Direct and inverse problems on conservation laws

Transcription

1 Direct and inverse problems on conservation laws Roman O. Popovych Wolfgang Pauli Institute, Vienna, Austria & Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine joint work with Alexander Bihlo, Artur Sergyeyev and Alexey Shevyakov Eighth Workshop Group Analysis of Differential Equations and Integrable Systems (June 12 17, 2016, Larnaca, Cyprus) R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 1 / 31

2 Direct problem on CLs Conservation laws (CLs) play an important role in mathematical physics and have different applications in numerical analysis, the integrability theory, the theory of normal forms and asymptotic integrability, etc. Direct problem on CLs Given a system of DEs, find the space of its CLs or at least a subspace of this space with certain additional constraints, e.g., on order of CLs. Standard tools for the solution of the direct problem on CLs: Noether s theorem different variations of the direct method techniques based on co-symmetries special techniques for integrable systems For a class of (systems of) differential equations, one should tackle the direct problem on conservation laws as classification problem. direct problem on CLs direct problem on symmetries? inverse problem on symmetries R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 2 / 31

3 Inverse problem on CLs: general discussion Various applications, e.g., the geometry-preserving parameterization of DEs need solving the inverse problem. The only paper with no citations! Galindo A., An algorithm to construct evolution equations with a given set of conserved densities, J. Math. Phys. 20 (1979), Inverse problem on CLs. Naive formulation Derive the general form of systems of DEs with a prescribed set of CLs. More generally, the inverse problem on CLs can be interpreted as the study of properties of DEs for which something is known about their CLs. Solving the inverse problem on conservation laws for a class of differential equations may help in the solution of the direct problem for this class.? What data on CLs should be given? R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 3 / 31

4 Systems of differential equations L: L(x, u (ρ) ) = 0 L 1 = 0,..., L l = 0 for u = (u 1,..., u m ) of x = (x 1,..., x n). u (ρ) = the set of all the derivatives of the functions u w.r.t. x of order no greater than ρ, including u as the derivatives of the zero order. L (k) = the set of all algebraically independent differential consequences that have, as differential equations, orders no greater than k. the jet space J (k) : x, uα a α u a x α, xn αn α = (α 1,..., α n), α i N {0}, α : = α α n k, i = 1,..., n, a = 1,..., m We associate L (k) with the manifold L (k) determined by L (k) in J (k). a differential function G[u] = a smooth function on a domain in J (k) a smooth function of x and a finite number of derivatives of u ord G = the maximal order of derivatives involved in G, ord G = if G depends only on x R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 4 / 31

5 Conserved currents Empiric definition of CLs: A conservation law of L is a divergence expression Div F := D i F i [u] which vanishes for all solutions of L. Definition A (local) conserved current of the system L is an n-tuple F = (F 1 [u],..., F n [u]) for which the total divergence Div F := D i F i vanishes for all solutions of L, i.e. DivF L = 0. D i = xi + uα+δ a i u a α, uα a α u a x α, u a xn αn α+δ i ua α, x i α = (α 1,..., α n), α i N {0}, α : = α α n, i = 1,..., n, a = 1,..., m n = 1: conserved current = first integral R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 5 / 31

6 Trivial conserved currents ˆF L = 0 = DivˆF L = 0, null divergence: DivˇF 0 = DivˇF L = 0 Definition A conserved current F is trivial if F = ˆF + ˇF, where ˆF L = 0 and DivˇF 0. The triviality concerning with vanishing conserved currents on solutions of the system can be easily eliminated by confining on the manifold of the system, taking into account all its necessary differential consequences. Lemma characterizing all null divergences The n-tuple F = (F 1,..., F n ), n 2, is a null divergence (Div F 0) iff there exist differential functions v ij = v ij [u], i, j = 1,..., n, such that v ij = v ji and F i = D j v ij. The functions v ij are called potentials corresponding to the null divergence F. n = 1: any null divergence is constant. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 6 / 31

