = 1 , respectively. [ABSTRACT FROM AUTHOR]

This article utilizes the Lie symmetry method to analyze the (n + 1) -dimensional modified Zakharov–Kuznetsov (mZK) equation, which characterizes weakly nonlinear traveling waves in plasma with a constant magnetic field, comprising cold ions and hot isothermal electrons. The model is also applicable to dusty and magnetized plasma. Lie point symmetries and associated group invariant solutions are computed using Lie group theory, with the underlying equation. The improved tan ( Φ (ξ) 2) -expansion method is then employed to derive soliton solutions, including hyperbolic, trigonometric, rational, and exponential forms. Graphic interpretations of specific solutions are provided. Nonlinear self-adjointness is used to compute non-local conservation laws in lower dimensions, and conservation laws are developed based on the equation's formal Lagrangian structure, applying Ibragimov's theorem. These findings highlight the novelty and reliability of the methodology employed. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to time fractional Caudrey–Dodd–Gibbon equation. We obtain a symmetric group spanned by two generators for the governing equation. The obtained group generators are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative, thereby getting one trivial solution and one convergent power series solution for the reduced equations. Then we present the dynamic behavior of the obtained analytical solutions graphically. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

In this research, the (2+1)‐dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much‐improved results. This analysis includes computing one‐dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

This study undertakes a comprehensive examination of the classical Lonngren wave equation, a fundamental computational model used for simulating electrical signals in semiconductor materials, with specific emphasis on the tunnel diode. The primary objective of this study is to attain novel and more comprehensive solutions beyond those documented in existing literature. To achieve this goal, we have employed well-established mathematical methods, specifically analysis via Lie symmetry, coupled with other specialized techniques such as the power series method and Jacobi elliptic expansion technique. Notably, this marks the inaugural application of these methodologies to the classical Lonngren wave equation, signifying a pioneering endeavor in the exploration of this equation using these analytical tools. These methodologies yield solutions characterized by elliptic functions. The results are visually presented through 3D, 2D, and density plots, effectively illustrating the characteristics of these solutions. The visual representations reveal a range of patterns, including periodic and singular periodic solutions. Furthermore, the paper applies the multiplier method and leverages the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a pivotal role in the examination of physical quantities, such as energy and momentum conservation, thereby enhancing our understanding of the underlying physics within the system. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear (4 + 1)-dimensional Fokas model is studied and its dynamic character is predicted. This inquiry aims to accomplish two main objectives. First, the Lie symmetries are created and the associated transformation is applied to reduce the model to ordinary differential equations. Plots are used to demonstrate and establish invariant solutions. Second, several methods such as bifurcation, sensitivity analysis, and chaos are used to investigate the dynamic behavior of the model. Bifurcation theories are used to analyse a dynamical system's bifurcation at equilibrium points. The nonperturbed system introduces the perturbation period, which deviates from regular patterns. Different initial conditions are used to investigate the sensitivity analysis the model is found to be highly sensitive. The findings are unique, stimulating and mathematically helpful to comprehend the model. It is essential to comprehend the systems and processes that behave dynamically to investigate new technology, suggest interventions and forecast the future. [ABSTRACT FROM AUTHOR]

An evolution equation is a partial differential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. To this end, this article investigates the analytical studies of a generalized geophysical Korteweg-de Vries equation in ocean physics. The examination of this model is conducted via the Lie group theory of differential equations. In the first place, point symmetries, which are constituent elements of a four-dimensional Lie algebra, are systematically computed. Thereafter, one-parameter transformation groups for the algebra are calculated. Besides, going forward, a one-dimensional optimal system of subalgebras is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction process which enables the derivation of nonlinear ordinary differential equations associated with the generalized geophysical Korteweg-de Vries equation under study. Most of the achieved nonlinear ordinary differential equations are further solved either via direct integration or using a power series approach. Furthermore, travelling wave solutions are initially obtained. This is attained via direct integration and the use of Jacobi elliptic function approach. These techniques enable the attainment of various exact soliton solutions, including non-topological soliton solutions as well as general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the under-study generalized geophysical Korteweg-de Vries equation in ocean physics. In the end, the investigation further gives attention to the calculation of conserved vectors for the model using Ibragimov’s theorem for conservation laws, as well as Noether’s theorem. [ABSTRACT FROM AUTHOR]

In this article, the nonlinear (3+1) dimensional Wazwaz Benjamin Bona Mahony (WBBM) equation is considered for analysis which is related to some specific undular bore evolution through a long wave in shallow water. One-dimensional optimal system of Lie infinitesimal generators, associated vector field, commutation relations, and adjoint representation for the WBBM equation are illustrated. Moreover, the symmetry reductions are made, and closed-form solutions of the WBBM equation are obtained based on the optimal system. In addition, we make use of the transnational symmetries to reduce the governing equation to a nonlinear ordinary differential equation which is solved using the new extended direct algebraic method (NEDAM) to obtain the traveling wave profile of the WBBM equation.

