Lie symmetry analysis and conservation laws of non-linear (2+1) elastic wave equation

The elastic wave propagation in inelastic media gives rise to non-linear wave equation. We study such a nonlinear wave in two dimensions using Murnaghan model. Lie symmetries, invariant exact solutions and conservation laws using the Noether theorem have been found. The nonlinear elastic wave equation with a damping term has been dealt with using the partial Noether approach.

under two dimensional subalgebras are discussed. Furthermore, some invariant solutions are given explicitly by solving the reduced ODEs. In Sect. 5, Noether symmetries are formulated and conservation laws are established corresponding to these Noether symmetries. In Sect. 6, two dimensional non-linear damped elastic wave equation is presented and in Sect. 7, Lie symmetries and their commutator table for damped elastic wave equation is given. In Sect. 8, symmetry reductions and corresponding invariant solutions of this equation are presented. In Sect. 9, partial Noether symmetries of this equation are formulated and conservation laws are established corresponding to these symmetries.

Nonlinear elastic wave equation in two dimensions
The linear theory of elasticity is based upon the assumption that the strain tensor i j depends linearly on the displacement vector u i as In this study, we focus on non-linear elastic model based upon the non-linear strain, We consider the two dimensional motion in which (x 1 , x 3 ) = (x, y) and u 1 = u and the nonlinear strain (2) has the components given by 11 = u x + Murnaghan potential [14] for the Cauchy-Green strain tensor is where λ and μ are Lame's coefficients. After substituting strain values, Murnaghan potential up to third order becomes Stress can be calculated using t nm = ∂ W ∂u m,n , and so we get the stress components given as Cauchy equation of motion [1] in our case can be written as Now, after inserting stress values, the equation of non-linear wave motion is given as where, . ρ is the density of the elastic medium.

Lie symmetry analysis
The Lie symmetry technique was primarily concerned with establishing symmetries of DEs and then using them to find transformations that would transform the under study DE in terms of a lower number of independent variables or reduce it to a linear one to make it easier.The method for determining the symmetries of a PDE is well-known and covered in numerous literature, e.g., [4,12,15].
The Lie symmetries of Eq. (6) are found in this section.To find out the symmetries, we consider the one parameter Lie group of infinitesimal transformations We want to find conditions on ξ 1 , ξ 2 , ξ 3 and ζ that transform the one-parameter Lie group into the Lie group of Eq. (6).Then the corresponding vector field will be the Lie point symmetry generator of Eq. (6) along the solutions of (6) if where X [2] is the second prolongation of X and is given by where, where the operator D i is defined as From Eq. (8), after comparing coefficients of the various derivatives of the dependent variable, we get an overdetermined linear PDE system, and after performing some calculations we get the following set of determining equations Solving above system of partial differential equations, we get the following infinitesimals Now, we construct the symmetry generators for each of the constants involved. There are six generators in total, as listed by The commutator table for the symmetry generators listed above is given by

Reduction and invariant solution
In this section, we perform reductions [2] of Eq. (6) by two-dimensional subalgebras and find invariant solutions in some interesting cases.
Reduction with respect to < X 1 , X 2 > We begin with The similarity variables for this generator are α = y, β = t, f (α, β) = u By using these transformations (6) reduces to Now, we perform reduction using Using these values, Eq. (9) reduces to the ODE Moving back to original variables, we get an invariant solution Using these values, Eq. (10) reduces to the ODE Using these values, Eq. (11) reduces to the ODE The invariant solution is Using these values, Eq. (13) reduces to the ODE ⇒ −w (−Cr 3 w + Cr 2 w + Ar 2 − 1) = 0 Table 2 Reductions of Eq. (6) Algebra Reductions Reduction with respect to < X 3 , X 6 > We begin with X 3 = ∂ ∂t . The similarity variables are x = α, y = β, u = f (α, β). By these transformations (6) reduces to PDE Using these values, Eq. (14) reduces to the ODE Now, we make a table of reductions corresponding to different two-dimensional subalgebras.

Conservation laws via Noether's approach
A non-trivial conservation law of (6) exists if there exist a vector (T t , T x , T y ) whose divergence D t T t + D x T x + D y T y vanishes on the solutions of (6). We will use the Noether's theorem [10] to construct the conserved vectors. The Lagrangian of (6) is The vector field is called Noether symmetry of Eq. (6) corresponding to Lagrangian (15), if it satisfies the condition where G 1 , G 2 and G 3 are gauge terms independent of the derivatives. From (16) we get a set of determining equations Solving above equations, we get Noether symmetries are given as Now, we will find the conserved vectors corresponding to above listed Noether's symmetries using the formula [9] where δ δu α i is the Euler operator defined as: where D i is the total derivative operator with respect to x i given by The first component of conserved vector is The second component of conserved vector is The third component of conserved vector is So, we have the conserved vectors corresponding to above listed Noether symmetries given by

Nonlinear damped elastic wave equation
Bokhari, Kara and Zaman [5] in their paper appended one-dimensional non-linear elastic wave equation with a small damping term u t and introduced non-linear damped elastic wave equation. Non-linear damped elastic wave equation in two dimensions can be written as where, γ is a damping coefficient.

Lie symmetry analysis
Lie symmetries of Eq. (18) are found in this section. The invariance condition is given by From Eq. (19), after comparing coefficients of the various derivatives of the dependent variable, we get an over-determined linear PDE system, and after performing some calculations we get the following set of determining equations Solving above system of linear partial differential equations, we get the following infinitesimals Finally, we get the symmetry generators given as The commutator table for the symmetry generators listed above is given by

Reduction and invariant solution
Reduction with respect to < X 1 , X 2 > We begin with X 1 = ∂ ∂ x . Similarity variables are y = α, t = β, u = f (α, β). Using these transformations, Using these values, Eq. (20) reduces to the ODE Reduction with respect to < X 1 , X 3 > We begin with The new invariants are α = r , f (α, β) = w(r ). Using these values, Eq. (21) reduces to the ODE The invariant solution is Reduction with respect to < X 2 , X 3 > We begin with

Conservation laws via partial Noether approach
A non-trivial conservation law of (18) exists if there exist a vector (T t , T x , T y ) whose divergence D t T t + D x T x + D y T y vanishes on the solutions of (18). We will use partial Noether's approach [7] to construct the conserved vectors. The partial Lagrangian of (18) is where, δ δu is the Euler operator. The vector field is called partial Noether symmetry of Eq. (18) corresponding to Lagrangian (23), if it satisfies the condition where G 1 , G 2 and G 3 are gauge terms independent of the derivatives. From (24) we get a set of determining equations Solving above equations, we get Partial noether symmetries are Now using (17) we will find the conserved vectors corresponding to partial Noether symmetries. The first component of conserved vector is The second component of conserved vector is The third component of conserved vector is So, we have the conserved vectors corresponding to above listed partial Noether symmetries given as y , e γ t (Bu x u y + Du 2 x u y ) (T t , T x , T y ) = − e γ t u y u t , e γ t Au x u y + C 2 u 2 x u y +

Conclusion
A non-linear elastic wave equation has been studied from symmetry stand point. The conservation laws using Noether's theorem have been obtained. The damped equation and conservation laws of this equation using partial Noether approach has also been studied. The conservation laws are useful in understanding the model.
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