7 Equivalence of conserved currents and spaces of CLs Definition Conserved currents F and F are called equivalent if F F is a trivial conserved current. For any system L of differential equations CC(L) = the linear space of conserved currents of L CC 0(L) = the set of trivial conserved currents of L, a linear subspace of CC(L) CL(L) = CC(L)/ CC 0(L) = the set of equivalence classes of CC(L) Definition The elements of CL(L) are called conservation laws of the system L, and the whole factor space CL(L) is called as the space of conservation laws of L. For F CC(L) and F CL(L): F F F is a conserved current of the conservation law F R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 7 / 31

8 Characteristics of CLs Let the system L be weakly totally nondegenerate, i.e. for any k ρ = ord L L (k) is of maximal rank in each point of L (k) and L is locally solvable in each point (x 0, u 0 (k)) L (k), which means that there exists a local solution u = u(x) of L with u (k) (x 0 ) = u 0 (k). Applying the Hadamard lemma to the definition of conserved current and integrating by parts imply, up to equivalence of conserved currents, Div F = λ µ L µ (1) with differential functions λ µ, µ = 1,..., l. Definition The equality (1) and the l-tuple λ = (λ 1,..., λ l ) are called the characteristic form and the characteristic of the conservation law F F, respectively. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 8 / 31

9 Equivalence of characteristics of CLs λ is trivial if λ L = 0. λ and λ are equivalent iff λ λ is a trivial characteristic. Ch(L) = the linear space of characteristics of CLs of L Ch 0(L) = the set of trivial characteristics, a linear subspace in Ch(L) Ch f (L) = Ch(L)/ Ch 0(L) = the set of equivalence classes of Ch(L) Theorem Let L be a normal system of DEs = system reduced to Cauchy Kovalevskaya form by a point transformation of independent variables = a totally nondegenerate analytical system of the same number of DEs as dependent variables. Then the equivalence of CL characteristics is well consistent with the equivalence of conserved currents. CL characteristics of L are equivalent if and only if the respective conserved currents are equivalent. representation of CLs of L in the characteristic form generates a one-to-one linear mapping between CL(L) and Ch f (L). R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 9 / 31

10 Criteria for characteristics f = Div F E(f ) = 0. The Euler operator E = (E 1,..., E m ): E a = ( D) α u a α, a = 1, m, α = (α 1,..., α n) (N {0}) n, ( D) α = ( D 1) α 1... ( D m) αm. Necessary and sufficient condition on λ Ch(L): Div F = λ µ L µ E(λ µ L µ ) = D λ (L) + D L (λ) = 0 (2) D λ and D L are matrix differential operators adjoint to the Fréchet derivatives Dλ and D L, i.e. ( ) ( ( )) λ µ Dλ(L) = D α L µ, uα a D λ (L) = ( D) α λ µ L µ. uα a Since D λ (L) L = 0 = a necessary condition for λ Ch(L): D L (λ) L = 0. (3) Condition (3) are considered as adjoint to the criteria D L (η) L = 0 for infinitesimal invariance of L w.r.t. an evolutionary vector field having the characteristic η = (η 1,..., η m ). That is why solutions of (3) are called cosymmetries or adjoint symmetries. L is an Euler Lagrange system = D L = DL = symmetry = cosymmetry CL characeristic = variational symmetry R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 10 / 31

11 System equivalence and CL characteristics System equivalence We call systems of differential equations equivalent if they are defined on the same space of independent and dependent variables and can be obtained from each other using the following operations: recombining equations with coefficients that are differential functions and constitute a nondegenerate matrix (this gives so-called linearly equivalent systems); supplementing a system with its differential consequences or excluding equations that are differential consequences of other equations. Equivalent systems have the same set of solutions. In contrast to conserved currents, symmetries and cosymmetries, CL characteristics are not class invariants under the system equivalence. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 11 / 31

12 Choice of data Conserved currents. Counter-arguments: Complex structure of the equivalence F F = F + ˆF + ˇF, where ˆF L = 0 and DivˇF 0 The determination whether a conserved current is trivial or not even for a fixed system of DEs is not so obvious and reduces, at least implicitly, to the consideration of related characteristics. Conserved currents may be trivial for some systems from a class and nontrivial for another systems from the same class. Weak consistency Not all n-tuples of differential functions (e.g. tuples with single nonvanishing components) are suitable candidates for conserved currents. Different candidates for conserved currents may be inconsistent. Fixing conserved currents considerably restricts the systems form. = Even minor variations within systems form lead to varying the form of conserved currents. =! Entire conserved currents are not appropriate initial data for the inverse problem on CLs! R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 12 / 31

