Abstract
A complete group classification for the Klein-Gordon equation is presented. Symmetry generators, up to equivalence transformations, are calculated for each f (u) when the principal Lie algebra extends. Further, considered equation is investigated by using Noether approach for the general case n  2. Conserved quantities are computed for each calculated Noether operator. At the end, a brief conclusion is presented.
Highlights
The (1+n)-dimensional Klein-Gordon equation utt 2u f (u), fuu 0, (1)n where u u(t, x1,..., xn ) with 2u uxixi . i 1In the past, the authors of [3, 10] have studied Eq (1) for different values of n, for exact solutions, compatibility of the conditions for the reduction and reduced equations by consideration of an ansatz which reduces the dimension of the corresponding PDE
The authors of [3, 10] have studied Eq (1) for different values of n, for exact solutions, compatibility of the conditions for the reduction and reduced equations by consideration of an ansatz which reduces the dimension of the corresponding PDE
Fushchych [10] invoked an ansatz of the form u f ( x) ( ) g( x) to analyze exact solutions of Eq (1)
Summary
The (1+n)-dimensional Klein-Gordon equation utt 2u f (u), fuu 0, (1) Lie symmetry analysis is a systematic way to construct an ansatz which further reduces the dimension of the differential equation. There are nonlinear equations with arbitrary coefficients which possess nontrivial Lie point symmetries. Such nonlinear differential equations can be classified, with respect to unknown functions, according to the nontrivial Lie point symmetries they admit.
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