Group invariant transformations for the Klein-Gordon equation in three   dimensional flat spaces

Sameerah Jamal; Andronikos Paliathanasis

arXiv:1703.01471·math-ph·April 26, 2017

Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces

Sameerah Jamal, Andronikos Paliathanasis

PDF

TL;DR

This paper classifies all potential functions for the Klein-Gordon equation in three-dimensional flat spaces that admit Lie and Noether symmetries, linking symmetries to the conformal algebra of the geometry and deriving invariant solutions.

Contribution

It provides a complete symmetry classification of the Klein-Gordon equation in symmetric spacetimes, identifying potentials with specific symmetry properties and solutions.

Findings

01

Classified all potentials $V(t,x,y)$ admitting Lie and Noether symmetries.

02

Derived invariant solutions for certain symmetric potentials.

03

Linked symmetries of the equation to the conformal algebra of the underlying geometry.

Abstract

We perform the complete symmetry classification of the Klein-Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions $V (t, x, y)$ that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. For some of the potentials, we use the admitted Lie algebras to determine corresponding invariant solutions to the Klein-Gordon equation. An integral part of this analysis is the problem of the classification of Lie and Noether point symmetries of the wave equation.

Tables2

Table 1. Table 4: Conservation Laws corresponding to X 1 − 10 superscript 𝑋 1 10 X^{1-10} .

$T_{j}$	$T_{j}^{t}, T_{j}^{x}, T_{j}^{y}$
$T_{1}$	$T^{t} = \frac{1}{2 ϵ} (ϵ u^{2} V (x, y) + ϵ u (u_{y y} + u_{x x}) + u_{t} {}^{2}), T^{x} = \frac{1}{2} (u_{x} u_{t} - u u_{t x}), T^{y} = \frac{1}{2} (u_{y} u_{t} - u u_{t y})$
$T_{2}$	$T^{t} = \frac{1}{2 ϵ} (u_{x} u_{t} - u u_{t x}), T^{x} = \frac{1}{2 ϵ} (ϵ u^{2} V (t, y) + ϵ u_{x} {}^{2}+ u (ϵ u_{y y} + u_{t t})), T^{y} = \frac{1}{2} (u_{y} u_{x} - u u_{x y})$
$T_{3}$	$T^{t} = \frac{1}{2 ϵ} (u_{y} u_{t} - u u_{t y}), T^{x} = \frac{1}{2} (u_{y} u_{x} - u u_{x y}), T^{y} = \frac{1}{2 ϵ} (ϵ u^{2} V (t, x) + ϵ u_{y} {}^{2}+ u (ϵ u_{x x} + u_{t t}))$
$T_{4}$	$T^{t} = \frac{1}{2 t ϵ} (ϵ u^{2} V (\frac{x}{t}, \frac{y}{t}) + t u_{t} (y u_{y} + x u_{x} + t u_{t}) + t u (t ϵ u_{y y} + t ϵ u_{x x} - u_{t} - y u_{t y} - x u_{t x})),$
	$T^{x} = \frac{1}{2 t^{2} ϵ} (x ϵ u^{2} V (\frac{x}{t}, \frac{y}{t}) + t^{2} ϵ u_{x} (y u_{y} + x u_{x} + t u_{t}) - t^{2} u (- x ϵ u_{y y} + ϵ u_{x} + y ϵ u_{x y} + t ϵ u_{t x} - x u_{t t})),$
	$T^{y} = \frac{1}{2 t^{2} ϵ} (y ϵ u^{2} V (\frac{x}{t}, \frac{y}{t}) + t^{2} ϵ u_{y} (y u_{y} + x u_{x} + t u_{t}) - t^{2} u (ϵ u_{y} + x ϵ u_{x y} - y ϵ u_{x x} + t ϵ u_{t y} - y u_{t t}))$
$T_{5}$	$T^{t} = \frac{1}{2 ϵ} (- x u_{y} u_{t} + y u_{x} u_{t} + u (x u_{t y} - y u_{t x})),$
	$T^{x} = \frac{1}{2 ϵ} (y ϵ u^{2} V (t, x^{2} + y^{2}) + ϵ u_{x} (- x u_{y} + y u_{x}) + u (ϵ u_{y} + y ϵ u_{y y} + x ϵ u_{x y} + y u_{t t})),$
	$T^{y} = - \frac{1}{2 ϵ} (x ϵ u^{2} V (t, x^{2} + y^{2}) + ϵ u_{y} (x u_{y} - y u_{x}) + u (ϵ u_{x} + y ϵ u_{x y} + x ϵ u_{x x} + x u_{t t}))$
$T_{6}$	$T^{t} = \frac{1}{2 ϵ} (x ϵ u^{2} V (x^{2} + t^{2} ϵ, y) + u_{t} (- t ϵ u_{x} + x u_{t}) + ϵ u (x u_{y y} + u_{x} + x u_{x x} + t u_{t x})),$
	$T^{x} = \frac{1}{2} (- t ϵ u^{2} V (x^{2} + t^{2} ϵ, y) + u_{x} (- t ϵ u_{x} + x u_{t}) - u (t ϵ u_{y y} + u_{t} + x u_{t x} + t u_{t t})),$
	$T^{y} = \frac{1}{2} (u_{y} (- t ϵ u_{x} + x u_{t}) + u (t ϵ u_{x y} - x u_{t y}))$
