Invariants and Symmetries for Partial Differential Equations and Lattices

Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as Mathematica. Invariants and symmetries are shown for several well-known equations. Our Mathematica package allows one to automatically compute invariants and symmetries. Applied to systems with parameters, the package determines the conditions on these parameters so that a sequence of invariants or symmetries exists. The software can thus be used to test the integrability of model equations for wave phenomena. 1 The Key Concept: Scaling Invariance The ubiquitous Korteweg-de Vries (KdV) equation from soliton theory, ut = 6uux + u3x, (1) is invariant under the dilation (scaling) symmetry (t, x, u) → (λt, λx, λu), where λ is an arbitrary parameter. Obviously, u corresponds to two derivatives in x, i.e. u ∼ ∂/∂x. Introducing weights, w(u) = 2 if we set w(∂/∂x) = 1. Similarly, ∂/∂t ∼ ∂/∂x, thus w(∂/∂t) = 3. The rank of a monomial equals the sum of all of its weights. Observe that (1) is uniform in rank since all the terms have rank R = 5. Likewise, the Volterra lattice, which is one of the discretizations of (1), u̇n = un (un+1 − un−1), (2) is invariant under (t, un) → (λ t, λun). So, un ∼ d/dt, or w(un) = 1 if we set w(d/dt) = 1. Every term in (2) has rank R = 2, thus (2) is uniform in rank. Scaling invariance, which is a special Lie-point symmetry, is common to many integrable nonlinear partial differential equations (PDEs) such as (1), and nonlinear differentialdifference equations (DDEs) like (2). Both equations have infinitely many polynomial invariants [1, 5] and symmetries [3]. In this paper we show how to use the scaling invariance to explicitly compute polynomial invariants and symmetries of PDEs and DDEs. 2 Computation of Invariants For PDE systems as in Table 1, the conservation law Dtρ = DxJ connects the invariant (conserved density) ρ and the associated flux J. As usual, Dt and Dx are total derivatives. Most polynomial density-flux pairs only depend on u,ux, etc. (not explicitly on t and x). Integration of the conservation law with respect to x yields that P = ∫ +∞ −∞ ρ dx is constant in time, provided J vanishes at infinity. P is a conserved quantity. Research supported in part by NSF under Grant CCR-9625421. Colorado School of Mines, Dept. of Mathematical and Computer Sciences, Golden, CO 80401-1887 The first three (of infinitely many) conservation laws for (1) are Dt(u) = Dx(3u 2 + u2x), Dt(u ) = Dx(4u 3 − u2x + 2uu2x), (3) Dt(u 3 − 1 2 u2x) = Dx( 9 2 u − 6uu2x + 3u u2x + 1 2 u22x − uxu3x). (4) The densities ρ = u, u, u − 1 2 u2x have ranks 2, 4 and 6, respectively. Conserved densities of PDEs like (1) can be computed as follows: • Require that each equation in the PDE system is uniform in rank. Solve the resulting linear system to determine the weights of the dependent variables. For (1), solve w(u) + w(∂/∂t) = 2w(u) + 1 = w(u) + 3, to get w(u) = 2 and w(∂/∂t) = 3. • Select the rank R of ρ, say, R = 6. Make a linear combination of all the monomials in the components of u and their x-derivatives that have rank R. Remove ‘equivalent’ monomials, that is, those that are total x-derivatives (like u4x) or differ by a total xderivative. For example, uu2x and u 2 x are equivalent since uu2x = 1 2 (u)2x − u 2 x. For (1), one gets ρ = c1u 3 + c2u 2 x of rank R = 6. • Substitute ρ into the conservation law, eliminate all t-derivatives, and require that the resulting expression is a total x-derivative. Apply the Euler operator [1] to avoid integration by parts. The remaining part must vanish identically. This yields a linear system for the constants ci. Solve the system. For (1), one gets c1 = 1, c2 = −1/2. See [1] for the complete algorithm and its implementation. See [4] for an integrated Mathematica Package that computes invariants (and also symmetries) of PDEs and DDEs. Table 1 Invariants and Symmetries Continuous Case (PDEs) Semi-discrete Case (DDEs) System ut = F(u,ux,u2x, ...) u̇n = F(...,un−1,un,un+1, ...) Cons. Law Dtρ = DxJ ρ̇n = Jn − Jn+1 Symmetry DtG=F (u)[G]= ∂ ∂ǫ F(u+ ǫG)|ǫ=0 DtG=F (un)[G]= ∂ ∂ǫ F(un + ǫG)|ǫ=0 For DDEs like (2), compute the weights in a similar way. Determine all monomials of rank R in the components of un and their t-derivatives. Use the DDE to replace all the t-derivatives. Monomials are ‘equivalent’ if they belong to the same equivalence class of shifted monomials. Keep only the main representatives (centered at n) of the various classes. Combine these representatives linearly with coefficients ci, and substitute the form of ρn into the conservation law ρ̇n = Jn−Jn+1. Remove all t-derivatives and pattern-match the resulting expression with Jn−Jn+1. Set the non-matching part equal to zero, and solve the linear system for the ci. Determine Jn from the pattern Jn − Jn+1. For (2), the first three (of infinitely many) densities ρn are listed in Table 2. Details about the algorithm and its implementation are in [2, 4, 5]. 