Nonlinear Self-adjointness Research Articles

Lie symmetry analysis, power series solutions and conservation laws of the time-fractional breaking soliton equation

The main attention of this work focus on extending the Lie symmetry and conservation law theories to the fractional partial differential equations involving the mixed derivative of the Riemann-Liouville time-fractional and first-order x-derivatives. More specifically, we first present a new prolongation formula of the infinitesimal generators of Lie symmetries for the time-fractional breaking soliton equation since the equation involves the mixed derivative, then perform Lie symmetry analysis for the equation. Furthermore, we construct an optimal system of one-dimensional Lie subalgebras and use them to reduce the equation to lower-dimensional fractional partial differential equations involving the Erdélyi-Kober operator. In order to construct the power series solution of the equation, we introduce the Hadamard's finite-part integral to deal with the divergence of the integrals. The convergence and error estimate of the power series solution are proved. Finally, a new conservation law formula for the equation is given by means of the nonlinear self-adjointness method and nontrivial conservation laws are found.

Symmetry Analysis and Conservation Laws for a Time-Fractional Generalized Porous Media Equation

The symmetry group method is applied to study a class of time-fractional generalized porous media equations with Riemann–Liouville fractional derivatives. All point symmetry groups and the corresponding optimal subgroups are determined. Then, the similarity reduction is performed to the given equation and some explicit solutions are derived. The asymptotic behaviours for the solutions are also discussed. Through the concept of nonlinear self-adjointness, the conservation laws arising from the admitted point symmetries are listed.

Lie analysis, conserved vectors, nonlinear self-adjoint classification and exact solutions of generalized $ \left(N+1\right) $-dimensional nonlinear Boussinesq equation

<abstract><p>In this article, the generalized $ \left(N+1\right) $-dimensional nonlinear Boussinesq equation is analyzed via Lie symmetry method. Lie point symmetries of the considered equation and accompanying invariant groups are computed. After transforming the equation into a nonlinear ordinary differential equation (ODE), analytical solutions of various types are obtained using the $ \left(G^\prime/G, 1/G\right) $ expansion method. The concept of nonlinear self-adjointness is used in order to determine nonlocal conservation laws of the equation in lower dimensions. By selecting the appropriate parameter values, the study provides a graph of the solutions to the equation under study.</p></abstract>

LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS

In this paper, the method of Lie symmetry is used to study the incompressible ideal MHD equations. We derive the infinitesimal generators of MHD by using Lie symmetry method. Then, we get the group invariant solutions to MHD based on the group transformation and known solutions. An one-dimensional optimal system which only depends on the commutator of Lie algebras is constructed. Through the similarity transformation of the optimal system, we obtain the 1 + 1-dimensional reduced partial differential equations, and further obtain the exact solutions of these equations, such as Bell soliton solution, junction soliton solution and seed solution. In addition, we prove the nonlinear self-adjoint and conservation laws of the MHD equations using the Ibragimov method. Finally, the classical symmetry and solutions to incompressible ideal MHD equations with initial conditions left invariant are discussed.

Lie symmetry transformation, conservation laws and nonlinear self-adjointness of (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

Lie symmetry transformation, conservation laws and nonlinear self-adjointness of (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

Lie symmetry analysis, optimal system and conservation law of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Symmetry analysis of the generalized space and time fractional Korteweg–de Vries equation

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

Invariant analysis and conservation laws of the time-fractional [formula omitted]-family peakon equations

In this paper, we further extend the theories of Lie symmetry group and conservation law to study the time-fractional b-family peakon equations. The main distinction of the equations with the usual time-fractional partial differential equations is the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order x-derivative. Thus we first give a prolongation formula of the infinitesimal generator for the case of mixed derivative, then after finding the Lie symmetries, we use them to transform the equations into fractional and integer-order ordinary differential equations respectively. Some exact solutions and power series solutions are constructed. Finally, a general conservation law formula is given based on the idea of nonlinear self-adjointness and some nontrivial conservation laws of the equations are presented.

Fundamental analysis of the time fractional coupled Burgers-type equations

Fractional differential equations play an essential role in describing the shallow water wave phenomena. In this article, we discussed the coupled Burgers-type equations in the sense of the fractional derivative of Riemann-Liouville. In return, a series of new results of this considered models, were obtained. In the first place, the formulation of the time fractional coupled Burgers-type equations by applying the Euler-Lagrange variational technology, was carried out. Then, the symmetry and one-parameter group of point transformations of this researched goals through the symmetry analysis scheme, were obtained. Subsequently, the time fractional coupled Burgers-type equations can be reduced into the fractional ordinary differential equations with the help of the Erdélyi-Kober fractional differential/integral operators. Next, the approximate solution and convergence analysis, were considered. At the same time, the stability analysis of the solitary wave, was also studied. Lastly, conservation laws of this discussed equations by using a new conservation theorem and nonlinear self-adjointness, were found. The series of results obtained above can provide strong support for us to reveal the mystery of the equation.

Conserved quantities and travelling wave profiles to the nonlinear transmission line via Lie group analysis

The nonlinear transmission line (NLTL) equations are significant nonlinear evolution equations (NLEEs) in nonlinear electrical transmission line (NLETL) regulation. The method is employed to compute some traveling wave patterns of the NLTL equation. The new extended direct algebraic method (NEDAM) is a viable and successful mathematical method to construct the traveling wave patterns of science and engineering problems. The NEDAM is effectively utilized to obtain the traveling wave structures of a considered model in the form of trigonometric and hyperbolic functions containing parameters. The Lie symmetry technique is used to analyze the NLTL equation and compute the Infinitesimal generators. Moreover, we have shown graphically obtained wave profiles by using the different suitable values of the parameters involved. Further, the nonlinear transmission line equation is described through nonlinear self-adjointness, and conserved quantities are computed for each vector.