$T_{7}$	$T^{t} = \frac{1}{2 ϵ} (y ϵ u^{2} V (x, y^{2} + t^{2} ϵ) + u_{t} (- t ϵ u_{y} + y u_{t}) + ϵ u (u_{y} + y u_{y y} + y u_{x x} + t u_{t y})),$
	$T^{x} = \frac{1}{2} (- t ϵ u_{y} u_{x} + y u_{x} u_{t} + u (t ϵ u_{x y} - y u_{t x})),$
	$T^{y} = \frac{1}{2} (- t ϵ u^{2} V (x, y^{2} + t^{2} ϵ) + u_{y} (- t ϵ u_{y} + y u_{t}) - u (t ϵ u_{x x} + u_{t} + y u_{t y} + t u_{t t}))$
$T_{8}$	$T^{t} = \frac{1}{2 t ϵ} (2 y ϵ u^{2} V (\frac{x}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t}) + t u_{t} (- (x^{2} - y^{2} + t^{2} ϵ) u_{y} + 2 y (x u_{x} + t u_{t}))$
	$+ t u (2 t ϵ u_{y} + 2 t y ϵ u_{y y} + 2 t y ϵ u_{x x} - 2 y u_{t} + x^{2} u_{t y} - y^{2} u_{t y} + t^{2} ϵ u_{t y} - 2 x y u_{t x})),$
	$T^{x} = \frac{1}{2 t^{2} ϵ} (2 x y ϵ u^{2} V (\frac{x}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t}) + t^{2} ϵ u_{x} (- (x^{2} - y^{2} + t^{2} ϵ) u_{y} + 2 y (x u_{x}$
	$+ t u_{t})) + t^{2} u (2 x ϵ u_{y} + 2 x y ϵ u_{y y} - 2 y ϵ u_{x} + x^{2} ϵ u_{x y} - y^{2} ϵ u_{x y} + t^{2} ϵ^{2} u_{x y} - 2 t y ϵ u_{t x} + 2 x y u_{t t})),$
	$T^{y} = - - \frac{1}{2 t^{2} ϵ} (ϵ u^{2} (t^{2} + (x^{2} - y^{2} + t^{2} ϵ) V (\frac{x}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t})) + t^{2} ϵ u_{y} ((x^{2} - y^{2} + t^{2} ϵ) u_{y}$
	$- 2 y (x u_{x} + t u_{t})) + t^{2} u (2 y ϵ u_{y} + 2 x ϵ u_{x} + 2 x y ϵ u_{x y} + x^{2} ϵ u_{x x} - y^{2} ϵ u_{x x}$
	$+ t^{2} ϵ^{2} u_{x x} + 2 t ϵ u_{t} + 2 t y ϵ u_{t y} + x^{2} u_{t t} - y^{2} u_{t t} + t^{2} ϵ u_{t t}))$
$T_{9}$	$T^{t} = \frac{1}{2 t ϵ} (2 x ϵ u^{2} V (\frac{y}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t}) + t u_{t} (2 x y u_{y} + (x^{2} - y^{2} - t^{2} ϵ) u_{x} + 2 t x u_{t})$
	$+ t u (2 t x ϵ u_{y y} + 2 t ϵ u_{x} + 2 t x ϵ u_{x x} - 2 x u_{t} - 2 x y u_{t y} - x^{2} u_{t x} + y^{2} u_{t x} + t^{2} ϵ u_{t x})),$
	$T^{x} = - \frac{1}{2 t^{2} ϵ} (ϵ u^{2} (t^{2} + (- x^{2} + y^{2} + t^{2} ϵ) V (\frac{y}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t})) - t^{2} ϵ u_{x} (2 x y u_{y}$
	$+ (x^{2} - y^{2} - t^{2} ϵ) u_{x} + 2 t x u_{t}) + t^{2} u (2 y ϵ u_{y} + ϵ (- x^{2} + y^{2} + t^{2} ϵ) u_{y y} + 2 x ϵ u_{x} + 2 x y ϵ u_{x y}$
	$+ 2 t ϵ u_{t} + 2 t x ϵ u_{t x} - x^{2} u_{t t} + y^{2} u_{t t} + t^{2} ϵ u_{t t})),$
	$T^{y} = \frac{1}{2 t^{2} ϵ} (2 x y ϵ u^{2} V (\frac{y}{t}, \frac{x^{2} + y^{2} + t^{2} ϵ}{t}) + t^{2} ϵ u_{y} (2 x y u_{y} + (x^{2} - y^{2} - t^{2} ϵ) u_{x}$
	$+ 2 t x u_{t}) + t^{2} u (- 2 x ϵ u_{y} + 2 y ϵ u_{x} - x^{2} ϵ u_{x y} + y^{2} ϵ u_{x y} + t^{2} ϵ^{2} u_{x y} + 2 x y ϵ u_{x x} - 2 t x ϵ u_{t y} + 2 x y u_{t t}))$
$T_{10}$	$T^{t} = - \frac{1}{2 x^{2} ϵ^{2}} (ϵ u^{2} (x^{2} + (x^{2} + y^{2} - t^{2} ϵ) V (\frac{y}{x}, \frac{x^{2} + y^{2} + t^{2} ϵ}{x ϵ}))$
	$+ x^{2} u_{t} (- 2 t y ϵ u_{y} - 2 t x ϵ u_{x} + (x^{2} + y^{2} - t^{2} ϵ) u_{t}) + x^{2} ϵ u (2 y u_{y} + (x^{2} + y^{2} - t^{2} ϵ) u_{y y} + 2 x u_{x}$
	$+ x^{2} u_{x x} + y^{2} u_{x x} - t^{2} ϵ u_{x x} + 2 t u_{t} + 2 t y u_{t y} + 2 t x u_{t x})),$
	$T^{x} = \frac{1}{2 x ϵ} (2 t ϵ u^{2} V (\frac{y}{x}, \frac{x^{2} + y^{2} + t^{2} ϵ}{x ϵ}) + x u_{x} (2 t y ϵ u_{y} + 2 t x ϵ u_{x} -$
	$(x^{2} + y^{2} - t^{2} ϵ) u_{t}) + x u (2 t x ϵ u_{y y} - 2 t ϵ u_{x} - 2 t y ϵ u_{x y} + 2 x u_{t} + x^{2} u_{t x} + y^{2} u_{t x} - t^{2} ϵ u_{t x} + 2 t x u_{t t})),$
	$T^{y} = \frac{1}{2 x^{2} ϵ} (2 t y ϵ u^{2} V (\frac{y}{x}, \frac{x^{2} + y^{2} + t^{2} ϵ}{x ϵ}) + x^{2} u_{y} (2 t y ϵ u_{y} + 2 t x ϵ u_{x}$
	$- (x^{2} + y^{2} - t^{2} ϵ) u_{t}) + x^{2} u (- 2 t ϵ u_{y} - 2 t x ϵ u_{x y} + 2 t y ϵ u_{x x} + 2 y u_{t} + x^{2} u_{t y} + y^{2} u_{t y} - t^{2} ϵ u_{t y} + 2 t y u_{t t}))$

