Some exact explicit solutions and conservation laws of Chaffee-Infante equation by Lie symmetry analysis

Abstract

In this work, the tanh method is employed to compute some traveling wave patterns of the nonlinear third-order (2+1) dimensional Chaffee-Infante (CI) equation. The tanh technique is successfully used to get the traveling wave solutions of a considered model in the form of some hyperbolic functions. The Lie symmetry technique is used to analyze the Chaffee-Infante (CI) equation and compute the Infinitesimal generators under the invariance criteria of Lie groups. Then we construct the commutator table, adjoint representation table, and we have represented symmetry groups for each Infinitesimal generator. The optimal system and similarity reduction method is used to obtain some analytical solutions of the considered model. With the help of the similarity reduction method, we have converted the nonlinear partial differential equation into nonlinear ordinary differential equations (ODEs). Moreover, we have shown graphically obtained wave solutions by using the different values of involving parameters. Conserved quantities of nonlinear CI equation are obtained by the multiplier approach.

The residual symmetry, Bäcklund transformations, CRE integrability and interaction solutions: (2+1)-dimensional Chaffee–Infante equation

“…In figure 2, we presented the 3D, contour and density representations of the u 2 (x, y, t) interaction given in equation (36), respectively.…”

“…In figure 3, the solution u 5 (x, y, t) given in equation (36), revealed by the interaction of a rational elliptical form solution of the equation (32) and a linear function, is presented together with s 0 = 1,…”

“…Here, bifurcation in a system parameter that indicates the potential's steepness also increases the model's attractiveness [35]. The CI equation has been studied using the modified Khater method [35], Lie symmetry analysis [36], the first integral method [37], and a variety of other approaches [38,39].…”

“…We know that kink waves are rising or descending waves that go from one asymptotic state to another. The kink solution approaches an infinite constant[41].In figure2, we presented the 3D, contour and density representations of the u 2 (x, y, t) interaction given in equation(36), respectively.In figure3, the solution u 5 (x, y, t) given in equation(36), revealed by the interaction of a rational elliptical form solution of the equation (32) and a linear function, is presented together with s 0 = 1, s 1 = 3, s 2 = 2, r 2 = 2,…”

Conservation Laws, Solitary Wave Solutions, and Lie Analysis for the Nonlinear Chains of Atoms

“…The Lie symmetry analysis approach [1][2][3][4][5][6][7][8] has many applications in different fields, including physics, engineering, and mathematical modeling. It can be used to study a wide range of nonlinear PDEs, including those that are difficult to solve using other methods.…”