[Paper Review] Reduction of non-linear d'Alembert equations to two-dimensional equations

Irina Yehorchenko|ArXiv.org|Oct 14, 2009

Nonlinear Waves and Solitons31 references3 citations

TL;DR

This paper establishes necessary compatibility conditions for reducing nonlinear multidimensional d'Alembert equations to two-dimensional equations via ansatzes with two new independent variables. It proves that such reductions yield only specific classes of two-dimensional equations—parabolic, hyperbolic, or elliptic—and extends the applicability of symmetry-based reduction methods to arbitrary Poincaré-invariant equations, including wave, Dirac, and Maxwell equations.

ABSTRACT

We study conditions of reduction of the multidimensional wave equation - a system of the d'Alembert and Hamilton equations. We prove necessary conditions for compatibility of such system of the reduction conditions. Possible types of the reduced equations represent interesting classes of two-dimensional parabolic, hyperbolic and elliptic equations. Ansatzes and methods used for reduction of the d'Alembert (n-dimensional wave) equation can be also used for arbitrary Poincare-invariant equations. This seemingly simple and partial problem involves many important aspects in the studies of the PDE.

Motivation & Objective

• To derive necessary compatibility conditions for the system of d'Alembert–Hamilton equations arising from ansatz-based reduction of nonlinear multidimensional d'Alembert equations.
• To classify the possible types of two-dimensional reduced equations that can emerge from such reductions.
• To extend the applicability of symmetry-based reduction techniques beyond standard Lie symmetries to include Q-conditional symmetries and Poincaré-invariant equations.
• To provide a foundation for further study of conditional symmetries and exact solutions of reduced equations.
• To explore the relationship between the symmetry of the original equation and the symmetry of the reduced two-dimensional system.

Proposed method

• Uses a general ansatz $ u = \varphi(y,z) $, where $ y $ and $ z $ are functions of the original $ n+1 $ spacetime variables.
• Derives the d'Alembert–Hamilton system: $ y_\mu y_\mu = r(y,z), \, y_\mu z_\mu = q(y,z), \, z_\mu z_\mu = s(y,z) $, and $ \Box y = R(y,z), \, \Box z = S(y,z) $.
• Applies compatibility analysis to the overdetermined system of PDEs to derive necessary conditions for consistent reduction.
• Employs Cartan's algorithm and ad hoc techniques to handle the complexity of compatibility conditions in arbitrary dimensions.
• Analyzes explicit ansatz examples (e.g., radial, linear, and mixed forms) to derive specific reduced equations like the radial wave equation.
• Relates the results to Q-conditional symmetries and discusses implications for hidden symmetries in reduced systems.

Experimental results

Research questions

• RQ1 What are the necessary compatibility conditions for reducing a nonlinear $ n $-dimensional d'Alembert equation to a two-dimensional PDE via a two-variable ansatz?
• RQ2 What specific classes of two-dimensional equations (parabolic, hyperbolic, elliptic) can arise from such reductions?
• RQ3 How do the symmetries of the reduced equations relate to those of the original d'Alembert equation?
• RQ4 Can the same reduction framework be generalized to other Poincaré-invariant equations, such as the Dirac or Maxwell equations?
• RQ5 What is the role of Q-conditional symmetries in generating exact solutions through this reduction process?

Key findings

• Necessary compatibility conditions for the d'Alembert–Hamilton system were derived, which are not obtainable via standard symmetry reduction procedures.
• The reduced equations resulting from the ansatz are restricted to specific types: parabolic, hyperbolic, or elliptic, depending on the form of the ansatz.
• Explicit examples of reduced equations were constructed, including the radial wave equation $ \varphi_{yy} - \varphi_{zz} - \frac{2}{z}\varphi_z = F(\varphi) $, which arises from a quadratic form of the new variables.
• The method applies not only to the d'Alembert equation but also to other Poincaré-invariant equations, such as the Dirac and Maxwell equations.
• The reduction process reveals hidden symmetries and Q-conditional symmetries in the original equation, which are not visible through classical Lie symmetry analysis.
• The compatibility conditions are shown to be independent of the number of spatial dimensions, making the results applicable to equations in arbitrary dimensions.