Table 2. Table 7: Case III: The Lie algebra is spanned by linear combinations of the CKVs.

Potential Function	Lie/Noether Algebra
$V ((ϵ a_{1} t - x) a_{3} + y)$	${\frac{1}{ϵ} X^{1} + a_{1} X^{2}, X^{2} + a_{3} X^{3}}$
$\frac{1}{{(ϵ a_{1} t - x)}^{2}} V (\frac{y}{ϵ a_{1} t - x})$	${\frac{1}{ϵ} X^{1} + a_{1} X^{2}, X^{4}}$
$V (\frac{(ϵ^{2} t^{2} + ϵ y^{2}) a_{1}^{2} - 2 ϵ a_{1} t x + x^{2} + y^{2}}{ϵ a_{1}^{2} + 1})$	${\frac{1}{ϵ} X^{1} + a_{1} X^{2}, X^{5} - a_{1} X^{7}}$
$V (- ϵ a_{1} t + x)$	${\frac{1}{ϵ} X^{1} + a_{1} X^{2} + a_{5} X^{3}, X^{3}}$
$V (- ϵ a_{2} t - x a_{3} + y)$	${\frac{1}{ϵ} X^{1} + a_{2} X^{3}, X^{2} + a_{3} X^{3}}$
$\frac{1}{x^{2}} V (\frac{- ϵ t a_{2} + y}{x})$	${\frac{1}{ϵ} X^{1} + a_{2} X^{3}, X^{4}}$
$V ((ϵ^{2} t^{2} + ϵ x^{2}) a_{2}^{2} - 2 ϵ a_{2} t y + x^{2} + y^{2})$	${\frac{1}{ϵ} X^{1} + a_{2} X^{3}, X^{5} + a_{2} X^{6}}$
$V (- ϵ a_{2} t + y)$	${\frac{1}{ϵ} X^{1} + a_{4} X^{2} + a_{2} X^{3}, X^{2}}$
$\frac{1}{t^{2}} V (\frac{- a_{3} x + y}{t})$	${X^{2} + a_{3} X^{3}, X^{4}}$
$V (t (a_{3} a_{4} - a_{5}) ϵ - a_{3} x + y)$	${X^{2} + a_{3} X^{3}, X^{1} + a_{4} X^{2} + a_{5} X^{3}}$
$V (\frac{(ϵ t^{2} + y^{2}) b_{3}^{2} + 2 b_{3} x y + x^{2} + t^{2} ϵ}{b_{3}^{2}})$	${X^{2} - \frac{1}{b_{3}} X^{3}, \frac{1}{ϵ} (X^{6} + b_{3} X^{7})}$
$\frac{1}{{(ϵ a_{4} t - x)}^{2}} V (\frac{- ϵ a_{5} t + y}{ϵ a_{4} t - x})$	${\frac{1}{ϵ} X^{1} + a_{4} X^{2} + a_{5} X^{3}, X^{4}}$
$V (\frac{ϵ (b_{1}^{2} t^{2} - 2 b_{1} t y + x^{2} + y^{2}) + x^{2} b_{1}^{2}}{ϵ})$	${\frac{1}{ϵ} X^{1} + \frac{b_{1}}{ϵ} X^{3}, X^{5} + \frac{b_{1}}{ϵ} X^{6}}$
$V (\frac{ϵ (b_{2}^{2} t^{2} + 2 b_{2} t x + x^{2} + y^{2}) + y^{2} b_{2}^{2}}{ϵ + b_{2}^{2}})$	${\frac{1}{ϵ} X^{1} - \frac{b_{2}}{ϵ} X^{2}, X^{5} + \frac{b_{2}}{ϵ} X^{7}}$
$\frac{1}{{(ϵ t - b_{2} x)}^{2}} V (\frac{(ϵ t^{2} + y^{2}) b_{2}^{2} + 2 b_{2} ϵ x t - ϵ^{2} t^{2}}{b_{2}^{2} {(- ϵ t + b_{2} x)}^{2}})$	${X^{4}, X^{5} + \frac{b_{2}}{ϵ} X^{7}}$
$\frac{1}{t^{2}} V (\frac{y c_{2} - x}{c_{2} t})$	${X^{4}, X^{8} + \frac{c_{2}}{2} X^{9}}$
$\frac{1}{{(- c_{5} ϵ t + c_{4} x)}^{2}} V (\frac{ϵ t - c_{4} y}{c_{4} (- ϵ c_{5} t + c_{4} x)})$	${X^{4}, X^{8} + \frac{c_{4}}{2} X^{10} + \frac{c_{5}}{2} X^{9}}$
$\frac{1}{{(c_{3} ϵ t - x)}^{2}} V (\frac{ϵ t^{2} + x^{2} + y^{2}}{c_{3} ϵ t - x})$	${X^{8}, X^{10} + c_{3} X^{9}}$
$\frac{1}{x^{2}} V (\frac{- ϵ t + c_{1} y}{c_{1} x})$	${X^{4}, \frac{1}{ϵ} X^{1} + \frac{1}{c_{1}} X^{3}, X^{8} + \frac{c_{1}}{2} X^{10}}$
$\frac{1}{{(c_{3} ϵ t^{2} - x)}^{2}} V (\frac{y}{c_{3} ϵ t - x})$	${X^{4}, \frac{1}{ϵ} X^{1} + c_{3} X^{2}, X^{10} + \frac{c_{3}}{2} X^{9}}$
$\frac{c}{(ϵ a_{1} t - x) a_{3} + y}$	${\frac{1}{ϵ} X^{1} + a_{1} X^{2}, X^{4}, X^{2} + a_{3} X^{3}}$
$V (- a_{3} x + y)$	${X^{1}, X^{2} + a_{3} X^{3}, \frac{1}{ϵ} (X^{6} + a_{3} X^{7})}$
$\frac{c}{{(ϵ a_{2} t + a_{3} x - y)}^{2}}$	${\frac{1}{ϵ} X^{1} + a_{2} X^{3}, X^{4}, X^{2} + a_{3} X^{3}}$
$V (\frac{ϵ t + b_{1} y}{b_{1}})$	${X^{1} - \frac{1}{b_{1}} X^{3}, X^{2}, X^{5} + \frac{b_{1}}{ϵ} X^{6}}$
$V (\frac{c}{(ϵ^{2} t^{2} + ϵ y^{2}) a_{1}^{2} - 2 ϵ a_{1} t x + x^{2} + y^{2}})$	${X^{1} + a_{1} X^{2}, X^{4}, X^{5} - a_{1} X^{6}}$
$V (\frac{c}{(ϵ^{2} t^{2} + ϵ x^{2}) a_{2}^{2} - 2 ϵ a_{2} t y + x^{2} + y^{2}})$	${X^{1} + a_{2} X^{3}, X^{4}, X^{5} + a_{2} X^{6}}$
$V (ϵ^{2} t^{2} (a_{4}^{2} + a_{5}^{2}) + ϵ (y^{2} a_{4}^{2} + a_{5}^{2} x^{2}) -$	${\frac{1}{ϵ} X^{1} + a_{4} X^{2} + a_{5} X^{3}, X^{5} + a_{5} X^{6} - a_{4} X^{7}}$
$2 ϵ (a_{4} a_{5} x y + a_{4} t x + a_{5} t y) + x^{2} + y^{2})^{\frac{1}{2}}$

Equations78

G_{α} (x, u^{(k)}) = 0, α = 1, \dots, q,

G_{α} (x, u^{(k)}) = 0, α = 1, \dots, q,

\overset{x}{ˉ} = Ξ (x, u; ε) \overset{u}{ˉ} = Φ (x, u; ε) .

\overset{x}{ˉ} = Ξ (x, u; ε) \overset{u}{ˉ} = Φ (x, u; ε) .

\begin{array}[]{ll}&\bar{x}^{i}=x^{i}+\varepsilon\xi^{i}(x,u)+{\cal O}(\varepsilon^{2}),\quad i=1,\ldots,p,\\ &\bar{u}^{\alpha}=u^{\alpha}+\varepsilon\eta^{\alpha}(x,u)+{\cal O}(\varepsilon^{2}).\end{array}

\begin{array}[]{ll}&\bar{x}^{i}=x^{i}+\varepsilon\xi^{i}(x,u)+{\cal O}(\varepsilon^{2}),\quad i=1,\ldots,p,\\ &\bar{u}^{\alpha}=u^{\alpha}+\varepsilon\eta^{\alpha}(x,u)+{\cal O}(\varepsilon^{2}).\end{array}

X = ξ^{i} \partial_{x^{i}} + η^{α} \partial_{u^{α .}}

X = ξ^{i} \partial_{x^{i}} + η^{α} \partial_{u^{α .}}

X [LHS Eq. (\ref main)] ∣_{E q . (\ref main)} = 0.

