Reductions and conservation laws for BBM and modified BBM equations

1 Introduction

Finding the exact solutions of nonlinear evolution equations (NEEs) plays a fundamental role in the study of nonlinear phenomena in physics and applied mathematics[1–6]. Solving these types of partial differential equations by using symmetry methods was originally developed by S. Lie and, without any doubt, this is one of the most effective algorithmic ways for analyzing and solving linear and nonlinear differential equations [1, 2]. During the last few decades some semi analytical methods have been developed for solving these type of equations.

One such NEEs is Benjamin-Bona-Mahony (BBM) equation

first introduced by Benjamin et al. (1972), as an improvement of the Korteweg-de Vries equation (KdV), and it is well known in applied sciences, where a and b are arbitrary constants. Originally KdV derived to describe shallow water waves of long wavelength and small amplitude. BBM and KdV are useful for modelling surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluids, hydromagnetics waves in cold plasma and acoustic waves in anharmonic crystals [3, 4, 7–10]. Peregrine also derived the BBM equations to regularized long-wave equation [11]. Another equation in this article that we will study is the nonlinear dispersive modified Benjamin- Bona- Mahony equation (MBBM)

where a is an arbitrary constant, which was first derived to describe an approximation for surface long waves in nonlinear dispersive media. This equation can also characterize the hydro-magnetic wave in cold plasma, acoustic-gravity waves in compressible fluids and acoustic waves in an-harmonic crystals. Physical applications and mathematical properties of these equations have been discussed in other papers [3, 4, 7, 10]. Khalique presented some result regarding BBM equation and we cover all of them [12].

This paper is organized as follows: Section 2 is devoted to some concepts needed to construct Lie point symmetries and general form of infinitesimal generators of BBM and MBBM equations and exact solutions associated with the symmetries. We will find an optimal system of one-dimensional subalgebras for symmetry algebras of these systems in Section 3. In Section 4, we compute the invariants associated with the symmetry operators by integrating the characteristic equations to reduced BBM and MBBM equations and differential invariant are exhibited. Finally in the last section, conservation laws of BBM and MBBM equations are obtained.

2 Calculation of symmetry groups

Suppose △(x,un) is a differential equation defined over the total space M=X×U, whose coordinates represent the independent and dependent variables and the derivatives of dependent variables up to order n, that is called the n-th order jet space on the underlying space X×U. If f(x):X→U is a smooth real-valued function whit p independent variables, the n-jet or n-th prolongation of f is

a typical point in U⁽ⁿ⁾ will be denoted by U⁽ⁿ⁾, with q.C(p + n, n) different components uJα=∂Jfα(x), where J=(j1,⋯,jk),1<_jk<_p and 0≤k≤n, note u^α refers to component u^α of u itself. The graph of Pr⁽ⁿ⁾f(x) lies in the n-jet space M(n)=X×U×U1×U2×⋯×Un and from this point of view, a smooth solution of the given system of differential equations is a smooth function u = f(x) such that

A local group transformation acting on M is given by one parameter Lie group

Let C-curve be the graph of G on M, that at each it's point the tangent vector

is infinitesimal transformation of G in g that acts on X×U×U(1).G is a symmetry group of (3), which transforms solutions of the system to other solutions. For determining symmetry group through the classical infinitesimal symmetry condition, we must check the following system

where Pr⁽ⁿ⁾v called n-th prolongation of v is

where J = (j₁,…, j_k) with 1≤k≤p is a multi-indices, and D_i represents the total derivative with respect to the i-th independent variable x^j[1, 13].

The commutator tables of Lie algebras g_BBM and g_MBBM for BBM and MBBM equations are given in Table 1 and 2 respectively, where the entry in the i-th row and j -th column is defined as [X_j, X_j] = X_jX_j−X_jX_j. The relations between these vector fields in Table 1 and 2 say that the Lie algebras g_BBM and gMBBM are solvable, because in g_BBM, we have

Similarly, in gMBBM we have

To obtain the group transformation, which is generated by the infinitesimal generators (11) and (12), we need to solve the following systems of first order ordinary differential equations for each of them, separately

where η = ξ¹,ξ²,φ and τ = x, t, u. Consequently, we conclude the following theorems:

In general a family of solutions, called group-invariant solutions, will correspond to each parameter subgroup of the full symmetry group of a system.

3 Classification of group invariant solutions

The Lie group theory plays an important role in finding the exact solutions of differential equations and any transformation in the full symmetry group will take a solution to another solution, so it is sufficient to find invariant solutions which are not related to transformations in the full symmetry group. There are a set of group-invariant solutions for every s-parameter subgroup of the full symmetry group G. Since for every full symmetry group there are infinite number of such subgroups, it's very difficult to know all of them. The problem of finding an optimal system of subgroups is equivalent to finding an optimal system of subalgebras. For one-dimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representations. Therefore, by taking a general element in the Lie algebra and subjecting it to various adjoint transformations we simplify it as much as possible, because two connected s-dimensional subqroups of the Lie group G are conjugate if and only if their subalgebras are conjugate subalgebras [1, 2].