In this paper, we attempt to obtain exact and novel solutions for Date-Jimbo-Kashiwara-Miwa equation (DJKM) via two different techniques: Lie symmetry analysis and generalized Kudryashov method (GKM). This equation has applications in plasma physics, fluid mechanics, and other fields. The Lie symmetry method is applied to reduce the governing equation to five different ordinary differential equations (ODEs). GKM is used to obtain general and various periodic solutions. These solutions have different behaviors such as kink wave, anti-kink wave, double soliton, and single wave solution. The physical behavior of the solutions was reviewed through 2-D and 3-D graphs.

In this study, the generalized perturbed-KdV partial differential equation is examined. Furthermore, symmetry generators address the Lie invariance criteria. The suggested approach produces the Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. The optimal system of the obtained system developed. By using Lie Group methods, the generalized perturbed-KdV partial differential equation is changed using suitable similarity transformations through a system of highly nonlinear ordinary differential equations. The new extended direct algebraic approach is applied to get the soliton solutions. As a result, a plane solution, periodic stumps, compacton, smooth soliton, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, peakon soliton, anti-peaked with decay, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are developed. The behavior of certain solutions is shown in 3-D and 2-D for specific values of the physical components in the studied equation. The outcomes hold significance for elevating research to a more impactful and effective level. The whole set of local conservation laws for the generalized perturbed-KdV equation for any arbitrary constant coefficients is found by applying the conservation laws multiplier. These findings are pivotal for advancing the current understanding and pushing the boundaries of knowledge to new heights. [ABSTRACT FROM AUTHOR]

The primary goal of this work is to derive newly constructed invariant solutions, conservation laws, and modulation instability in the context of the (2+1)-dimensional cubic nonlinear Schrödinger's equation (cNLSE), which explains the phenomenon of soliton propagation along optical fibers. The nonlinear Schrödinger equations and their various formulations hold significant importance across a wide spectrum of scientific disciplines, especially in nonlinear optics, optical fiber, quantum electronics, and plasma physics. In this context, we utilize Lie symmetry analysis to determine the vector fields and assess the optimality of the governing equation. Upon establishing the optimal system, we derive similarity reduction equations, thereby converting the system of partial differential equations into a set of ordinary differential equations. Through the simplification of these ordinary differential equations, we are able to construct several optically invariant solutions for the governing equation. Furthermore, through the utilization of the generalized exponential rational function (GERF) approach, we have derived additional intriguing closed-form solutions. Conservation laws are derived for the governing equation by utilizing the resulting symmetries introduced by the Ibragimov scheme. Furthermore, modulation instability and gain spectrum are derived for this equation to understand the correlation between nonlinearity and dispersive effects. To provide a comprehensive and insightful portrayal of our findings, we have created three-dimensional (3D) visualizations of these solutions, which reveal the periodic waves and the solitary wave structures. The form of cubic nonlinear Schrödinger's equation discussed in this article and the optical solutions obtained have never been studied before. Also, these attained solutions can be beneficial to study analytically the identical models arising in fluid dynamics, birefringent fibers, plasma physics, and other optical areas. [ABSTRACT FROM AUTHOR]

In this article, the nonlinear (3+1) dimensional Wazwaz Benjamin Bona Mahony (WBBM) equation is considered for analysis which is related to some specific undular bore evolution through a long wave in shallow water. One-dimensional optimal system of Lie infinitesimal generators, associated vector field, commutation relations, and adjoint representation for the WBBM equation are illustrated. Moreover, the symmetry reductions are made, and closed-form solutions of the WBBM equation are obtained based on the optimal system. In addition, we make use of the transnational symmetries to reduce the governing equation to a nonlinear ordinary differential equation which is solved using the new extended direct algebraic method (NEDAM) to obtain the traveling wave profile of the WBBM equation. [ABSTRACT FROM AUTHOR]

