On Simple Solutions of Some Equations of Mathematical Physics
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On Simple Solutions of Some Equations of Mathematical Physics V. K. Beloshapka∗,1 ∗
Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991 Russia, E-mail: 1 [email protected] Received March 10, 2020; Revised March 14, 2020; Accepted March 20, 2020
Abstract. All solutions to the Burgers, Hopf, Helmholtz, Klein–Gordon, sine-Gordon, Schr¨ odinger, and Monge–Ampere equations having analytical complexity one (simple solutions) are described. It turns out that all simple solutions of the Burgers and Hopf equation are represented by elementary functions. An example of a family of solutions of complexity two to the Burgers equation is presented. Simple solutions to the Helmholtz (or Klein–Gordon) equation are expressed in terms of Bessel functions and elementary functions. For the Laplace and wave equations, an explicit description is given for the simple solutions that are expressed in terms of Jacobi elliptic functions. Open problems of the theory of analytic complexity (the analytical spectrum of an equation) are discussed. DOI 10.1134/S1061920820030036
Among the questions that arose in connection with the discussion of the 13th Hilbert’s problem, many remain open [1, 2], despite the existing successes. The notion of analytical complexity [3], developed by the author of the present paper, is related to these questions. One can act on an analytic function of two variables z(x, y) using functions of one variable. This naturally gives rise to the pseudogroup G which acts as follows. If g = (a, b, c) ∈ G, where (a, b, c) are three nonconstant analytic functions of one variable, and z(x, y) is an analytic function of two variables, then (g ◦ z)(x, y) = c(z(a(x), b(y))). This pseudogroup plays a fundamental role in problems of measuring the complexity of analytic functions of two variables. From the point of view of the theory of analytical complexity, the functions z and g ◦ z are indistinguishable (equivalent). In terms of this pseudogroup, one can, in particular, formulate a unique property of arithmetic operations. Since all four arithmetic operations are equivalent modulo G, and this property can be formulated in terms of the pseudogroup, it follows that we are talking about a unique property of all functions of this class. By defining this class as the orbit of the function z = x + y, we obtain its description in the form Cl1 = {z(x, y) = c(a(x) + b(y))}. Further, for every analytic function z(x, y), we can consider the pseudosubgroup Stab(z) = {g ∈ G : (g ◦ z)(x, y) = z(x, y)}; let d(z) be the dimension of Stab(z), understood as the dimension of the corresponding Lie algebra. The following theorem was proved in [5]. For an arbitrary analytic function z(x, y) depending on both the variables (the partial derivatives are not equal to zero identically), d(z) can take only three values: 0, 1, and 3. Moreover, the dimension of the stabilizer takes the value 3 if and only if the function z belongs to Cl1 \ Cl0 . This theorem shows that the functio
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