Adv Syst Sci Appl 2022; 04:1–10 Published online at https://ijassa.ipu.ru. Dynamics of Evolutionary Equations with 1+2 Independent Variables Alexei G. Kushner1,3*, Elena N. Kushner2, Tao Sinian1 1Lomonosov Moscow State University, Moscow, Russia 2Moscow State Technical University of Civil Aviation, Moscow, Russia 3Moscow Pedagogical State University, Moscow, Russia Abstract: The paper is devoted to a method for constructing exact solutions of evolutionary differential equations with two space variables.The method uses the theory of symmetries of completely integrable distributions. As an example, we consider a linear parabolic equation that arises in filtration theory, thermodynamics, and mathematical biology. Dynamics are calculated for this equation and classes of their exact solutions are constructed. Keywords: integrability, completely integrable distributions, symmetry, evolutionary equa- tions, filtration, thermodynamics 1. INTRODUCTION Consider the following partial differential equation: ∂u ∂t = f ( x, u, ∂|σ|u ∂xσ ) , (1.1) where σ = (σ1, . . . , σn) is a multi-index whose elements are non-negative integers, |σ| = σ1 + · · ·+ σn, x = (x1, . . . , xn), ∂|σ|u ∂xσ = ∂|σ|u ∂xσ1 1 , . . . ∂xσn n . Here the variables x1, . . . , xn are called spatial, and the variable t is called temporal. For equations (1.1) with one spatial variable, the method of finite-dimensional dynamics was proposed in [1, 2]. This method was further developed in [3, 4]. It allows us to select finite-dimensional submanifolds of solutions from the infinite set of all solutions of evolutionary equations. These submanifolds are “numbered” by solutions of ordinary differential equations. However, this method does not allow direct generalization to equations with several spatial variables. In this case, using of ordinary differential equations are not enough anymore. In [5], a method was proposed in which systems of partial differential equations of finite type are used instead of ordinary differential equations. ∗Corresponding author: kushner@physics.msu.ru 2 A. KUSHNER, E. KUSHNER, S. TAO 2. SYMMETRIES OF COMPLETELY INTEGRABLE DISTRIBUTIONS Here we give the necessary information about a symmetry of distributions (see [6, 7]). Let M be a smooth manifold and P a completely integrable distribution on M . A vector field X on M is called an infinitesimal symmetry of a distribution P if the local group of translations Φt along trajectories of X preserves this distribution, i.e. (Φt)∗ (P) = P . In what follows, for brevity, an infinitesimal symmetry will be called symmetry. The set of all symmetries of the distribution P forms the Lie R-algebra SymP with respect to the Lie bracket. This Lie algebra contains the ideal of characteristic symmetries CharP that consists of vector fields lying in the distribution P . The quotient Lie algebra ShufP = SymP/CharP is called the Lie algebra of shuffling symmetries of the distribution P . Elements of this Lie algebra “shuffle” the maximal integral manifolds of the distribution. 3. SYMMETRIES OF FINITE TYPE DIFFERENTIAL EQUATIONS A system of differential equations is called a system of finite type if the space of its solutions is finite-dimensional [6]. Examples of systems of finite type are systems of ordinary differential equations. A more important example is given by overdetermined systems of differential equations. For simplicity, we consider the case n = 2 and finite type equations of the second order. Consider the following overdetermined system of three differential equations ∂2v ∂x2 = P ( x, y, v, ∂v ∂x , ∂v ∂y ) , ∂2v ∂x∂y = Q ( x, y, v, ∂v ∂x , ∂v ∂y ) , ∂2v ∂y2 = R ( x, y, v, ∂v ∂x , ∂v ∂y ) . (3.2) In the space of 1-jets J1 = J1(R2) with canonical coordinates x1 = x, x2 = y, v, p1, p2, this system defines the two-dimensional distribution P : a ∋ J1 7→ P(a) = 2⋂ i=0 kerωi,a ⊂ TaJ 1, (3.3) where differential 1-forms ω0 = dv − p1dx1 − p2dx2, ω2 = dp1 − P (x1, x2, v, p1, p2)dx1 −Q(x1, x2, v, p1, p2)dx2, ω2 = dp2 −Q(x1, x2, v, p1, p2)dx1 −R(x1, x2, v, p1, p2)dx2. This distribution is completely integrable if the conditions of the Frobenius theorem are satisfied: dωi ∧ ω0 ∧ ω1 ∧ ω2 = 0 (i = 0, 1, 2). (3.4) Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF PARABOLIC EQUATIONS 3 If these conditions are satisfied, then system (3.2) is a system of finite type. In what follows, we assume that these conditions are satisfied. The distribution P can be given by two vector fields D1 = ∂ ∂x1 + p1 ∂ ∂v + P ∂ ∂p1 +Q ∂ ∂p2 , D2 = ∂ ∂x2 + p2 ∂ ∂v +Q ∂ ∂p1 +R ∂ ∂p2 . Let a vector field S be a shuffling symmetry of the distribution P . It means that LS(ωi) ∧ ω0 ∧ ω1 ∧ ω2 = 0 (i = 0, 1, 2). Here L is the symbol of Lie’s derivative. Theorem 3.1: Each shuffling symmetry of distribution (3.3) is uniquely determined by a function φ on the space J1 and has the form Sφ = φ ∂ ∂v +D1(φ) ∂ ∂p1 +D2(φ) ∂ ∂p2 . (3.5) The function φ is called generating function of the vector field Sφ. Theorem 3.2: The generating function φ satisfies the following system of differential equations: D2 1(φ)− Sφ(P ) = 0, D2 2(φ)− Sφ(R) = 0, D1D2(φ)− Sφ(Q) = 0, D2D1(φ)− Sφ(Q) = 0. (3.6) 4. DYNAMICS Let Φt be a shift transformation along the trajectories of the vector field Sφ from t = 0 to t and L be an integral manifold of the distribution P . Then the manifold Φt(L) is also integral manifold of this distribution. A completely integrable distribution P is called a dynamics of the equation ∂u ∂t = f ( x, y, u, ∂u ∂x , ∂u ∂y , ∂2u ∂x2 , ∂2u ∂x∂y , ∂2u ∂y2 ) (4.7) if φ = f ( x1, x2, v, p1, p2, P̄ , Q̄, R̄ ) (4.8) is a generating function of a shuffling symmetry of the distribution P . Here x1 = x, x2 = y, P̄ = P (x1, x2, v, p1, p2), Q̄ = Q(x1, x2, v, p1, p2), R̄ = R(x1, x2, v, p1, p2) (4.9) are restrictions of the functions to the distribution P . Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) 4 A. KUSHNER, E. KUSHNER, S. TAO The following theorem allows us to obtain solutions of equation (4.7) from the known integral manifolds of the distribution P . Theorem 4.1: Let L be a maximal integral manifold of the distribution P . Then the three-dimensional manifold Φt(L) is the 1-graph of a solution of evolutionary equation (4.7). Let us show how we can calculate the dynamics. 1. First, we construct function (4.8) with unknown P,Q and R. 2. Second, we construct vector field (3.5) and the differential 1-forms ω0, ω1, ω2. 3. Third, we construct equations (3.4) and add to them equations (3.6). Since equation (3.4) with i = 0 is trivial, we obtain a system of six equations for the unknown functions P,Q,R. 4. Solving it we find dynamics (3.2). Since system (3.2) is involutive, it has a three-parameter family of solutions. In order to solve this system, we can use the Lie–Bianchi theorem (see [7]). According to this theorem, a completely integrable distribution P is integrable by quadratures if the Lie algebra SymP is solvable and, moreover, dimSymP = codimP . In our case, codimP = 3, and the vector field Sφ is a symmetry of the distribution P . Therefore, we need two more symmetries. For example, if the function f in equation (4.7) does not explicitly depend on variables x, y. Then we obtain these two symmetries automatically. Namely, these are vector fields with generating functions p1 and p2. Due to Theorem 3, to construct an explicit solution of equation (3.2) we should calculate the flow Φt of the vector field Sφ. Suppose that an integral manifold of the distribution P is defined by the following system: H1(x1, x2, v, p1, p2) = 0, H2(x1, x2, v, p1, p2) = 0, H3(x1, x2, v, p1, p2) = 0. (4.10) Acting to this system by the transformation Φ−1 t , we get the system U1(t, x1, x2, v, p1, p2) = 0, U2(t, x1, x2, v, p1, p2) = 0, U3(t, x1, x2, v, p1, p2) = 0. (4.11) Solving this system concerning the function v(t, x), we get an explicit solution u = v(t, x) of equation (3.2). If the flow Φt cannot be found explicitly, then we can construct solutions numerically. 