13 Choice of data Characteristics. Arguments: Their equivalence is much simpler. They are more stable under varying systems within a class of differential equations. Obstacles (can be overcome): They are defined only for systems that are weakly totally nondegenerate. There may be no 1-to-1 correspondence between equivalence classes of CL characteristics and conserved currents. They are not invariant under system equivalence. Confining characteristics to solutions of the considered systems is not trivial. If representatives of the equivalence classes of DE systems under consideration are fixed, CL characteristics are appropriate initial data then!! for the inverse problem on CLs R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 13 / 31

14 Choice of data Form for systems. Definition A system of PDEs L is of extended Kovalevskaya form if it can be written as u a r aδ n := ra u a x ra n = H j (x, ũ(r) ), a = 1,..., m. (4) where 0 r a r and ũ(r) denotes all derivatives of u w.r.t. x up to order r, where each u b is differentiated with respect to x n at most r b 1 times, b = 1,..., m. The extended Kovalevskaya form is less restrictive than the usual Kovalevskaya form. Zero-order derivatives may appear in the left-hand side of equations. There are no restrictions on the order of derivatives w.r.t. x 1,..., x n 1 depending on r a s. Systems of the form (4) with all r a > 0 are called normal systems Martínez-Alonso L., On the Noether map, Lett. Math. Phys. 3 (1979), Cauchy Kowalevsky systems in a weak sense (resp., pseudo CK systems in short) Tsujishita T., On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 14 / 31

15 Reformulation of inverse problem on CLs Inverse problem on CLs. Enhanced naive formulation Derive the general form of systems of DEs with a prescribed set of CL characteristics. Inverse problem on CLs. Extended Kovalevskaya form Given a class of systems of DEs in the extended Kovalevskaya form, find its subclass of systems admitting a prescribed set of reduced CL characteristics (resp. reduced densities). Inverse problem on CLs. Low-order characteristics Derive the form of systems of DEs with prescribed set of low-order (in particular zero-order) CL characteristics. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 15 / 31

16 Inverse problem on first integrals for ODEs Problem Find the general form of systems of ordinary differential equations that admit a prescribed set of integrating factors. Theorem An ODE H(x, u, u,..., u (n) ) = 0, H u (n) 0, admits p totally linearly independent first integrals with integrating factors γ s iff Here H = DT[γ 1,..., γ p ] G for some G=G[u]. W (γ 1,..., γ p ) = det(d s 1 x γ s ) p s,s =1 denotes the Wronskian of γ1,..., γ p w.r.t. the operator of total derivative D x, DT[γ 1,..., γ p ] is the operator in D x associated with the Darboux transformation DT[γ 1,..., γ p ]G = W (γ1,..., γ p, G) W (γ 1,..., γ p ). denotes formally adjoint, Q = Q j D j x, Q = ( D x) j Q j. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 16 / 31

17 Preliminaries on evolution equations An evolution equation in two independent variables, E : u t = H(t, x, u 0, u 1,..., u n), n 2, H un 0, where u j j u/ x j, u 0 u, and H uj = H/ u j, u x = u 1, u xx = u 2, and u xxx = u 3. D t = t + u t u + u tt ut + u tx ux +, D x = x + u x u + u tx ut + u xx ux + the total derivatives w.r.t. the variables t and x. The subscripts like t, x, u, u x, etc. stand for the partial derivatives in the respective variables. Without loss of generality, for any evolution equation the associated quantities like symmetries, cosymmetries, densities and characteristics of CLs can be assumed independent of the t-derivatives or mixed derivatives of u. We refer to a (smooth) function of t, x and a finite number of u j as to a differential function. Given a differential function f, its order (denoted by ord f ) is the greatest integer k such that f uk 0 but f uj = 0 for all j > k. For f = f (t, x) we assume that ord f =. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 17 / 31