3 Computation of Symmetries As summarized in Table 1, G(x, t,u,ux,u2x, ...) is a symmetry of a PDE system iff it leaves it invariant for the change u → u + ǫG within order ǫ. Hence, Dt(u + ǫG) = F(u + ǫG) must hold up to order ǫ. Thus, G must satisfy the linearized equation DtG = F (u)[G], where F is the Fréchet derivative: F(u)[G] = ∂ ∂ǫ F(u+ ǫG)|ǫ=0. Using the dilation invariance, generalized symmetries G can be computed as follows: • Determine the weights of the dependent variables as in Section 2. • Select the rank R of the symmetry. Make a linear combination of all the monomials involving u and its x-derivatives of rank R. For example, for (1), G = c1 u ux + c2 uxu2x + c3 uu3x + c4 u5x is the generalized symmetry of rank R = 7. • Compute DtG. Use the PDE system to remove all t-derivatives. Equate the result to the Fréchet derivative F(u)[G]. Treat the different monomial terms in u and its x-derivatives as independent, to get the linear system for ci. Solve that system. For (1), one obtains G = 30uux + 20uxu2x + 10uu3x + u5x. (5) The symmetries of rank 3, 5, and 7 are listed in Table 2. They are the first three of infinitely many. For DDEs like (2), G(...,un−1,un,un+1, ...) is a symmetry iff Table 2 Prototypical Examples Korteweg-de Vries Equation Volterra Lattice Equation ut = 6uux + u3x u̇n = un (un+1 − un−1) Invariants ρ=u ρ = u ρn = un ρn = un( 1 2 un + un+1) ρ = u − 1 2 u2x ρn= 1 3 u3n+unun+1(un+un+1+un+2) Symmetries G = ux G = 6uux + u3x G = unun+1(un + un+1 + un+2) G=30uux+20uxu2x+10uu3x+u5x −un−1un(un−2 + un−1 + un) the infinitesimal transformation un → un + ǫG leaves the DDE invariant within order ǫ. Consequently, G must satisfy dG dt = F(un)[G], where F ′ is the Fréchet derivative, F(un)[G] = ∂ ∂ǫ F(un + ǫG)|ǫ=0. Algorithmically, one performs the following steps: First compute the weights of the variables in the DDE. Determine all monomials of rank R in the components of un and their t-derivatives. Use the DDE to replace all the t-derivatives. Make a linear combination of the resulting monomials with coefficients ci. Compute DtG and remove all u̇n−1, u̇n, u̇n+1, etc. Equate the resulting expression to the Fréchet derivative F(un)[G] and solve the system for the ci, treating the monomials in un and its shifts as independent. Details are in [2, 5]. For (2), the symmetry G of rank R = 3 is listed in Table 2. There are infinitely many symmetries, all with different ranks. See [3] for the complete algorithm and its implementation in Mathematica, and [4] for an integrated Mathematica Package that computes symmetries of PDEs and DDEs. Notes: (i) If PDEs or DDEs are of second or higher order in t, like the Boussinesq equation in [1], we assume that they can be recast in the form given in Table 1. (ii) A slight modification of the methods in Section 2 and 3 allows one to find invariants and symmetries that explicitly depend on t and x. See next section for an example. (iii) Applied to systems with free parameters, the linear system for the ci will depend on these parameters. A careful analysis of the eliminant leads to conditions on these parameters so that a sequence of invariants or symmetries exists. (iv) For equations that lack uniformity in rank, we introduce one or more auxiliary (constant) parameters with weights. After the form of the invariant or symmetry is determined, the auxiliary parameters can be reset to one. (v) Higher-order symmetries, such as (5) lead to new integrable evolution equations. For example, ut = 30u ux + 20uxu2x + 10uu3x + u5x is the completely integrable fifth-order KdV equation due to Lax. Details about these 5 items are given in [1, 2, 3, 5]. 4 Examples 4.1 Vector Modified KdV Equation In [7, Eq. (4)], Verheest investigates the integrability of a vector form of the modified KdV equation (vmKdV), which upon projection, reads ut + 3u ux + v ux + 2uvvx + u3x = 0, vt + 3v vx + u vx + 2uvux + v3x = 0. (6) With our software InvariantsSymmetries.m [4] we computed the following invariants: ρ1 = u, ρ2 = v, ρ3 = u 2 + v, (7) ρ4 = 1 2 (u + v) − (u2x + v 2 x), (8) ρ5 = 1 3 x(u + v)− 1 2 t(u + v) + t(u2x + v 2 x). (9) Note that the latter invariant depends explicitly on x and t. Verheest [7] has shown that (6) is non-integrable for it lacks a bi-Hamiltonian structure and recursion operator. We were unable to find additional polynomial invariants. Polynomial higher-order symmetries for (6) do not appear to exist. 4.2 Extended Lotka-Volterra Equation Itoh [6] studied the following extended version of the Lotka-Volterra equation (2),