5. DYNAMICS AND SOLUTIONS OF LINEAR PARABOLIC EQUATIONS The linear equation ∂u ∂t = n∑ i=1 ∂ ∂xi ( ki(x) ∂u ∂xi ) (5.12) is among the most commonly used equations in mathematical physics. Such equations are often encountered in biology [8], thermodynamics, diffusion theory, and filtration theory. These equations describe, for example, linear conservation laws. Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF PARABOLIC EQUATIONS 5 5.1. Case n = 1 For n = 1 equation (5.12) takes the form (see [9]) ∂u ∂t = ∂ ∂x ( k(x) ∂u ∂xi ) . (5.13) Theorem 5.1: Suppose that the function k is not vanishing. Then equation (5.13) has a finite dimension second-order dynamics (k(x)v′) ′ + αv − β ∫ dx k(x) − γ = 0, (5.14) where α, β, and γ are constants. If α = 0 then general solution of equation (5.13) is v(x) = ∫ ∫ ( β ∫ dx k(x) + γ ) dx+ δ k(x) dx. (5.15) Here δ is an arbitrary constant. The corresponding vector field Sφ = ( β ∫ dx k(x) + γ ) ∂ ∂v . Applying translations along its trajectories to solution (5.15), we obtain a three-parameter family of solutions of equation (5.13): u(t, x) = ( β ∫ dx k(x) + γ ) t+ ∫ ∫ ( β ∫ dx k(x) + γ ) dx+ δ k(x) dx. Note that this solution was obtained for any non-zero function k(x). 5.2. Case n = 2 For n = 2 equation (5.12) takes the form ∂u ∂t = ∂ ∂x1 ( k1(x) ∂u ∂x1 ) + ∂ ∂x2 ( k2(x) ∂u ∂x2 ) . (5.16) Let us look for dynamics (3.2) in the form ∂2v ∂x2 = P (x, y), ∂2v ∂x∂y = Q(x, y), ∂2v ∂y2 = R(x, y). (5.17) Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) 6 A. KUSHNER, E. KUSHNER, S. TAO The distribution P integrability condition has the form P (x1, x2) = ∫ ∂Q ∂x1 dx2 + p(x1), R(x1, x2) = ∫ ∂Q ∂x2 dx1 + q(x2), where p, q are arbitrary functions. Here x1 = x, x2 = y. Note that in this case [D1,D2] = 0. Consider the case when the functions are quadratic: k1 = k120x 2 1 + k111x1x2 + k102x 2 2 + k110x1 + k101x2 + k100, k2 = k220x 2 1 + k211x1x2 + k202x 2 2 + k210x1 + k201x2 + k200, where kins are constants. The index i means the number of the function, and the indices n and s mean the powers of x1 and x2, respectively. 5.2.1. Trivial dynamics P = Q = R = 0 Equation (5.16) admits trivial dynamics when the functions P,Q, and R are zero. We can construct the corresponding explicit solution to equation (5.16). However, the form of this solution is very cumbersome, and we will consider the special case when k211 ̸= 0 and k2 120 − 2k202k120 + k2 202 + k111k211 = 1. The differential 1-forms are ω1 = dp1, ω2 = dp2. Linear functions v = C1x1 + C2x2 + C0 (5.18) form a 3-parameter family of solutions to equation (3.2). The vector field Sφ = (2k120x1p1 − k−1 211(k 2 120 − 2k202k120 + k2 202 − 1)x2p1 + p1k110 + 2p2k202x2 + p2k211x1 + p2k201) ∂ ∂v + (2k120p1 + k211p2) ∂ ∂p1 + (−k−1 211(k 2 120 − 1− 2k202k120 + k2 202)p1 + 2k202p2) ∂ ∂p2 is a shuffling symmetry. Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF PARABOLIC EQUATIONS 7 Fig. 5.1. Vector field (k1, k2). Acting to function (5.18) by the transformation, we obtain the following explicit solution to equation (5.16): u =− [(bk211 + a(k120 − 1− k202)(k120 + k202 + 1) (k120 + k202 + 1)((k120 x1 + k110 + x1 k202 − x1)k211 − (k120 + 1− k202)(x2 k120 + k201 − x2 + k202 x2)) × e(k120+k202−1)t − (k120 + k202 − 1) (bk211 + a(k120 + 1− k202))((k120 x1 + k110 + x1 k202 + x1)k211 − (k120 − 1− k202)(x2 k120 + k201 + x2 + k202 x2))e (k120+k202+1)t − 2 bk2 211k110 + (4 ak110 k202 + 4 bk120 k201)k211 + 2 ak201 (k120 + 1− k202)(k120 − 1− k202)] × (2k211(k120 + k202 + 1)(k120 + k202 − 1))−1 , where a = C1, b = C2 are arbitrary constant. For brevity, we assume C0 = 0. 