In this article, the ( 1 + 1 )-dimensional Pochhammer–Chree (PC) model, which illustrates a nonlinear scheme of longitudinal wave propagation of elastic bars, is investigated to determine its analytical solutions using Lie infinitesimal generators. The solutions obtained using the symmetries of the PC model allow one to investigate the propagation of longitudinal deformation waves within an elastic rod. The infinitesimal generators are obtained by implementing the Lie symmetry technique and the geometric method. In the geometric method, the Estabrook and extended Harrison differential forms are used for establishing the infinitesimal generators. Using Olver’s conventional method, a system of optimal subalgebras has been built because there are an infinite number of possible linear combinations of infinitesimal generators. Furthermore, it is demonstrated by the use of formal Lagrangian that the aforementioned model satisfies the quasi-self-adjointness criteria. Additionally, the conservation laws associated with the symmetries of the PC model are derived using the quasi-self-adjointness condition and Ibragimov’s ‘new conservation theorem’. Finally, the three-dimensional surfaces of the acquired solutions have been plotted with the corresponding density and contour plots. The newly developed results show the effectiveness, dependability and validity of the symmetry techniques for obtaining invariant solutions to this nonlinear governing model. The novelty of symmetry analysis is that the group of transformations under which the differential equations remain invariant can be used to simplify the given model. [ABSTRACT FROM AUTHOR]

The lie group invariance method is used to study a cylindrical shock wave in a self-gravitating, rotating perfect gas in the presence of monochromatic radiation and an azimuthal or axial magnetic field. The density of the ambient medium is taken as variable according as the law of shock path. For the system of equations of motion, the one-dimensional optimal system of subalgebra is determined by using Lie group analysis. We have utilized optimal classes of infinitesimal generators to acquire the flow variable transformation and the similarity variable, which are important prerequisites for obtaining the system of ordinary differential equations from the system of partial differential equations. In detail, we have numerically solved and discussed the results in two cases: with power and exponential laws shock path. The effects of variation of the rotational parameter, gravitational parameter, Alfvén Mach number, adiabatic exponent, and dimensionless parameter that characterize the interaction between incident radiation flux and gas are studied in depth. A comparative study is done between power law and exponential law in respect of the strength of shock wave and the flow variables distribution in the flow-field region behind the shock front. The shock is stronger with an axial magnetic field in a power law case; whereas shock is stronger with an azimuthal magnetic field in an exponential law case. The shock strength is observed to decline when the adiabatic index of the gas or the Alfvén Mach number increases. The shock decay with rotational parameter in case of exponential law, but it strength enhanced in case of power law. Also, the rotational parameter and gravitational parameter have an exact opposite impact on the strength of shock in power law and exponential law cases. [ABSTRACT FROM AUTHOR]

We present a novel computational approach for computing invariant manifolds that correspond to equilibrium solutions of nonlinear parabolic partial differential equations (or PDEs). Our computational method combines Lie symmetry analysis with the parameterization method. The equilibrium solutions of PDEs and the solutions of eigenvalue problems are exactly obtained. As the linearization of the studied nonlinear PDEs at equilibrium solutions yields zero eigenvalues, these solutions are non-hyperbolic, and some invariant manifolds are center manifolds. We use the parameterization method to model the infinitesimal invariance equations that parameterize the invariant manifolds. We utilize Lie symmetry analysis to solve the invariance equations. We apply our framework to investigate the Fisher equation and the Brain Tumor growth differential equation. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of bi-fractional Black–Scholes equation derived by the fractional G-Brownian motion. Then some exact solutions are obtained and the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, the new conservation theorem and the generalization of the Noether operators are developed to construct the conservation laws for bi-fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

In this paper, we give the form of the solutions of the following rational difference equation where an and bn are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to obtain exact and novel solutions for Date-Jimbo-Kashiwara-Miwa equation (DJKM) via two different techniques: Lie symmetry analysis and generalized Kudryashov method (GKM). This equation has applications in plasma physics, fluid mechanics, and other fields. The Lie symmetry method is applied to reduce the governing equation to five different ordinary differential equations (ODEs). GKM is used to obtain general and various periodic solutions. These solutions have different behaviors such as kink wave, anti-kink wave, double soliton, and single wave solution. The physical behavior of the solutions was reviewed through 2-D and 3-D graphs. [ABSTRACT FROM AUTHOR]

