Abstract
In 1773 Laplace obtained two fundamental semi‐invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations (PDEs) in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semi‐invariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting a complex scalar ordinary differential equation (ODE) into its real and imaginary parts PDEs for two functions of two variables were obtained and their symmetry structure studied. In this work we revisit semi‐invariants under equivalence transformations of the dependent variables for systems of two linear hyperbolic PDEs in two independent variables when such systems correspond to scalar complex linear hyperbolic equations in two independent variables, using the above‐mentioned splitting procedure. The semi‐invariants under linear changes of the dependent variables deduced for this class of hyperbolic linear systems correspond to the complex semi‐invariants of the complex scalar linear (1 + 1) hyperbolic equation. We show that the adjoint factorization corresponds precisely to the complex splitting. We also study the reductions and the inverse problem when such systems of two linear hyperbolic PDEs arise from a linear complex hyperbolic PDE. Examples are given to show the application of this approach.
Highlights
In the study of scalar linear second-order partial differential equations PDEs in two independent variables, x and y,A x, y uxx 2B x, y uxy C x, y uyy D x, y ux E x, y uy F x, y u G x, y, 1.1Mathematical Problems in Engineering where A to G are C2 functions in some domain and ux ∂u/∂x and so forth, it is well known that there are three canonical forms, namely, hyperbolic, parabolic, and elliptic see, e.g. 1, according to the sign of the discriminant Δ B2 − AC
The general system 2.4 arises from the complex continuation of the scalar linear (1 1) hyperbolic PDE 1.1 if and only if its coefficients are precisely of the form 2.8 or equivalently if 2.4 has quantities Kis which are written solely in terms of the semi-invariants h1, h2, k1, and k2 as
In this work we have used complex splitting of the scalar linear 1 1 hyperbolic equation to transform it into a system of two linear hyperbolic equations by a complex split
Summary
In the study of scalar linear second-order partial differential equations PDEs in two independent variables, x and y,. These are subclasses of systems of the form considered by Tsaousi and Sophocleous 6. The construction of the mappings that relate the two systems of hyperbolic PDEs of the class considered is done using the explicit dependent variable change for the scalar complex case which is in terms of the coefficients of the original and target PDEs see Theorem 3.1 This is not the case for the general system studied in 6 as these explicit formulas are not as yet known for the general case.
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