5.2.2. Dynamics P = Q = R = 1 Consider the following partial case of equation (5.16) with k1(x1, x2) = x1x2 − x2 2, k2(x1, x2) = −x1x2 + x2 1 (see Fig. 5.1). Then equation (5.16) admits the dynamics with P = Q = R = 1. The corresponding solution of equation (3.2) is v = 1 2 (x2 1 + x2 2) + x1x2 + ax1 + bx2 + c, (5.19) Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) 8 A. KUSHNER, E. KUSHNER, S. TAO where a, b and c are arbitrary constants. The generating function is φ = x2 1 − x2 2 + x2p1 − x1p2. Therefore, Sφ = (x2 1 − x2 2 + x2p1 − x1p2) ∂ ∂v + (x1 + x2 − p2) ∂ ∂p1 + (p1 − x1 − x2) ∂ ∂p2 . The translation along this vector field is Φt :  x1 7→x1, x2 7→x2, v 7→v + (−x2 2 + (−2x1 + p2)x2 − x1(−p1 + x1)) cos t + (−x2 2 + x2p1 + x1(x1 − p2)) sin t + x2 2 + (2x1 − p2)x2 + x2 1 − x1p1, p1 7→(x1 + x2) cos 2 t+ (p1 − x2 − x1) cos t + ((x2 + x1) sin t+ x1 + x2 − p2) sin t, p2 7→x2 + x1 + (p2 − x1 − x2) cos t + (p1 − x2 − x1) sin t. The inverse transformation is Φ−1 t :  x1 7→x1, x2 7→x2, v 7→v + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) cos 2 t + (−x2 1 + (p1 − 2x2)x1 − x2(x2 − p2)) cos t + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) sin 2 t + (−x2 1 + x1p2 + x2(−p1 + x2)) sin t, p1 7→(p1 − x2 − x1) cos t+ (p2 − x1 − x2) sin t + x1 + x2, p2 7→(p2 − x1 − x2) cos t+ (x1 + x2 − p1) sin t + x1 + x2. Applying the transformations Φ−1 t to the vector function [ v − 1 2 (x2 1 + x2 2) + x1x2 + ax1 + bx2 + c, p1 − x1 − x2 − a, p2 − x2 − x1 − b ] we get system (4.11): Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) DYNAMICS OF PARABOLIC EQUATIONS 9 Fig. 5.2. The graph of solution (5.20) at t = 0.  (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) cos 2 t + (−x2 1 + (p1 − 2x2)x1 − x2(x2 − p2)) cos t + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) sin 2 t + (−x2 1 + x1p2 + x2(−p1 + x2)) sin t + v − x2 1 + x2 2 2 − x1x2 − ax1 − bx2 − c = 0, − a+ (p1 − x2 − x1) cos t+ (p2 − x1 − x2) sin t = 0, − b+ (p2 − x1 − x2) cos t+ (x1 + x2 − p1) sin t = 0. Solving this system, we get the following 3-parameter solutions family to evolutionary equation (5.16): u = 1 2 ( x2 1 + x2 2 ) + (ax1 + bx2) cos t+ (ax2 − bx1) sin t+ x1x2 + c, (5.20) where a, b and c are arbitrary constants. The evolution of a graph of function (5.20) in time, see Fig. 5.2, 5.3. Here a = b = 1, c = 0. REFERENCES 1. Kruglikov, B.S. & Lychagina, O.V. (2005) Finite dimensional dynamics for Kolmogorov – Petrovsky – Piskunov equation. Lobachevskii Journal of Mathematics, 19, 13–28. Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) 10 A. KUSHNER, E. KUSHNER, S. TAO Fig. 5.3. The graph of solution (5.20) at t = 5. 2. Lychagin, V.V. & Lychagina, O.V. (2007) Finite Dimensional Dynamics for Evolutionary Equations. Nonlinear Dyn., 48, 29–48. 3. Akhmetzyanov, A.V., Kushner, A.G. & Lychagin, V.V. (2017) Attractors in Models of Porous Media Flow. Doklady. Mathematics, 472:6, 627–630. 4. Kushner, A.G. & Matviichuk, R.I. (2020) Exact solutions of the Burgers – Huxley equation via dynamics. Journal of Geometry and Physics, 151, 103615. 5. Kushner, A.G. (2022) Dynamics of evolutionary equations with multiple spatial variables. Proc. of Conf. “Symmetries: theoretical and methodological aspects, 9(33), Astrakhan, 15-16 Sept. 2022, 37-43, [in Russian]. 6. Duzhin, S.V, Lychagin, V.V. (1991) Symmetries of distributions and quadrature of ordinary differential equations. Acta Appl. Math. 24(1), 29–57. 7. Kushner, A.G., Lychagin, V.V. & Rubtsov, V.N. (2007) Contact geometry and nonlinear differential equations. Cambridge: Cambridge University Press, xxii+496 pp. 8. Pethame, B. (2015) Parabolic Equations in Biology. Springer, Heidelberg, 203 p. 9. Kushner, A.G., Kushner, E.N. & Tao, S. (2022) Finite-dimensional dynamics of filtration equation. Differential equations and mathematical modeling, 4, 62–63, [in Russian]. Copyright © 2022 ASSA. Adv Syst Sci Appl (2022) Introduction Symmetries of Completely Integrable Distributions Symmetries of Finite Type Differential Equations Dynamics Dynamics and Solutions of Linear Parabolic Equations Case n=1 Case n=2 Trivial dynamics P=Q=R=0 Dynamics P=Q=R=1