Lie Symmetry Reductions Research Articles

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

Lie symmetry reductions, exact solutions and soliton dynamics to Burgers equation

Lie symmetry reductions, exact solutions and soliton dynamics to Burgers equation

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

On solution of Schrödinger–Hirota equation with Kerr law via Lie symmetry reduction

On solution of Schrödinger–Hirota equation with Kerr law via Lie symmetry reduction

Lie symmetry analysis, optimal system, symmetry reductions and analytic solutions for a (2＋1)-dimensional generalized nonlinear evolution system in a fluid or a plasma

Nonlinear evolution equations are used to describe such nonlinear phenomena as the solitons, travelling waves and breathers in fluid mechanics, plasma physics and optics. In this paper, we investigate a (2＋1)-dimensional generalized nonlinear evolution system in a fluid or a plasma. Via the Lie symmetry analysis, we acquire the Lie point symmetry generators and Lie symmetry groups of that system. Via the optimal system method, we derive the optimal system of the 1-dimensional subalgebras. Based on the symmetry generators in that optimal system, we give some symmetry reductions for the (2＋1)-dimensional generalized nonlinear evolution system. Finally, via those symmetry reductions, we acquire some soliton, rational-type and power-series solutions.

Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability

The purpose of this study is to investigate the deeper characteristics of the system of shallow water wave equations that is used to model the turbulence in the atmosphere and oceans from different standpoints. The multi-soliton structures such as 1-soliton, 2-soliton, 3-soliton solutions are successfully generated with the help of simplified Hirota’s method. For the sake of physical demonstration and visual presentation, we graphically illustrate the identified solutions in 3D, 2D, and contour plots using Mathematica software. The Lie symmetry technique is used to identify the Lie group invariant transformations and symmetry reductions of the examined system, which aid in reducing the dimension by one or into an ordinary differential equation. Further, the qualitative behavior of the time-dependent dynamical system is observed using the bifurcation and chaos theory. The phase portraits of bifurcation are observed at the equilibrium point of a planar dynamical system. We discuss various tools to identify chaos (random and unpredictable behavior) in autonomous dynamic systems, such as 3D phase portraits, 2D phase portraits, time series, and Poincaré maps. At various initial conditions, the sensitivity and modulation instability analyses are also presented, and it is discovered that the investigated system is stable, as a small change in the initial conditions does not cause an abrupt change in solutions. The findings of this investigation will contribute in the overall depiction of soliton theory and nonlinear dynamical systems.

Use of optimal subalgebra for the analysis of Lie symmetry, symmetry reductions, invariant solutions and conservation laws of the (3 + 1)-dimensional extended Sakovich equation

This paper investigates the [Formula: see text]-dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the [Formula: see text]-dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie’s continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver’s standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov’s generalized theorem and quasi-self-adjointness condition.

Periodic Hunter–Saxton equation parametrized by the speed of the Galilean frame: Its new solutions, Nucci’s reduction, first integrals and Lie symmetry reduction

The main purpose of this study is to introduce symmetry analysis and Nucci’s reduction method for revealing the exact solutions of the periodic Hunter–Suxon equation, which is parameterized by the speed of the Galilean frame. Under one-point similarity transformations, we obtained a set of generators that transformed the given equation into an ordinary differential equation, which can be conveniently taken into consideration for exact solutions. Under special choices, we got three separate families of vector fields out of the problem and constructed solutions for every family. In each case, the Nucci reduction technique generated a solution with a first integral. 3D, density, and contour graphics are given for a better understanding of the solutions.

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

<abstract><p>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.</p></abstract>

An investigation of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov system: Lie symmetry reductions, invariant solutions, dynamical behaviors and conservation laws

In this study, we develop the asymmetric Nizhnik–Novikov–Veselov (ANNV) system in (2+1)-dimensions, which has applications in processes of interaction of exponentially localized wave structures, as well as the infinitesimal generators, Lie symmetries, vector fields, and the commutator table. The link between Lie symmetry vectors and conserved vectors is constructed using symmetry conservation principles once Lie point symmetries are first deduced. Using the aforementioned Lie symmetry technique, two-stage symmetry reductions are used to obtain the precise analytical answers. These analytical solutions all incorporate a number of different functional parameters as well as arbitrary constant parameters. The diversity of the physical phenomena of the obtained soliton solutions is illustrated by the inclusion of arbitraryness of functional parameters and constants. By using Noether’s method, conservation laws have subsequently been attained. The innovative aspect of the work described in this paper is an attempt to use 3-dimensional and 2-dimensional visuals, along with appropriate arbitrary parameter selections and functional parameter values, to represent the dynamical behavior of the solutions that have been produced. In order to make this research more intriguing, stripe solitons, dark-bright solitons, solitary waves, singular wave-form soliton, and other types of soliton wave profiles of the achieved solutions are described. The effectiveness, benefits, and utility of the employed approach are demonstrated by the physical and graphical interpretation of the answers attained.