X [LHS Eq. (\ref main)] ∣_{E q . (\ref main)} = 0.

D_{i} = \frac{\partial}{\partial x ^{i}} + u_{i}^{α} \frac{\partial}{\partial u ^{α}} + u_{ij}^{α} \frac{\partial}{\partial u _{j}^{α}} + \dots .

D_{i} = \frac{\partial}{\partial x ^{i}} + u_{i}^{α} \frac{\partial}{\partial u ^{α}} + u_{ij}^{α} \frac{\partial}{\partial u _{j}^{α}} + \dots .

W^{α} = η^{α} - ξ^{j} u_{j}^{α} .

W^{α} = η^{α} - ξ^{j} u_{j}^{α} .

\frac{δ}{δ u ^{α}} = \frac{\partial}{\partial u ^{α}} + s \geq 1 \sum (- 1)^{s} D_{i_{1}} \dots D_{i_{s}} \frac{\partial}{\partial u _{i_{1} \dots i_{s}}^{α}} .

\frac{δ}{δ u ^{α}} = \frac{\partial}{\partial u ^{α}} + s \geq 1 \sum (- 1)^{s} D_{i_{1}} \dots D_{i_{s}} \frac{\partial}{\partial u _{i_{1} \dots i_{s}}^{α}} .

X (L) + L D_{i} (ξ^{i}) = D_{i} (f_{i}) .

X (L) + L D_{i} (ξ^{i}) = D_{i} (f_{i}) .

T^{i} = f^{i} - N^{i} (L) .

T^{i} = f^{i} - N^{i} (L) .

D_{i} T^{i} = 0

D_{i} T^{i} = 0

N^{i} = ξ^{i} + W^{α} \frac{δ}{δ u ^{α}} + s \geq 1 \sum D_{i_{1}} \dots D_{i_{s}} W^{α} \frac{δ}{δ u _{i i_{1} \dots i_{s}}^{α}},

N^{i} = ξ^{i} + W^{α} \frac{δ}{δ u ^{α}} + s \geq 1 \sum D_{i_{1}} \dots D_{i_{s}} W^{α} \frac{δ}{δ u _{i i_{1} \dots i_{s}}^{α}},

\frac{d x ^{i}}{ξ ^{i}} = \frac{d u}{η} = \dots

\frac{d x ^{i}}{ξ ^{i}} = \frac{d u}{η} = \dots

W^{[r]} (x^{i}, u) .

W^{[r]} (x^{i}, u) .

L_{X} g_{ab} = 2 ψ g_{ab},

L_{X} g_{ab} = 2 ψ g_{ab},

\begin{array}[]{lcl}\psi_{;ab}=0&\Longleftrightarrow&X\,\textrm{is a special CKV (sCKV)},\\ \psi_{,a}=0&\Longleftrightarrow&X\,\textrm{is a homothetic vector (HV)},\\ \psi=0&\Longleftrightarrow&X\,\textrm{is a Killing vector (KV)}.\end{array}

\begin{array}[]{lcl}\psi_{;ab}=0&\Longleftrightarrow&X\,\textrm{is a special CKV (sCKV)},\\ \psi_{,a}=0&\Longleftrightarrow&X\,\textrm{is a homothetic vector (HV)},\\ \psi=0&\Longleftrightarrow&X\,\textrm{is a Killing vector (KV)}.\end{array}

\begin{array}[]{ll}\square u+V(x^{i})u=&{1\over{\sqrt{|-g|}}}{\partial\over\partial x^{i}}\left({\sqrt{|-g|}}g^{ik}{\partial u\over\partial x^{k}}\right)\\ &+V(x^{i})u=0,\end{array}

\begin{array}[]{ll}\square u+V(x^{i})u=&{1\over{\sqrt{|-g|}}}{\partial\over\partial x^{i}}\left({\sqrt{|-g|}}g^{ik}{\partial u\over\partial x^{k}}\right)\\ &+V(x^{i})u=0,\end{array}

X = ξ^{i} (x^{k}) \partial_{i} + (\frac{2 - n}{n} ψ (x^{k}) u + a_{0} u + b (x^{k})) \partial_{u}

X = ξ^{i} (x^{k}) \partial_{i} + (\frac{2 - n}{n} ψ (x^{k}) u + a_{0} u + b (x^{k})) \partial_{u}

ξ^{k} V_{, k} + 2 ψ V - \frac{2 - n}{2} △ ψ = 0.

ξ^{k} V_{, k} + 2 ψ V - \frac{2 - n}{2} △ ψ = 0.

X = ξ^{i} (x^{k}) \partial_{i} + (a_{0} u + b (x^{k})) \partial_{u}

X = ξ^{i} (x^{k}) \partial_{i} + (a_{0} u + b (x^{k})) \partial_{u}

ξ^{k} V_{, k} + 2 ψ V = 0.

ξ^{k} V_{, k} + 2 ψ V = 0.

d s^{2} = ϵ d t^{2} + d x^{2} + d y^{2}, ϵ = \pm 1.

d s^{2} = ϵ d t^{2} + d x^{2} + d y^{2}, ϵ = \pm 1.

X^{1} = \partial_{t} X^{2} = \partial_{x} X^{3} = \partial_{y},

X^{1} = \partial_{t} X^{2} = \partial_{x} X^{3} = \partial_{y},

X^{4} = x \partial_{x} + y \partial_{y} + t \partial_{t},

X^{4} = x \partial_{x} + y \partial_{y} + t \partial_{t},

X^{5} = y \partial_{x} - x \partial_{y}, X^{6} = - ϵ t \partial_{x} + x \partial_{t}, X^{7} = - ϵ t \partial_{y} + y \partial_{t},

