Construction Of Conservation Laws Research Articles

Optimal System of 1-D Subalgebras and Conserved Quantities of A Nonlinear Wave Equation in Three Dimensions Arising in Engineering Physics

Abstract The construction of explicit structures of conserved vectors plays diverse crucial roles in the study of nonlinear science inclusive of the fact that they are invoked in developing appropriate numerical schemes and for other mathematical analyses. Therefore, in this paper, we examine the conserved quantities of a nonlinear wave equation, existing in three dimensions, and highlight their applications in physical sciences. The robust technique of the Lie group theory of differential equations (DEs) is invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of one-dimensional (1-D) Lie subalgebras related to the equation is obtained. Thereafter, we engage the formal Lagrangian of the nonlinear wave equation in conjunction with various gained subalgebras to construct conservation laws of the equation under study using Ibragimov’s theorem for conserved vectors.

Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates.

An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx+2uxxyy+uyyyy=f(u) . To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE.

Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

<abstract><p>We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.</p></abstract>

Lie symmetry, numerical solution with spectral method and conservation laws of Degasperis–Procesi equation by homotopy and direct methods

ABSTRACT Partial differential equations are widely used to describe complex phenomena in various branches of science, including: physics, mechanics, etc. Therefore, obtaining high-precision target solutions and, if possible, finding exact analytical solutions of these equations play an important role in these sciences. In this paper, we first examine Lie symmetry and invariant transformations. Next, we obtain the equation solutions with the help of symmetries. We compute invariants and similarity solutions. Spectral method was used to numerically solve the equation. Then, we introduce two approaches, the Homotopy method and the direct method, which are used to construct conservation laws. Then we construct three conservation laws for the Degasperis-Procesi equation by using the direct method. Numerical solution and analysis of the Degasperis-Procesi equation by the methods proposed in this research have not been done in previous studies. Also, in Iran, this is the first time that this research has been done, which is a combination of mathematics, physics and computer science, and is considered interdisciplinary and applied.

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

Exact solutions of a quintic dispersive equation

We present novel exact solutions for a class of Rosenau’s quintic dispersive equations. The variational derivative approach is employed to construct conservation laws for a slightly generalized version of the quintic equation. The double reduction theory, based on the association of symmetries and conservation laws, is utilized to obtain a fourth-order nonlinear ODE, which is then solved to compute exact solutions of the quintic equation. In particular, the G′G-expansion method for the reduced fourth-order nonlinear ODE is applied to construct new exact solutions of the quintic PDE with a cubic nonlinearity.

Non‐classical Lie symmetries for nonlinear time‐fractional Heisenberg equations

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications, more precisely in physics, biology, chemistry and engineering fields. This domain serves as a point of interest to a large extent in the world’s mathematical community. Thus, this paper purveys an analytical study of a generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model. We further reduce the equation using the subalgebras obtained. Besides, more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique. Consequently, we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright, dark, periodic (cnoidal and snoidal), compact-type as well as singular solitons. The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time, are outlined with suitable diagrams. This can be of interest to oceanographers and ocean engineers for future analysis. Furthermore, to visualize the dynamics of the results found, we present the graphic display of each of the solutions. Conclusively, we construct conservation laws of the understudy equation via the application of Noether’s theorem.

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.

Symmetry solutions and conservation laws of a new generalized 2D Bogoyavlensky-Konopelchenko equation of plasma physics

<abstract><p>In physics as well as mathematics, nonlinear partial differential equations are known as veritable tools in describing many diverse physical systems, ranging from gravitation, mechanics, fluid dynamics to plasma physics. In consequence, we analytically examine a two-dimensional generalized Bogoyavlensky-Konopelchenko equation in plasma physics in this paper. Firstly, the technique of Lie symmetry analysis of differential equations is used to find its symmetries and perform symmetry reductions to obtain ordinary differential equations which are solved to secure possible analytic solutions of the underlying equation. Then we use Kudryashov's and $ (G'/G) $-expansion methods to acquire analytic solutions of the equation. As a result, solutions found in the process include exponential, elliptic, algebraic, hyperbolic and trigonometric functions which are highly important due to their various applications in mathematic and theoretical physics. Moreover, the obtained solutions are represented in diagrams. Conclusively, we construct conservation laws of the underlying equation through the use of multiplier approach. We state here that the results secured for the equation understudy are new and highly useful.</p></abstract>