Lie symmetry reductions and generalized exact solutions of Date–Jimbo–Kashiwara–Miwa equation

The propagation of nonlinear waves with nonuniform velocities is described by nonlinear evolution equations and their solutions involving arbitrary functions. When a nonlinear evolution equation is integrated, it reveals the several existing features of natural phenomena with continuous and fluctuating background. The Date–Jimbo–Kashiwara–Miwa equation is long water wave equation, which describes the propagation of nonlinear and weakly dispersive waves in inhomogeneous media. This work aims to extend the previous results and derive symmetry reductions of Date–Jimbo–Kashiwara–Miwa equation via Lie symmetry method. The infinitesimals involving four arbitrary functions are constructed by preserving invariance property of Lie groups under one parameter transformations. Then, the first symmetry reduction of test equation is determined using symmetry variables. The commutative and adjoint relations of four dimensional subalgebra are presented for reduced equation. Thereafter, the repeated utilization of Lie symmetry method results into the ordinary differential equations. These determining ODEs are solved under numeric constraints and provide exact solutions. The derived solutions retain all the four arbitrary functions appeared in infinitesimals and several arbitrary constants. Due to existing arbitrary functions, these solutions are generalized than previous established results. The deductions of previous results (Wang et al., 2014; Ali et al., 2021; Chauhan et al., 2020; Kumar and Kumar, 2020; Tanwar and Kumar, 2021; Kumar and Manju, 2022) show the novelty and significance of these solutions. Moreover, the derived results are expanded systematically with numerical simulation to analyze their physical significance and thus doubly soliton, multisoliton, line soliton, bell shape, parabolic nature are discussed.

Different dynamics of invariant solutions to a generalized (3+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation arising in shallow water-waves

Using the Lie symmetry technique, this paper studies a (3+1)-dimensional generalized Camassa-Holm-Kadomtsev-Petviashvili (GCHKP) equation that can possess various dynamical structures of solitary waves. The GCHKP equation is a special case of the internal shallow-water wave equation, which has recently gained popularity in ocean physics and hydrodynamics engineering. By taking advantage of the Lie symmetry technique, we derive Lie infinitesimals, geometric vector fields, commutator table, adjoint table, and an optimal system for the considered equation. Based on the one-dimensional optimal system, various attractive Lie symmetry reductions are carried out, and the GCHKP equation is reduced to a number of nonlinear ordinary differential equations (NODEs). Consequently, innumerable explicit exact solutions of a (3+1)-D GCHKP equation are obtained employing symmetry reductions via a one-dimensional optimal system of Lie subalgebra. With the assistance of symbolic computation, several 3D and 2D graphics exhibited their dynamic structures, and graphical illustrations were physically interpreted. For the first time, we furnish exact invariant solutions in terms of a variety of functions such as exact rational solutions, trigonometric and hyperbolic function solutions, V-shaped solitons, dark and bright solitons, and solitary waves, singular solitons, breather waves, and parabolic waves. In addition, the obtained results have significantly supplemented the exact invariant solutions of the GCHKP equation in the literature, allowing us to gain a better understanding of the dynamics of complex physical systems. Finally, the proposed method will significantly boost obtaining the exact explicit solutions of the nonlinear evolution equations.

Lie group analysis for a higher-order Boussinesq-Burgers system

Boussinesq-Burgers (BB)-type equations have been proposed to model the shallow water waves. Under investigation in this Letter is a higher-order BB system. We obtain the Lie point symmetry generators, Lie symmetry groups and symmetry reductions for that system via the Lie group method. Hyperbolic-function, trigonometric-function and rational solutions for that system are derived.

Closed-form solutions and conserved quantities of a new integrable (2 + 1)-dimensional Boussinesq equation of nonlinear sciences

Abstract In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach.

Lie symmetry analysis and invariant solutions of (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.

Invariance analysis for determining the closed-form solutions, optimal system, and various wave profiles for a (2+1)-dimensional weakly coupled B-Type Kadomtsev-Petviashvili equations

In the case of negligible viscosity and surface tension, the B-KP equation shows the evolution of quasi-one-dimensional shallow-water waves, and it is growingly used in ocean physics, marine engineering, plasma physics, optical fibers, surface and internal oceanic waves, Bose-Einstein condensation, ferromagnetics, and string theory. Due to their importance and applications, many features and characteristics have been investigated. In this work, we attempt to perform Lie symmetry reductions and closed-form solutions to the weakly coupled B-Type Kadomtsev-Petviashvili equation using the Lie classical method. First, an optimal system based on one-dimensional subalgebras is constructed, and then all possible geometric vector yields are achieved. We can reduce system order by employing the one-dimensional optimal system. Furthermore, similarity reductions and exact solutions of the reduced equations, which include arbitrary independent functional parameters, have been derived. These newly established solutions can enhance our understanding of different nonlinear wave phenomena and dynamics. Several three-dimensional and two-dimensional graphical representations are used to determine the visual impact of the produced solutions with determined parameters to demonstrate their dynamical wave profiles for various examples of Lie symmetries. Various new solitary waves, kink waves, multiple solitons, stripe soliton, and singular waveforms, as well as their propagation, have been demonstrated for the weakly coupled B-Type Kadomtsev-Petviashvili equation. Lie classical method is thus a powerful, robust, and fundamental scientific tool for dealing with NPDEs. Computational simulations are also used to prove the effectiveness of the proposed approach.