18 Contact transformations of evolution equations The contact transformations mapping a (fixed) equation E: u t = H into another equation Ẽ: ũ t = H are well known [Magadeev,1993] to have the form t = T (t), x = X (t, x, u, u x), ũ = U(t, x, u, u x). The nondegeneracy assumptions: T t 0, rank ( Xx X u X ux U x U u U ux ) = 2 The contact condition: (U x + U uu x)x ux = (X x + X uu x)u ux = ũ x = V (t, x, u, u x), where Ux + Uuux X x + X uu x or V = Uux X ux if X x + X uu x 0 or X ux 0, respectively V = = ũ k k ũ x k = ( ) k 1 1 U u X uv D xx Dx Ut XtV V, H = H + T t T t The equiv. group Gc generates the whole set of admissible contact transformations in the class, i.e., the class is normalized [ROP & Kunzinger & Eshraghi,2010] w.r.t. contact transformations. Proposition The class of evolution equations is contact-normalized. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 18 / 31

19 Some basic results on conservation laws A conserved current of E is a pair of differential functions (ρ, σ) satisfying the condition D tρ + D xσ = 0 mod Ě, where Ě = E and all its differential consequences. Here ρ is the density and σ is the flux for the conserved current (ρ, σ). Let δ δu = ( D x) i ui, f = f ui Dx, i f = i=0 i=0 ( D x) i f ui denote the operator of variational derivative, the Fréchet derivative of a differential function f, and its formal adjoint, respectively. A conserved current (ρ, σ) is called trivial if D tρ + D xσ = 0 on the entire jet space ρ Im D x, i.e., there exists a differential function ζ: ρ = D xζ. Two conserved currents are equivalent if they differ by a trivial conserved current. A CL of E is an equivalence class of conserved currents of E. The set CL(E) of CLs of E is a vector space, and the zero element of this space is the CL being the equivalence class of trivial conserved currents. i=0 R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 19 / 31

20 Some basic results on conservation laws For any L CL(E) there exists a unique differential function γ called the characteristic of L such that for any conserved current (ρ, σ) L there exists a trivial conserved current ( ρ, σ): D t(ρ + ρ) + D x(σ + σ) = γ(u t H). The characteristic γ of any CL is a cosymmetry, i.e., it satisfies the equation D tγ + H γ = 0 mod Ě, or equivalently, γ t + γ H + H γ = 0. The characteristic of the CL associated with (ρ, σ) is γ = δρ/δu. Therefore, a cosymmetry γ to be a characteristic of a CL iff γ = γ. ρ t + γh = D x ˆσ for some ˆσ. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 20 / 31

21 Inverse problem on CLs for evolution equations Theorem An evolution equation E : u t = H(t, x, u 0, u 1,..., u n), n 2, H un 0, admits p linearly independent CLs with densities ρ s and characteristics γ s = δρ s /δu iff H = DT[γ 1,..., γ p ] G for some G = G[u]. Here p s=1 ( ) W (γ DT[γ 1,..., γ s,..., γ p ] 1,..., γ s,..., γ p ) ρ s W (γ 1,..., γ p t ) W (γ 1,..., γ p ) = det(d s 1 x γ s ) p s,s =1 denotes the Wronskian of γ1,..., γ p w.r.t. the operator of total derivative D x, DT[γ 1,..., γ p ] is the operator in D x associated with the Darboux transformation DT[γ 1,..., γ p ]G = W (γ1,..., γ p, G) W (γ 1,..., γ p ). denotes formally adjoint, Q = Q j D j x, Q = ( D x) j Q j. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 21 / 31