X^{5} = y \partial_{x} - x \partial_{y}, X^{6} = - ϵ t \partial_{x} + x \partial_{t}, X^{7} = - ϵ t \partial_{y} + y \partial_{t},

\begin{array}[]{lcl}X^{8}&=2{{x}{y}}\,\partial_{x}-\,({{t}^{2}\epsilon+{{x}}^{2}-{{y}}^{2}})\,\partial_{y}+2{{y}t}\,\partial_{t},&\psi^{1}=2y\\ X^{9}&=-(\,{{t}^{2}\epsilon-{{x}}^{2}+{{y}}^{2}})\,\partial_{x}+2{x}{y}\,\partial_{y}+2{x}t\,\partial_{t},&\psi^{2}=2x\\ X^{10}&\displaystyle{=2{x}t\,\partial_{x}+2{y}t\,\partial_{y}+\,{\frac{{t}^{2}\epsilon-{{x}}^{2}-{{y}}^{2}}{{\epsilon}}}}\,\partial_{t},&\psi^{3}=2t.\end{array}

\begin{array}[]{lcl}X^{8}&=2{{x}{y}}\,\partial_{x}-\,({{t}^{2}\epsilon+{{x}}^{2}-{{y}}^{2}})\,\partial_{y}+2{{y}t}\,\partial_{t},&\psi^{1}=2y\\ X^{9}&=-(\,{{t}^{2}\epsilon-{{x}}^{2}+{{y}}^{2}})\,\partial_{x}+2{x}{y}\,\partial_{y}+2{x}t\,\partial_{t},&\psi^{2}=2x\\ X^{10}&\displaystyle{=2{x}t\,\partial_{x}+2{y}t\,\partial_{y}+\,{\frac{{t}^{2}\epsilon-{{x}}^{2}-{{y}}^{2}}{{\epsilon}}}}\,\partial_{t},&\psi^{3}=2t.\end{array}

L (x^{i}, u, u_{, i}) = \frac{1}{2} V (t, x, y) u^{2} - \frac{1}{2} u_{, x}^{2} - \frac{1}{2} u_{, y}^{2} - \frac{1}{2 ϵ} u_{, t}^{2},

L (x^{i}, u, u_{, i}) = \frac{1}{2} V (t, x, y) u^{2} - \frac{1}{2} u_{, x}^{2} - \frac{1}{2} u_{, y}^{2} - \frac{1}{2 ϵ} u_{, t}^{2},

\frac{1}{ϵ} u_{, tt} + u_{, xx} + u_{, y y} + V (x, y, t) u = 0.

\frac{1}{ϵ} u_{, tt} + u_{, xx} + u_{, y y} + V (x, y, t) u = 0.

D_{t} T^{t} + D_{x} T^{x} + D_{y} T^{y} = 0

D_{t} T^{t} + D_{x} T^{x} + D_{y} T^{y} = 0

Y_{1} = X^{2} + κ_{1} X^{11} and Y_{2} = X^{7} + κ_{2} X^{11},

Y_{1} = X^{2} + κ_{1} X^{11} and Y_{2} = X^{7} + κ_{2} X^{11},

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces

Sameerah Jamal

School of Mathematics and Centre for Differential Equations,Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, South Africa

[email protected]

Andronikos Paliathanasis

Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile

Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa

[email protected]

Abstract

We perform the complete symmetry classification of the Klein-Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions $V(t,x,y)$ that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. For some of the potentials, we use the admitted Lie algebras to determine corresponding invariant solutions to the Klein-Gordon equation. An integral part of this analysis is the problem of the classification of Lie and Noether point symmetries of the wave equation.

keywords:

Lie symmetry, potential functions, Klein-Gordon equation, invariant solutions.

MSC:

[2010] 22E60, 76M60, 35Q75, 34C20

††journal: Journal of Geometry and Physics

1 Introduction

The group classification problem was initiated by Ovsiannikov [1] who analysed the nonlinear heat equation. Since then numerous studies have been devoted to group classifications of fundamental equations that model mathematical, relativistic, biological and physical phenomena [2, 3, 4, 5, 6, 7, 8] . In addition, there is now a rich body of literature surrounding Lie symmetry theory, its scheme and vast applications to differential equations [9, 10, 11, 12, 13]. In particular, wave and Klein-Gordon equations are of particular interest as they are two important equations in all areas of physics. A knowledge of the Lie symmetry structures of the Klein-Gordon equation in a Riemannian space enables the determination of solutions of this equation which is invariant under a given Lie symmetry.

Indeed, recent investigations [14, 15, 16, 17, 18, 19, 20] have revolved around wave, Klein-Gordon, Poisson and Schrödinger equations - showing that the Lie symmetry vectors are obtained directly from the collineations of a metric which defines the underlying geometry in which the evolution occurs. In [21] it was proved that for a linear (in derivatives) second-order partial differential equation (PDE), the Lie point symmetries are related to the conformal algebra of the geometry defined by the PDE. In [22], a geometric approach related the Lie symmetries of the Klein-Gordon equation to the conformal algebra of classes of the Bianchi I spacetime and a study of potential functions was performed. Whilst recently the connection between collineations and symmetries was established for a system of quasilinear PDEs [23]. A similar result has been proved for the Poisson equation [24].

In this paper, we use geometric results that transfer the problem of the Lie and Noether symmetry classification of the Klein-Gordon equation to the problem of determining the conformal Killing vectors that admit appropriate potential functions. Inspired from the symmetry classification of the two- and three-dimensional Newtonian systems in which a geometric approach was applied [25, 26], in this work in order to perform the classification, the conformal Killing vectors of the space are used to solve a constraint condition. The general results are applied to two practical problems viz., the classification of all potential functions in a three dimensional Euclidian and Minkowski space, for which the Klein-Gordon equation admits Lie and Noether point symmetries and secondly, the Lie point symmetries are used to determine invariant solutions of the equation.

The paper is organized as follows. Section 2 provides the geometrical preliminaries and the theoretical background about symmetry analysis. In section 3, we state the main theorem containing the constraint condition. Section 4 provides a short review about the spacetime and its properties and we perform the symmetry classification for the Klein-Gordon equation and the corresponding potentials. Section 5 illustrates some invariant solutions for the Klein-Gordon equation using particular potential functions. Finally in section 6 we draw our conclusions.

2 Preliminaries

In this section we review the definitions and properties of spacetime collineations and of the point symmetries of differential equations.

2.1 Lie and Noether Point Symmetries

Consider a system with $q$ unknown functions $u^{a}$ which depends on $p$ independent variables $x^{i}$ , i.e. we denote $u=(u^{1},\ldots,u^{q})$ and $x=(x^{1},\ldots,x^{p})$ , respectively. Let

[TABLE]

be a system of $m$ nonlinear differential equations, where $u^{(k)}$ represents the $k^{th}$ derivative of $u$ with respect to $x$ . A one-parameter Lie group of transformations ( $\varepsilon$ is the group parameter) that is invariant under (1) is given by

[TABLE]

Invariance of (1) under the transformation (2) implies that any solution $u=\Theta(x)$ of (1) maps into another solution $v=\Psi(x;\varepsilon)$ of (1). Expanding (2) around the identity $\varepsilon=0$ , we can generate the following infinitesimal transformations:

[TABLE]

The action of the Lie group can be recovered from that of its infinitesimal generators acting on the space of independent and dependent variables. Hence, we consider the following vector field

[TABLE]

The action of X is extended to all derivatives appearing in the equation in question through the appropriate prolongation. The infinitesimal criterion for invariance is given by

[TABLE]

Eq. (5) yields an overdetermined system of linear homogeneous equation which can be solved algorithmically, more details can be found in [9] among other texts.

