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Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations

Sergey P. Tsarev

TL;DR

The paper addresses factorization and complete integration of strictly hyperbolic LPDEs in the plane by generalizing Laplace factorization to $n\times n$ hyperbolic systems. It develops a recurrent framework based on converting to characteristic form, defining Laplace invariants, and applying multi-step transformations that progressively simplify the system toward (block-) triangularity and solvability. The approach yields closed-form solutions in many cases without solving auxiliary differential equations and is applicable over arbitrary differential fields. A concrete $3\times3$ example demonstrates the method and outlines a practical scheme for generalized factorization and integration of higher-order hyperbolic PDE systems.

Abstract

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.

Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations

TL;DR

The paper addresses factorization and complete integration of strictly hyperbolic LPDEs in the plane by generalizing Laplace factorization to hyperbolic systems. It develops a recurrent framework based on converting to characteristic form, defining Laplace invariants, and applying multi-step transformations that progressively simplify the system toward (block-) triangularity and solvability. The approach yields closed-form solutions in many cases without solving auxiliary differential equations and is applicable over arbitrary differential fields. A concrete example demonstrates the method and outlines a practical scheme for generalized factorization and integration of higher-order hyperbolic PDE systems.

Abstract

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.

Paper Structure

This paper contains 4 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The classical heritage: Laplace transformations
  3. Generalized Laplace transformations of $n\times n$ hyperbolic systems
  4. An example and the general factorization procedure

Key Result

Proposition 1

Any strictly hyperbolic LPDE is equivalent to a 2\times2 first-order characteristic system with \hat{X}_i = m_i(x,y)\hat{D}_x + n_i(x,y)\hat{D}_y, \hat{X}_1 \neq \gamma(x,y)\hat{X}_2, and any such system with non-diagonal matrix (\alpha_{ij}) is equivalent to a second-order strictly hyperbolic LPDE.

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4