22 Inverse problem on CLs for evolution equations F = DT[γ 1,..., γ p ] H p DT[γ 1,..., γ s,..., γ p ] (ϕ s ρ s t), (5) s=1 where ϕ s := ( 1) p s W (γ 1,..., γ s,..., γ p ) W (γ 1,..., γ p ) are adjoint diff. functions to γ 1,..., γ p, DT[γ 1,..., γ p ] = ( 1) p DT[ϕ 1,..., ϕ p ]. F = ( 1) p W ( ˆϕ 1,..., ˆϕ p, ( Ĥ) p W (γ1,..., γ p ) ) p DT[γ 1,..., γ s,..., γ p ] ˆϕ s ρ s t W (γ 1,..., γ p ), (6) s=1 where ˆϕ s := ( 1) p s W (γ 1,..., γ s,..., γ p ). (5) depends rather on ρ 1,..., ρ p than on {ρ 1,..., ρ p }. For fixed F and ρ 1,..., ρ p, H is defined up to H H + ζ s (t)ϕ s. If ρ s = ρ s + D xθ s for some diff. functions θ s, then H = H + (ζ s θt s )ϕ s for some ζ s = ζ s (t). Let q := max ord γ s ( {, 0} 2N) and ρ s : ord ρ s max(0, q + p 1). s (For best ρ s : ord ρ s = max(0, 1 ord 2 γs ).) Then ord Ĥ max(n p, q + p 2). If max ord γ s n 2p + 2, then ord Ĥ n p. s If max ord γ s n 2p, then ord H = ord Ĥ = n p. s R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 22 / 31

23 Second-order evolution equation E: u t = F (t, x, u, u x, u xx) Density order of CLs 1. Density order of CLs = 1 if F u2 u 2 = 0. Theorem [Bryant&Griffiths, 1995, F t = 0; ROP&Samoilenko, 2008, gen. case] dim CL(E) {0, 1, 2, } for any second-order (1 + 1)D evolution equation E. The equation E is (locally) reduced by a contact transformation 1 to the form u t = D xg(t, x, u, u x), where G ux 0, iff dim CL(E) 1; 2 to the form u t = D 2 x H(t, x, u), where H u 0, iff dim CL(E) 2; 3 to a linear evolution equation iff dim CL(E) =. If the equation E is quasi-linear (i.e., F uxx u xx = 0) then the contact transformation is a prolongation of a point transformation. Corollary The space of CLs of u t = D 2 x H(t, x, u) is 2D iff H is not linear in u. The space of CLs of u t = D xg(t, x, u, u x) is 1D if G is not a fractionally linear in u x. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 23 / 31

24 CLs of even-order evolution equations Let E be a (1 + 1)D even-order (n 2N) evolution equation. Lemma [Ibragimov, 1983; Abellanas & Galindo, 1979; Kaptsov, 1980] For any CL F of E we have ord char F n. Theorem Cosym f (E) = Ch f (E). Theorem If dim CL(E) p, then ord γ n 2p + 2 for any γ Ch f (E) (modg cont if 2p n + 2). Roughly speaking, the greater dimension of the space of CLs, the lower upper bound for orders of CL characteristics. In particular, ord γ = mod G cont if 2p > n + 2. Corollary dim CL(E) > n iff E is linearizable by a contact transformation and thus dim CL(E) =. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 24 / 31

25 CLs of fourth-order evolution equations E: u t = F (t, x, u, u 1, u 2, u 3, u 4), F u4 0 dim CL(E) 1: F = Dx ˆσ1 + ρ 1 t γ 1, ord ρ 1 2, ord ˆσ 1 3, γ 1 = δρ1 δu 0. dim CL(E) 2, modg cont: F = D xg, G = Dx ˇσ1 + ρ 1 t D xγ 1, ord ρ 1 1, ord ˇσ 1 2, γ 1 = δρ1 δu, Dxγ1 0 (ρ 2 = u, γ 2 = 1). dim CL(E) 3, modg cont: F = D 2 x G, G = Dx σ1 + ρ 1 t D 2 x ρ 1 u ord ρ 1 = 0, ord σ 1 1, D 2 x ρ 1 u 0 (ρ 2 = u, γ 2 = 1, ρ 3 = xu, γ 3 = x). dim CL(E) 4, modgcont: p ( ) W (γ F = DT[γ 1,..., γ 4 ] H DT[γ 1,..., γ s,..., γ 4 ] 1,..., γ s,..., γ p ) γ s W (γ 1,..., γ 4 t u, ) s=1 ord γ s =, γ 3 = x, γ 4 = 1, ord H = 0. dim CL(E) > 4, modg cont: E is a linear equation = dim CL(E) =., R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 25 / 31