The generalized total differentiation operator $D_{i}$ with respect to $x^{i}$ is given by

[TABLE]

and $W^{\alpha}$ is the characteristic function given by

[TABLE]

The Euler-Lagrange equations, if they exist, is the system ${\delta}L/{\delta u^{\alpha}}=0$ , where $\delta/\delta u^{\alpha}$ is the Euler-Lagrange operator given by

[TABLE]

$L$ is referred to as a Lagrangian. If we include point dependent gauge terms $f_{1},\ldots,f_{n}$ , the Noether symmetries $X$ are given by

[TABLE]

2.2 Conservation Laws

Corresponding to each $X$ , there exists a conserved vector $(T^{1},\ldots,T^{n})$ that may then be determined by Noether’s theorem [27]

[TABLE]

where

[TABLE]

along the solutions of the differential equation. Here $N^{i}$ is the Noether operator associated with the symmetry operator $X$ given by

[TABLE]

where $\delta/\delta u^{\alpha}$ is the Euler-Lagrange operator given by (8).

2.3 Invariant Solutions

The operator in Eq. (4) can be used to define the Lagrange system

[TABLE]

whose solution provides the invariant functions

[TABLE]

These invariants can be used in order to reduce the order of the PDE. Further details of the relevant equations and formulae can be found in, inter alia, [28].

2.4 Collineations of Riemannian Spaces

A one-parameter group of conformal motions, or Conformal Killing vectors (CKVs), generated by the vector field $X$ is defined by [29]

[TABLE]

where ${\cal L}_{X}$ is the Lie derivative operator along the vector field $X$ and $\psi=\psi(x^{a})$ is a conformal factor. If $\psi_{;ab}\neq 0$ , the CKV is said to be proper. The possible cases of $\psi$ provide special cases which form subalgebras on the conformal algebra of the space:

[TABLE]

3 Klein-Gordon Equations

The linear second order in derivatives Klein-Gordon equation is expressed as

[TABLE]

where $\square$ is the d’Alembertian operator. In a recent paper by Paliathanasis and Tsamparlis [17], it was shown that the Lie point symmetries of the Klein-Gordon equation in a general Riemannian space are elements of the conformal algebra of the space, modulo a constraint relation involving the Lie symmetry vector and the potential entering the Klein-Gordon equation. More specifically the following theorem was proved.

Theorem 1. The Lie point symmetries of the Klein-Gordon equation (16) in a Riemannian space of dimension $n$ are generated from the elements of the conformal algebra of the metric, as follows:

For $n>2$ the Lie symmetry vector is

[TABLE]

where $\xi^{k}$ is a CKV with conformal factor $\psi(x^{k}),b(x^{k})$ is a solution of (16) and the following condition involving the potential

[TABLE] 2. 2.

For $n=2$ the Lie symmetry vector is

[TABLE]

where $\xi^{k}$ is a CKV with conformal factor $\psi(x^{k}),b(x^{k})$ is a solution of (16) and the following condition involving the potential

[TABLE]

Constraint condition (17) acts as a double selection rule - selecting for each CKV a corresponding potential or, if this is not possible, abandoning the CKV for being a Lie point symmetry of the Klein-Gordon equation [22]. We remark that the case $V(x^{i})=0$ reduces the Klein-Gordon equation to that of the wave equation. Further, it is well known that a Lie algebra contains the Noether algebra, where the Noether algebra will exclude a symmetry involving the dependent variable, viz. $u\partial_{u}$ .

4 The Klein-Gordon Equation in three dimensional Maximal Symmetric Spacetimes

A Riemannian space which admits a Killing algebra of dimension $\frac{1}{2}n(n+1)$ is called a maximally symmetric metric space, for example, the Euclidian space $E^{3}$ and the Minkowski spacetime $M^{3}$ , defined by

[TABLE]

The conformal algebra of this space is 10-dimensional and consists of the following vectors.

3- gradient KVs:

[TABLE]

1- gradient HV:

[TABLE]

3- rotations:

[TABLE]

3- special CKVs with their respective conformal factors:

[TABLE]

In Table 1, we list real subalgebras of the conformal algebra, where the algebra $F_{r,k}$ denotes the $k$ -th algebra of dimension $r$ and excludes linear combinations.

The Klein-Gordon equation follows from the Lagrangian

[TABLE]

and is explicitly expressed as

[TABLE]

In this study, we apply elements of the conformal algebra in the solution of Eq. (17) to determine the form of the potentials $V(t,x,y)$ . That is, we consider three cases below, namely we apply (a) the vectors $X^{1-10}$ , (b) selected linear combinations of the vectors $X^{1-10}$ and (c) real subalgebras of the conformal algebra. For each CKV of the conformal algebra of the maximal symmetric spacetimes, we must solve constraint condition (17) where $n=3$ and find the potentials $V(x,y,t)$ for which it is satisfied. Before we proceed with the symmetry analysis, we mention that the Klein-Gordon equation is a linear equation which implies that it will always admit the linear symmetry $X^{11}=u\partial_{u}$ and the infinite dimensional abelian subalgebra of solutions $X^{\infty}=F(x,y,t)\partial_{u}$ , where $F(x,y,t)$ is a solution of Eq. (21). Due to the many results and for ease of reference, the results are presented in the form of tables.

4.1 Case I - The vectors $X^{1-10}$ .

Taking each of the vectors of the conformal algebra $X^{1-10}$ , we solved Eq. (17) and found the potentials $V(t,x,y)$ of Table 3. The columns of Table 3 contain the potential functions, corresponding Lie point symmetries, Lie invariant functions, Noether point symmetries and lastly, the associated conservation laws $T_{1-10}$ appear in Table LABEL:tab:table5. Hence, $(T^{t},T^{x},T^{y})$ is the conserved vector that satisfies

[TABLE]

on Eq. (21).

4.2 Case II - Linear combinations of $X^{1-10}$ .

In this case, for linear combinations, we take pairs of each of the vectors of the conformal algebra $X^{1-10}$ . In turn these linear combinations are applied to Eq. (17) and the potentials $V(t,x,y)$ of Table 5 are determined. Note that not all pairs of linear combinations of the vector fields provide us with potential functions. We do not consider all other possible linear combinations because the resulting Lie and Noether symmetries are too many but they can be computed in the standard way. In Table 5 below, $a$ and $b$ are arbitrary non-zero constants.

4.3 Case III - Real subalgebras of $X^{1-10}$ .

The real subalgebras contained within the conformal algebra $X^{1-10}$ which solve condition Eq. (17), are used in order to determine all the potentials in which the Klein-Gordon equation admits Lie and Noether symmetries. The list of potential functions appear in Table 6 - 7 together with the corresponding point symmetries. It is important to note that we display the smallest subalgebra that admits potentials in Tables 6. Moreover, we have that $a_{r},b_{r},c_{r}\neq 0\;(r=1,2,3,4,5)$ in Table 7 which include subalgebras containing linear combinations. It is necessary to remark for Case III, that when the Klein-Gordon equation admits a special CKV as its Lie/Noether point symmetry, then the form of the Lie point symmetry is expressed as a sum of the special CKV and its respective conformal factor, i.e. for instance we would have $X^{8}+\frac{1}{2}\psi^{1}u\partial_{u}$ .