Breather waves, analytical solutions and conservation laws using Lie–Bäcklund symmetries to the ([formula omitted])-dimensional Chaffee–Infante equation

The (2+1)-dimensional Chaffee–Infante has a wide range of applications in science and engineering, including nonlinear fiber optics, electromagnetic field waves, signal processing through optical fibers, plasma physics, coastal engineering, fluid dynamics and is particularly useful for modeling ion-acoustic waves in plasma and sound waves. In this paper, this equation is investigated and analyzed using two effective schemes. The well-known tanh-coth and sine-cosine function schemes are employed to establish analytical solutions for the equation under consideration. The breather wave solutions are derived using the Cole–Hopf transformation. In addition, by means of new conservation theorem, we construct conservation laws (CLs) for the governing equation by means of Lie–Bäcklund symmetries. The novel characteristics for the (2+1)-dimensional Chaffee–Infante equation obtained in this work can be of great importance in nonlinear sciences and ocean engineering.

New conservation laws and exact solutions of the special case of the fifth-order KdV equation

The current study deals with the Kaup-Kupershmidt (KK) equation to construct formal Lagrangian, conservation laws, and exact solutions. KK is basically a special case of the 5th-order KdV equation. The conservation laws obtained by using the conservation theorem are trivial conservation laws. In addition, exact solutions are found via the modified simple equation (MSE) method. For a suitable value of solutions, the 3D surfaces have been plotted using MAPLE. These plots giving novel exact solutions are made to reveal important wave characteristics. Our obtained results in this work concerning our investigated equation are essential to explain many physical and oceanographic applications involving ocean gravity waves and many other related phenomena.

Group classification and constructing conservation laws for time-fractional Sawada–Kotera–Parker–Dye equation

Group classification and constructing conservation laws for time-fractional Sawada–Kotera–Parker–Dye equation

Lie symmetries, exact solutions, and conservation laws of the nonlinear time-fractional Benjamin-Ono equation

In this work, we use the symmetry of the Lie group analysis as one of the powerful tools which that deals with the wide class of fractional order differential equation in the Riemann-Liouville concept. We employ the classical Lie symmetries to obtain similarity reductions of nonlinear time-fractional Benjamin-Ono equation and then, we find the related exact solutions for the derived generators. Finally, according to the Lie symmetry generators obtained, we construct conservation laws for related classical vector fields of time-fractional Benjamin-Ono equation.

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries (LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries (KdV) equation, and second, we find the related exact solutions for the derived generators. Finally, according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.

A RIGID PENDULUM IN A MICROGRAVITY: SOME SPECIAL PROPERTIES AND A TWO-SCALE FRACTAL MODEL

Under a microgravity condition, the microgravity-induced motion of a rigid pendulum has many special properties, which cannot be successfully revealed by any known theory. This work suggests a new fractal model for this purpose to describe the microgravity-induced motion by using the fractal derivative, and its variational principle is successfully established according to the semi-inverse method. The variational principle is very helpful to construct conservation laws and to suggest solution structures of the new fractal model. The approximate analytical solution of the model is gained by the two-scale transform method and the homotopy perturbation method.

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′/G,1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.

On the Solutions of a (3+1)-Dimensional Novel KP-Like Equation

This paper aims to study a ( $$3+1$$ )-dimensional novel KP-like equation, which arises in the analysis of versions of resonant phenomena. The classical symmetry approach will be employed to search for exact solutions. Thereafter, we will search for the admitted conserved vectors of the mentioned novel KP-like equation. Although Kuo and Ma (Wave Random Complex, https://doi.org/10.1080/17455030.2020.1792580 , 2020) have given a recommendable effort to solve a new ( $$3+1$$ )-dimensional novel KP-like equation, there is no unified method. The method employed in Kuo and Ma (Wave Random Complex, https://doi.org/10.1080/17455030.2020.1792580 , 2020), cannot be used to construct conservation laws or point symmetries and hence this prompted the utilization of the classical symmetry approach for the underlying equation.