An extended (3+1)-dimensional Jimbo–Miwa equation: Symmetry reductions, invariant solutions and dynamics of different solitary waves

The Lie symmetry method is used to obtain a variety of closed-form wave solutions for the extended (3[Formula: see text]+[Formula: see text]1)-dimensional Jimbo–Miwa (JM) Equation, which describes certain interesting higher-dimensional waves in ocean studies, marine engineering, and other fields. By applying the Lie symmetry technique, we explicitly investigate all the possible vector fields, commutation relations of the considered vectors, and various symmetry reductions of the equation. Based on three stages of Lie symmetry reductions, the JM equation is reduced to several nonlinear ordinary differential equations (NLODEs). Consequently, abundant closed-form wave solutions are achieved, including arbitrary functional parameters. Evolutionary dynamics of some analytic wave solutions are demonstrated through three-dimensional plots based on numerical simulation. Consequently, singular soliton, kink waves, periodic oscillating wave profiles, combined singular soliton profiles, curved-shaped multiple solitons, and periodic multiple solitons with parabolic wave profiles are demonstrated by taking advantage of symbolic computation work. The obtained analytical wave solutions, which include arbitrary independent functions and other constants of the governing equation, could be used to enrich the advanced dynamical behaviors of solitary wave solutions. Furthermore, the study of conservation laws is investigated via the Ibragimov technique for Lie point symmetries.

New exact solutions for the nonlinear Schrödinger's equation with anti-cubic nonlinearity term via Lie group method

The nonlinear Schrödinger's equation with anti-cubic nonlinearity was studied via the Lie group method. At first, we obtained the infinitesimal generators, Lie point symmetries and symmetry reduction by implementing the invariance criteria of the Lie group method on the nonlinear Schrödinger's equation with anti-cubic nonlinearity. After that, we used these symmetries for constructing the reduced ordinary differential equations. Corresponding to the reduced nonlinear ordinary differential equations, initially, the method of the general elliptic equations is described and is applied for constructing the exact travelling wave solutions. The basic idea is that the general elliptic equation containing five parameters. By specific choices of these parameters, we got sub-equations involving three parameters like the Riccati equation, generalized Riccati equation and the ordinary auxiliary equation. In contrast, we got sub-equations involving four parameters like an extended auxiliary function method with other choices. Consequently, many new families of rational formal solutions and other famous forms like bright solitons, singular solitons and dark –singular combo soliton are obtained. Finally, the differential transform (DTM) – Padé method is described and applied for constructing new solutions in closed form.

Analytic solutions and conservation laws of a (2+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation

This article analytically investigates a (2+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation. We secure the solutions of this equation via the engagement of Lie symmetry reductions alongside direct integration techniques. We gain non-topological 1-soliton as well as periodic function solutions of the equation. Series solution of the underlying equation is also achieved by exploiting power series technique. Furthermore, the utilization of (G′/G)-expansion method is done in procuring some closed-form solutions of the equation. Graphical exhibition of the dynamical character of the gained results is given in a bid to have a sound understanding of the physical phenomena of the underlying model. Conclusively, we give the conserved vectors of the aforementioned equation by employing both the standard multiplier approach as well as the Ibragimov’s conservation theorem.

A study of Bogoyavlenskii’s (2+1)-dimensional breaking soliton equation: Lie symmetry, dynamical behaviors and closed-form solutions

This paper employs the Lie symmetry analysis to investigate novel closed-form solutions to a (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. This Lie symmetry technique, used in combination with Maple’s symbolic computation system, demonstrates that the Lie infinitesimals are dependent on five arbitrary parameters and two independent arbitrary functions f1(t) and f2(t). The invariance criteria of Lie group analysis are used to construct all infinitesimal vectors, commutative relations of their examined vectors, a one-dimensional optimal system and then several symmetry reductions. Subsequently, Bogoyavlenskii’s breaking soliton (BBS) equation is reduced into several nonlinear ODEs by employing desirable Lie symmetry reductions through optimal system. Explicit exact solutions in terms of arbitrary independent functions and other constants are obtained as a result of solving the nonlinear ODEs. These established results are entirely new and dissimilar from the previous findings in the literature. The physical behaviors of the gained solutions illustrate the dynamical wave structures of multiple solitons, curved-shaped wave–wave interaction profiles, oscillating periodic solitary waves, doubly-solitons, kink-type waves, W-shaped solitons, and novel solitary waves solutions through 3D plots by selecting the suitable values for arbitrary functional parameters and free parameters based on numerical simulation. Eventually, the derived results verify the efficiency, trustworthiness, and credibility of the considered method.