In this paper, it is noted that three apparently disparate areas of mathematics—singularity analysis, complex symmetry analysis and the distributional representation of special functions—have a basic commonality in the underlying methods used. The insights obtained from the first of these provides a much-needed explanation for the effectiveness of the latter two. The consequent explanations are provided in the form of two theorems and their corollaries. [ABSTRACT FROM AUTHOR]

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

Strong waves known as tsunamis are caused by earthquakes, landslides, or volcanic eruptions that traverse oceans. This article examines the geophysical Korteweg–de Vries (GPKdV) equation, which controls the propagation of tsunami waves in seas. The study involves exploring symmetry diminution exerted Lie group analysis, examining the properties of the dynamical structure with the help of bifurcation phase pictures, and researching the dynamic demeanor of the perturbed dynamical system utilizing chaos theory. Techniques such as 3D and 2D phase portraits, time series analysis, poincaré maps, looking into the existence of multistability in the autonomous structure under diverse beginning circumstances, lyapunov exponent, and bifurcation diagram are applied to identify chaotic demeanor. Furthermore, the study establishes general forms of solitary wave results, containing periodic, trigonometric, and singular soliton results, by using the unified Riccati equation expansion approach to address the examined problem analytically. These results are visually represented as 2D and 3D graphs with cautiously chosen parameters, along with their accompanying constraint conditions. Moreover, the sensitivity evaluation of the investigated equation is discussed and demonstrated pictorially. The discoveries revealed are intriguing, novel, and potentially helpful in understanding a wide range of physical events in engineering and science.

This article presents a study on the generalized Korteweg-de Vries-Zakharov-Kuznetsov (gKdV-ZK) model, which is a nonlinear system that demonstrates the effect of magnetic fields on weak ion-acoustic waves in plasma consisting of cold and hot electrons. The research entails investigating the reduction of symmetry through Lie group analysis, scrutinizing the characteristics of the dynamic structure using bifurcation phase diagrams, and examining the dynamic behaviour of the perturbed dynamical system employing chaos theory. Methods such as 3D and 2D phase portraits, time series analysis, Poincaré maps, exploration of multistability in the autonomous structure across various initial conditions, Lyapunov exponents, and bifurcation diagrams are exercised to demonstrate chaotic behaviour. Additionally, the research establishes general forms of solitary wave solutions, encompassing hyperbolic, trigonometric, and rational soliton solutions, through the utilization of a modified auxiliary equation approach to analytically address the examined problem. These findings are visually depicted as 2D and 3D graphs with carefully selected parameters, accompanied by their corresponding constraint conditions. Furthermore, the sensitivity analysis of the studied equation is deliberated upon and visually illustrated. The uncovered findings are captivating, innovative, and potentially beneficial for comprehending various physical phenomena in engineering and science.

In this paper, we investigate the Kraenkel-Manna-Merle (KMM) system, which explains the nonlinear propagation of ultra-short wave pulses in specific saturated ferromagnetic materials. Ferromagnetic materials play a crucial role in data storage, manipulation, and telecommunications applications. Lie symmetry invariance analysis was conducted to identify infinitesimal generators of symmetries, and the adjoint representation was utilized to formulate an optimal system based on the identified Lie vectors. We obtained the analytical solution using the powerful power series method. To assess the impact of damping on the nonlinear Kraenkel-Manna-Merle (KMM) system, we evaluated its stability based on modulation instability criteria. Furthermore, we visually depicted wave solutions obtained by applying damping with varying parameter values. Additionally, we computed conserved quantities for the nonlinear KMM equation using the multiplier technique.

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.

The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.

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.

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

This paper proposed a novel dual-mode nonlinear Schrödinger equation (DMNLSE) with cubic law. The DMNLSE describes the propagations of two moving waves simultaneously. This nonlinear Schrödinger equation (NLSE) model appears in different areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. In the context of photonics, NLSE models the propagation of soliton pulses through inter-continental distances. In this study, the Lie symmetry procedure is successfully utilized on the DMNLSE to discover the infinitesimal generators using the invariance condition. Then, we transform the DMNLSE into an ordinary differential equation (ODE) employing accepted generators and similarity reduction concepts. Later, we consider conservation laws for the DMNLSE. Thanks to the multiplier technique, the conserved quantities' densities and fluxes are discovered. With the aid of the Mathematica package program, 3D, contour, and density graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. [ABSTRACT FROM AUTHOR]