5 Invariant Solutions

The above tables are useful because they provide the appropriate Lie point symmetries which can be used for the reduction of the Klein-Gordon equation and subsequently the determination of corresponding invariant solutions. In this section, we apply the Lie symmetries in order to reduce Eq. (21). We study the two cases: $V(t,x,y)=V(\epsilon t^{2}+y^{2})$ and $V(t,x,y)=V(-a_{3}x+y)$ .

${\bf a.V(t,x,y)=V(\epsilon t^{2}+y^{2})}.$ Based on Table 6, the subgroup labeled $F_{2,17}$ admits this particular potential. Thus, for the purpose of Lie reduction we may utilise the symmetries

[TABLE]

with Lie Bracket $\left[Y_{1},Y_{2}\right]=0.$ Reduction with respect to the Lie invariants of the symmetry vector $Y_{1}$ gives

[TABLE]

where $\zeta(t,y)$ satisfies the equation

[TABLE]

To this equation we apply $Y_{2}$ and obtain the second-order ordinary differential equation

[TABLE]

where $\sigma=\frac{1}{2}\left(\epsilon t^{2}+y^{2}\right)$ , $\phi(\sigma)$ and

[TABLE]

We continue with the determination of the invariant solutions for a second potential function.

${\bf b.V(t,x,y)=V(-a_{3}x+y)}.$ From Table 7, this potential function is admitted by the subgroup $\{X^{1},X^{2}+a_{3}X^{3},\frac{1}{\epsilon}(X^{6}+a_{3}X^{7}),X^{11},X^{\infty}\}$ . Hence, for reduction we may utilise the symmetries

[TABLE]

with Lie Bracket $\left[Z_{1},Z_{2}\right]=0.$ Reduction with respect to the Lie invariants of the symmetry vector $Z_{1}$ provides the solution

[TABLE]

where $\beta(t,y)$ satisfies the equation

[TABLE]

To Eq. (27) we apply $Z_{2}$ and obtain the second-order ordinary differential equation

[TABLE]

where $\alpha=-a_{3}x+y$ , $\rho(\alpha)$ and the solution of the Klein-Gordon equation (21) is

[TABLE]

6 Conclusion

We have determined the functional form of potential functions for which the resulting Klein-Gordon equation in Euclidean and Minkowski three dimensional space admits Lie and Noether point symmetries. The application of the conformal Killing algebra produces the classification of Lie and Noether point symmetries and potential functions. It is easily seen from Tables 3 and 5, that the generators of the Lie and Noether point symmetries are the CKVs and their linear combinations; while the real subalgebras generate interesting forms of the potential function in Table 6. Moreover, a consideration of subalgebras consisting of linear combinations of the vector fields produces a further list of potential functions in Table 7.

Naturally each of the Noether symmetries listed here can be used to determine a corresponding conservation law as displayed in Table LABEL:tab:table5. The usefulness of the tabular results can be seen in the reduction of the (1+2)-dimensional Klein-Gordon equation. We applied the zero-order invariants of the Lie symmetries and reduced the Klein-Gordon equation to a linear ordinary differential equation. In particular, concerning $V(t,x,y)=V(\epsilon t^{2}+y^{2})$ and $V(t,x,y)=V(-a_{3}x+y)$ , we found the closed forms of the group invariant solutions. In the analysis of $V(t,x,y)=0$ , the Klein-Gordon equation becomes the wave equation which has a Lie symmetry algebra that is identically the conformal algebra $X^{1-10}$ , and for a constant potential, i.e. $V(t,x,y)=V_{0}$ , the Lie symmetry algebra contains $X^{1-3,5-7}$ plus the the linear and infinite symmetry in both cases.

The results of this analysis can be used in various ways, such as to construct conservation laws for the equation of motions of a particle in the classical or the semi-classical approach. Last but not least, the results of this analysis holds for all the Yamabe equations (conformal Laplace equations) in which the underlying geometry, the metric which defines the Laplace operator, is conformally flat.

Acknowledgments: SJ would like to acknowledge the financial support from the National Research Foundation of South Africa with grant number 99279. AP acknowledges the financial support of FONDECYT grant no. 3160121.

References

Ovsiannikov [1982]

Ovsiannikov LV.

Group Analysis of Differential Equations.

New York: Academic Press; 1982.

Andriopoulos et al. [2009]

Andriopoulos K, Dimas S, Leach PGL, Tsoubelis D.

On the systematic approach to the classification of differential equations by group theoretical methods.

Journal of Computational Applied Mathematics 2009;230(1):224–32.

doi:10.1016/j.cam.2008.11.002.

Cicogna et al. [2006]

Cicogna G, Ceccherini F, Pedorado F.

Applications of symmetry methods to the theory of plasma physics.

SIGMA 2006;2(017).

doi:10.1063/1.1664886.

Paliathanasis [2016]

Paliathanasis A.

f(R)-gravity from Killing tensors.

Classical and Quantum Gravity 2016;33(7):075012.

URL: http://stacks.iop.org/0264-9381/33/i=7/a=075012.

Spichak and Stognii [1999]

Spichak S, Stognii V.

Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion.

Journal of Physics A: Mathematical and General 1999;32(47):8341.

URL: http://stacks.iop.org/0305-4470/32/i=47/a=312.

Ivanova et al. [2010]

Ivanova N, Sophocleous C, Leach PGL.

Group classification of a class of equations arising in financial mathematics.

Journal of Mathematics Analysis and Application 2010;372:723.

doi:10.1063/1.1664886.

Nucci and Sanchini [2015]

Nucci M, Sanchini G.

Symmetries, Lagrangians and conservation laws of an Easter Island population model.

Symmetry 2015;7(3):1613–32.

doi:10.3390/sym7031613.

Jamal et al. [2015]

Jamal S, Kara AH, Narain R, Shabbir G.

Symmetry structure of a wave equation on some classes of Bianchi cosmological models.

Indian Journal of Physics 2015;89(4):411–6.

doi:10.1007/s12648-014-0625-0.

Olver [1993]

Olver PJ.

Applications of Lie Groups to Differential Equations.

New York: Springer; 1993.

Baumann [2000]

Baumann G.

Symmetry Analysis of Differential Equations with MathematicaⓇ.

New York: Springer; 2000.

Bluman and Kumei [1989]

Bluman GW, Kumei S.

Symmetries and Differential Equations.

New York: Springer; 1989.

Hydon [2000]

Hydon PE.

Symmetry Methods for Differential Equations: A Beginner’s Guide.

Cambridge: Cambridge University Press; 2000.

Ibragimov [1994]

Ibragimov NH, ed.

CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1: Symmetries, Exact Solutions, and Conservation Laws.