26 Inverse problem on CLs with arbitrary parametric functions Let L: L[u] = 0 be a PDE for a single unknown function u of x = (x 1,..., x n). Theorem L admits an arbitrary function h = h(x 1) as a CL characteristic iff L[u] = D 2F 2 [u] + + D nf n [u] for some differential functions F 2,..., F n. Corollary If L admits N + 1 linearly independent CLs with characteristics depending only on x 1, h s = h s (x 1), s = 0,..., N, where N = max{α 1 L uα 0}, then the above representation holds and hence L admits CLs with characteristic being an arbitrary function of x 1. Corollary L admits an arbitrary h = h(x 1,..., x q) with q < n as a CL characteristic iff L = D q+1f q+1 [u] + + D nf n [u], for a tuple of n q differential functions F q+1,..., F n. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 26 / 31

27 Inverse problem on CLs with arbitrary parametric functions Theorem L admits characteristics of the form h(x 1) + i>1 f i (x 1)x i with arbitrary functions h = h(x 1) and f i = f i (x 1), i = 2,..., n, iff L = D i D j F ij [u]. 2 i j n for some differential functions F ij, 2 i j n. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 27 / 31

28 Conservative parameterization for the vorticity equation L: ζ t + ψ xζ y ψ y ζ x = 0, ζ := ψ xx + ψ yy. Here ψ = ψ(t, x, y) is the stream function, ζ is the vorticity, and we use a specific notation for the variables, t, x, y and ψ instead of x 1, x 2, x 3 and u. Ch f (L) h(t), f (t)x, g(t)y, ψ. We split ψ as well as ζ into a resolved (mean) parts ψ and ζ and an unresolved (sub-grid scale) parts ψ and ζ, i.e. ψ = ψ + ψ and ζ = ζ + ζ. The Reynolds averaging rule, ab = ā b + a b, gives the equation for the mean part, L: ζt + ψ x ζy ψ y ζx = ζ v, ζ := ψxx + ψ yy. The problem is that the right-hand side of the equation involves the divergence of the unknown vorticity flux ζ v for which no equation is given. In other words, the equation is underdetermined, which is the usual closure problem of fluid mechanics. We use the local closure ζ v = V [ ψ] of the equation, which gives ζ t + ψ xζ y ψ y ζ x = V [ψ], ζ := ψ xx + ψ yy, where bars are omitted. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 28 / 31

29 Conservative parameterization for the vorticity equation L: ζ t + ψ xζ y ψ y ζ x = V [ψ], ζ := ψ xx + ψ yy Ch f (L) h(t), f (t)x, g(t)y, ψ. Proposition If the unclosed vorticity flux ζ v is parameterized by V [ψ] in the Reynolds averaged vorticity equation, where V [ψ] is given through V [ψ] = D 2 x(ψ yy P 2 ψ xy P 3 ) + D xd y (ψ xxp 3 ψ yy P 1 ) + D 2 y (ψ xy P 1 ψ xxp 2 ) + (D 2 x + D 2 y )(ζ Div S + 2S 1 ζ t + 2S 2 ζ x + 2S 3 ζ y ) for some differential functions P i and S i, i = 1, 2, 3, of ψ the resulting closed equation L possesses the conservation laws associated with characteristics h(t), f (t)x, h(t)y and ψ. That is, the closed equation will preserve generalized circulation, generalized momenta in x- and y-direction and energy. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 29 / 31

30 CLs of abnormal systems Definition A system L is called abnormal if it has an identically vanishing differential consequence such that the corresponding tuple of differential operators acting on system s equations does not vanish on the manifold L ( ). Example Maxwell s equations give an example of an essentially abnormal system. E t = B, E = 0, = (E t B) D t E 0 B t = E, B = 0, = (B t + E) D t B 0 Theorem (generalized Noether s second theorem) The following statements on a weakly totally nondegenerate system L of DEs are equivalent: 1) the system L is abnormal; 2) it possesses a family of CLs with a characteristic λ[u, h] that depends, additionally to u as differential functions, on an arbitrary function h = h(x) and whose Fréchet derivative with respect to h does not vanish on solutions of L for a value of the parameter-function h; 3) it possesses a trivial conserved current corresponding to a nontrivial characteristic. R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 30 / 31

31 Thanks Thank you for your attention! R.O. Popovych GADEIS-VIII Direct and inverse problems on CLs 31 / 31