On the Construction and Classification of the Common Invariant Solutions for Some <i>P</i>(1,4) -Invariant Partial Differential Equations — Oak Academic Publishing
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On the Construction and Classification of the Common Invariant Solutions for Some <i>P</i>(1,4) -Invariant Partial Differential Equations
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
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Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
1 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
2 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
We consider the following (1 + 3)-dimensional P (1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P (1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional ( dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P (1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
KeywordsSymmetry ReductionClassification of Invariant SolutionsCommon Invariant SolutionsThe Eikonal EquationsThe Euler-Lagrange-Born-Infeld EquationsThe Monge-Ampère EquationsClassification of Lie AlgebrasNonconjugate SubalgebrasPoincaréGroup <i>P</i>
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