Boca Raton: CRC Press; 1994.

Jamal et al. [2012a]

Jamal S, Kara AH, Narain R.

Wave equations in Bianchi space-times.

Journal of Applied Mathematics 2012a;2012:765361.

doi:10.1155/2012/765361.

Jamal and Kara [2015]

Jamal S, Kara AH.

A classification of zero gauge Noether symmetries for the wave equation on cylindrically symmetric static manifolds.

Romanian Journal of Physics 2015;60(9-10):1328–36.

Bokhari et al. [2011]

Bokhari AH, Al-Dweik AY, Kara AH, Karim M, Zaman FD.

Wave equation on spherically symmetric Lorentzian metrics.

Journal of Mathematical Physics 2011;52:063511.

doi:10.1063/1.3597232.

Paliathanasis and Tsamparlis [2014]

Paliathanasis A, Tsamparlis M.

The geometric origin of Lie point symmetries of the Schrödinger and the Klein Gordon equations.

International Journal of Geometric Methods in Modern Physics 2014;11(04):1450037.

doi:10.1142/S0219887814500376.

Jamal et al. [2012b]

Jamal S, Kara AH, Bokhari AH.

Symmetries, conservation laws, reductions, and exact solutions for the Klein-Gordon equation in de Sitter space-times.

Canadian Journal of Physics 2012b;90:667–74.

doi:10.1139/p2012-065.

Jamal and Shabbir [2016]

Jamal S, Shabbir G.

Noether symmetries of vacuum classes of pp-waves and the wave equation.

International Journal of Geometric Methods in Modern Physics 2016;13(09):1650109.

doi:10.1142/S0219887816501097.

Jamal et al. [2013]

Jamal S, Kara AH, Bokhari AH, Zaman FD.

The symmetries and conservation laws of some Gordon-type equations in Milne space-time.

Pramana 2013;80(5):739–55.

doi:0.1007/s12043-013-0518-3.

Paliathanasis and Tsamparlis [2012]

Paliathanasis A, Tsamparlis M.

Lie point symmetries of a general class of PDEs: The heat equation.

Journal of Geometry and Physics 2012;62(12):2443 –56.

doi:10.1016/j.geomphys.2012.09.004.

Paliathanasis et al. [2015]

Paliathanasis A, Tsamparlis M, Mustafa MT.

Symmetry analysis of the Klein Gordon equation in Bianchi I spacetimes.

International Journal of Geometric Methods in Modern Physics 2015;12(03):1550033.

doi:10.1142/S0219887815500334.

Paliathanasis and Tsamparlis [2016]

Paliathanasis A, Tsamparlis M.

Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations.

Journal of Geometry and Physics 2016;107:45–59.

doi:10.1016/j.geomphys.2016.05.004.

Bozhkov and Freire [2010]

Bozhkov Y, Freire IL.

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation.

Journal of Differential Equations 2010;249(4):872–913.

Tsamparlis and Paliathanasis [2011]

Tsamparlis M, Paliathanasis A.

Two-dimensional dynamical systems which admit Lie and Noether symmetries.

Journal of Physics A: Mathematical and Theoretical 2011;44(17):175202.

URL: http://stacks.iop.org/1751-8121/44/i=17/a=175202.

Tsamparlis et al. [2012]

Tsamparlis M, Paliathanasis A, Karpathopoulos L.

Autonomous three-dimensional Newtonian systems which admit Lie and Noether point symmetries.

Journal of Physics A: Mathematical and Theoretical 2012;45(27):275201.

URL: http://stacks.iop.org/1751-8121/45/i=27/a=275201.

Noether [1918]

Noether E.

Invariante Variationsprobleme.

Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse 1918;1918:235–57.

URL: http://eudml.org/doc/59024.

Stephani [1989]

Stephani H.

Differential Equations: Their Solution Using Symmetries.

Cambridge: Cambridge University Press; 1989.

Katzin et al. [1969]

Katzin GH, Levine J, Davis WR.

Curvature collineations: A fundamental symmetry property of the space-times of General Relativity defined by the vanishing Lie derivative of the Riemann curvature tensor.

Journal of Mathematical Physics 1969;10:617.

doi:10.1063/1.1664886.

Bibliography29

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1Ovsiannikov [1982] Ovsiannikov LV. Group Analysis of Differential Equations. New York: Academic Press; 1982.
2Andriopoulos et al. [2009] Andriopoulos K, Dimas S, Leach PGL, Tsoubelis D. On the systematic approach to the classification of differential equations by group theoretical methods. Journal of Computational Applied Mathematics 2009;230(1):224–32. doi: 10.1016/j.cam.2008.11.002 . · doi ↗
3Cicogna et al. [2006] Cicogna G, Ceccherini F, Pedorado F. Applications of symmetry methods to the theory of plasma physics. SIGMA 2006;2(017). doi: 10.1063/1.1664886 . · doi ↗
4Paliathanasis [2016] Paliathanasis A. f(R)-gravity from Killing tensors. Classical and Quantum Gravity 2016;33(7):075012. URL: http://stacks.iop.org/0264-9381/33/i=7/a=075012 .
5Spichak and Stognii [1999] Spichak S, Stognii V. Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion. Journal of Physics A: Mathematical and General 1999;32(47):8341. URL: http://stacks.iop.org/0305-4470/32/i=47/a=312 .
6Ivanova et al. [2010] Ivanova N, Sophocleous C, Leach PGL. Group classification of a class of equations arising in financial mathematics. Journal of Mathematics Analysis and Application 2010;372:723. doi: 10.1063/1.1664886 . · doi ↗
7Nucci and Sanchini [2015] Nucci M, Sanchini G. Symmetries, Lagrangians and conservation laws of an Easter Island population model. Symmetry 2015;7(3):1613–32. doi: 10.3390/sym 7031613 . · doi ↗
8Jamal et al. [2015] Jamal S, Kara AH, Narain R, Shabbir G. Symmetry structure of a wave equation on some classes of Bianchi cosmological models. Indian Journal of Physics 2015;89(4):411–6. doi: 10.1007/s 12648-014-0625-0 . · doi ↗

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces

Abstract

keywords:

MSC:

1 Introduction

2 Preliminaries

2.1 Lie and Noether Point Symmetries

2.2 Conservation Laws

2.3 Invariant Solutions

2.4 Collineations of Riemannian Spaces

3 Klein-Gordon Equations

4 The Klein-Gordon Equation in three dimensional Maximal Symmetric Spacetimes

4.1 Case I - The vectors X1−10X^{1-10}X1−10.

4.2 Case II - Linear combinations of X1−10X^{1-10}X1−10.

4.3 Case III - Real subalgebras of X1−10X^{1-10}X1−10.

5 Invariant Solutions

6 Conclusion

References

4.1 Case I - The vectors $X^{1-10}$ .

4.2 Case II - Linear combinations of $X^{1-10}$ .

4.3 Case III - Real subalgebras of $X^{1-10}$ .