Volume 4, Issue 1-2, May 1997, Pages 49 - 61
On Integration of the Nonlinear d'Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations
Renat Z. ZHDANOV
Renat Z. ZHDANOV
Available Online 1 May 1997.
- https://doi.org/10.2991/jnmp.1997.4.1-2.6How to use a DOI?
- We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d'Alembert equation 2u = F(u) and nonlinear eikonal equation uxµ uxµ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d'Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d'Alembert, Dirac, and YangMills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation 2w = F0(w) in the fourdimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.
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Cite this article
TY - JOUR AU - Renat Z. ZHDANOV PY - 1997 DA - 1997/05 TI - On Integration of the Nonlinear d'Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations JO - Journal of Nonlinear Mathematical Physics SP - 49 EP - 61 VL - 4 IS - 1-2 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.1997.4.1-2.6 DO - https://doi.org/10.2991/jnmp.1997.4.1-2.6 ID - ZHDANOV1997 ER -