The adjoint action is given by the Lie series

where L_A(B) = ad_A(B) = [A, B] is the commutator for the Lie algebra, s is a parameter and i, j = 1,2,3. Adjoint representation of infinitesimal symmetries of BBM and MBBM equations are shown in Table 3 and 4.

we can proof

According to our optimal systems of one-dimensional subalgebras of the full symmetry algebra g, we only need to find group-invariant solutions for one-parameter subgroups generated by (15) and (16).

4 Symmetry reduction and differential invariants for BBM and MBBM equations

The fundamental theorem on group-invariant solutions roughly states that the solutions which are invariant under a given r-parameter symmetry group of the system can all be found by solving a system of differential equations involving r fewer independent variables than the original system. Assume G is a symmetry group that acts regularly on Δ, that is a system of differential equation defined on an open subset M ⊂ X × U, the space of independent and dependent variables. The reduced system Δ/G is a differential equation on M/G by p + q − s functionally independent invariant on M, that determine global coordinates on M/G, where s denotes the dimension of the orbits of G.

We can now compute the invariants associated with the symmetry operators (15) and (16) by integrating the characteristic equations. In continuation, we will find some group-invariant solutions for both BBM and MBBM equations. For example, take operator ∂_t, so we have dx0=dt1=du0 and the corresponding invariants are I₁ = x, I₂ = u and a similarity solution of the form y = y(x) . Now substitute it into (1) and (2) respectively, to determine the form of the function y. Finally we have y′ − 2ayy′ = 0 and y′ + ay²y′ + y′′′ = 0 as reduced equations of BBM and MBBM. Solving these ODEs leads to a solutions for BBM and MBBM. For example, by solving y′ − 2ayy′ = 0, we have

where c is an arbitrary constant. Another results are in Table 5, 6, 7 and 8.

Now we turn the whole procedure around and ask the complementary question: What is the most general type of differential equation which admits a given group as a group of symmetries? For an answer we introduce differential invariants, that play an important role in this procedure. Let S_ΔMⁿ be the corresponding subvariety to Δ(x, uⁿ), so S_Δ is invariant under the n-th prolongation Pr⁽ⁿ⁾(G) . Furthermore, there is an equivalent equation Δ~=0 describing the subvariety S_Δ, where Δ~ depends only on the invariants of the group action Pr⁽ⁿ⁾(G) . An n-th order differential invariant of G is a smooth function η : Mⁿ → R, depending on x, u and derivatives of u such that η is invariant on the prolonged group action Pr⁽ⁿ⁾(G), namely

If G is a symmetry group for a system of PDEs with functionally differential invariants, we can rewrite the system in terms of these invariants. To calculate differential invariants of a system the following systems of PDEs must be solved

where I, I₁ and I₂ are smooth functions of (x, t, u), (x, t, u, u_x, u_t) and (x, t, u, u_x, u_t, u_xx, u_xt, u_tt) respectively. The solutions of (17) for (1) and (2) are shown in 9 and 10. Note in Table 10, m = LambertW(−(2t+x+6)6exp(−(2t+x+6)6)) that LambertW(x).exp(LambertW(x)) = x.

5 Conservation laws for BBM and MBBM equations

A conservation law of differential equation Δ(x, u⁽ⁿ⁾) = 0 is a divergence expression

which vanishes for all solutions of the system. Here P = (P₁(x, u⁽ⁿ⁾),..., P_p(x, u⁽ⁿ⁾)) is a p-tuple of smooth functions of x = (x¹,..., x^p), u = (u¹,..., u^q) and derivative of u on the jet space X × U⁽ⁿ⁾; this requires

P(x, u⁽ⁿ⁾) to be constant for all solutions of the system. Thus a conservation law for a system is equivalent to the classical notion of a first integral or constant of the motion of the system. This definition in characteristic form is

L-tuple Q is characteristic of this conservation law. In general, the characteristics are determined up to the equivalence and are not unique. In other words, if Q and Q¯ are characteristics of one conservation law, then also is Q−Q¯ and is called trivial characteristic, and two characteristics Q and Q−Q¯ are equivalent. The Euler operator is defined as bellow:

To find all local conservation law multipliers for the system (1) and (2) of the form ξ(x, t, u, u_x, u_xx, u_xt, u_tt) and η(x, t, u, u_x, u_xx, u_xt, u_tt), the determining equations (18) become

where U(x, t) is an arbitrary function. The solutions of (19) and (20) or the set of local multipliers of all nontrivial local conservation laws for BBM and MBBM equations are

where c_i, i = 1, 2, 3, 4, 5 are arbitrary constants. So local multipliers are given by:

Each pair of the local multipliers ξ and φ determine a nontrivial local conservation law for BBM and MBBM equations, respectively, that in characteristic form are

and

To calculate Ψ₁, φ₁, Ψ₂ and φ₂, we must apply integration of an expression in multi-dimensions involving arbitrary functions and its derivatives, which is very complicated, but we can do it by the homotopy operator tool (explicit formula) [15].

The results of applying (21) for BBM and MBBM are in Table 11.

Thus we have a set of conservation laws for BBM and MBBM equations, as shown below:

and