Lie symmetry analysis, optimal system and conservation laws of a new (2+1)-dimensional KdV system

In this paper, Lie point symmetries of a new (2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system, associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear self-adjointness is provided and conservation laws of this KdV system are constructed.

Nonlinear self-adjointness and conservation laws for the modified dissipative hyperbolic geometric flow equation

The nonlinear self-adjointness of the modified dissipative hyperbolic geometric flow equation is studied. Furthermore, based on the method of Ibragimov, we can obtain conservation laws and nonlocal conservation laws for the modified dissipative hyperbolic geometric flow equation.

Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu))

The present exploration is concerned with fundamental elements corresponding to the phase decomposition in (Fe-Cr-Mo) and (Fe-Cr-Cu) ternary composites. For the ternary composites of iron, we examine the dynamical behavior of the phase separation. The dynamic of this separation is depicted by a model known as the Cahn-Hilliard equation. The nonlinear self-adjointness for the model under consideration is taken into account. The conserved quantities are calculated with the help of the direct method. For each symmetry generator, we have reduced the considered equation into non-linear ordinary differential equations (ODEs). Also, we have computed the optimal system of the equation under study to find the similarity reduction. Also, the traveling wave structures of the Cahn-Hilliard equation are obtained with the modified simple equation (MSE) technique. Moreover, solitary wave structures is exhibited graphically in the form of 3D, 2D and contour plots.

Some generalized isospectral-nonisospectral integrable hierarchies

In this article, with the aid of the Lie algebra A1 composed of second order matrices and Lie algebra A2 composed of third order matrices, some new soliton hierarchies of evolution equations are deduced and the corresponding Hamiltonian structures are also worked out by utilizing the trace identity. Specially, one of the integrable soliton hierarchicy is reduced to the generalized Fokker–Plank equation (gFP) and special bond pricing equation. Next, the nonlinear self-adjointness of the generalized Fokker–Plank equation is verified and conservation laws are constructed with the aid of Ibragimov’ method. Moreover, we investigate the coverings and the nonlocal symmetries of the generalized Fokker–Plank equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product. Besides, we apply the Li’s method to obtain two basic Darboux transformations of (gFP) with two different forms of T.

A new fourth-order integrable nonlinear equation: breather, rogue waves, other lump interaction phenomena, and conservation laws

In this study, we investigate a new fourth-order integrable nonlinear equation. Firstly, by means of the efficient Hirota bilinear approach, we establish novel types of solutions which include breather, rogue, and three-wave solutions. Secondly, with the aid of Lie symmetry method, we report the invariance properties of the studied equation such as the group of transformations, commutator and adjoint representation tables. A differential substitution is found by nonlinear self-adjointness (NSA) and thereafter the associated conservation laws are established. We show some dynamical characteristics of the obtained solutions through via the 3-dimensional and contour graphs.

Computing wave solutions and conservation laws of conformable time-fractional Gardner and Benjamin–Ono equations

This paper presents travelling wave solutions for the nonlinear time-fractional Gardner and Benjamin–Ono equations via the exp( $$- \Phi ( \varepsilon ))$$ -expansion approach. Specifically, both the models are studied in the sense of conformable fractional derivative. The obtained travelling wave solutions are structured in rational, trigonometric (periodic solutions) and hyperbolic functions. Further, the investigation of symmetry analysis and nonlinear self-adjointness for the governing equations are discussed. The exact derived solutions could be very significant in elaborating physical aspects of real-world phenomena. We have 2D and 3D illustrations for free choices of the physical parameter to understand the physical explanation of the problems. Moreover, the underlying equations with conformable derivative have been investigated using the new conservation theorem.

Lie analysis, conserved quantities and solitonic structures of Calogero-Degasperis-Fokas equation

The paper investigates Calogero-Degasperis-Fokas (CDF) equation, an exactly solvable third order nonlinear evolution equation (Fokas, 1980). All possible functions for the unknown function F(ν) in the considered equation are listed that contains the nontrivial Lie point symmetries. Furthermore, nonlinear self-adjointness is considered and for the physical parameter A≠0 the equation is proved not strictly self-adjoint equation but it is quasi self-adjoint or more generally nonlinear self-adjoint equation. In addition, it is remarked that CDF equation admits a minimal set of Lie algebra under invariance test of Lie groups. Subsequently, Lie symmetry reductions of CDF equation are described with the assistance of an optimal system, which reduces the CDF equation into different ordinary differential equations. Besides, Lie symmetries are used to indicate the associated conservation laws. Also, the well-known (G′/G)-expansion approach is applied to obtain the exact solutions. These new periodic and solitary wave solutions are feasible to analyse many compound physical phenomena in the field of sciences.

Dynamics of solitons to the coupled sine-Gordon equation in nonlinear optics

This paper employs the principle of undetermined coefficients to establish the hyperbolic and trigonometric function solutions of the coupled sine-Gordon equation (CSGE) which describes the propagation of an optical pulse in fiber waveguide. Lie point symmetry of the CSGE is derived. Previously, it was noticed that the concept of nonlinear self-adjointness (NSA) was not applied on the equation under consideration. Here, we apply the concept of NSA to find an explicit form of the differential substitution. By means of the obtained substitution, we establish a new variant of conserved vectors by a new conservation theorem.

Exact solutions and conservation laws of multi Kaup–Boussinesq system with fractional order

The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup–Boussinesq system with $$l=2$$ by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness.