Equation 22 always has time translation symmetries

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Equation (2.2) always has time translation symmetries and we may multiply solutions by a constant. In our context, we term these trivial symmetries. We require richer, non trivial (at least four dimensional) symmetry groups. For γ 6 = 2, introduce h ( x ) = x 1 - γ f ( x ). It can be shown (see [8]) that the Lie algebra of symmetries of (2.2) is non trivial if and only if for a given g , h satisfies any one of the following Riccati equations. σxh 0 - σh + 1 2 h 2 + 2 σx 2 - γ g ( x ) = 2 σAx 2 - γ + B, σxh 0 - σh + 1 2 h 2 + 2 σx 2 - γ g ( x ) = Ax 4 - 2 γ 2(2 - γ ) 2 + Bx 2 - γ 2 - γ + C, σxh 0 - σh + 1 2 h 2 + 2 σx 2 - γ g ( x ) = Ax 4 - 2 γ 2(2 - γ ) 2 + Bx 3 - 3 2 γ 3 - 3 2 γ + Cx 2 - γ 2 - γ - κ, with κ = γ 8 ( γ - 4) σ 2 . The constant factors multiplying A , B and C above are included to simplify our later notation. Remark 2.2 . Clearly we could fix f in advance and these equations would then give us conditions on g which guarantee the existence of non trivial symmetries. For the first Riccati equation Craddock and Lennox proved in [8] the following result.
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FUNDAMENTAL SOLUTIONS 7 Theorem 2.3. Suppose that γ 6 = 2 and h ( x ) = x 1 - γ f ( x ) is a solution of the Riccati equation σxh 0 - σh + 1 2 h 2 + 2 σx 2 - γ g ( x ) = 2 σAx 2 - γ + B. (2.3) Then the PDE (2.2) has a symmetry solution of the form U ² ( x, t ) = 1 (1 + 4 ²t ) 1 - γ 2 - γ exp - 4 ² ( x 2 - γ + (2 - γ ) 2 t 2 ) σ (2 - γ ) 2 (1 + 4 ²t ) × exp ( 1 2 σ ˆ F ˆ x (1 + 4 ²t ) 2 2 - γ ! - F ( x ) !) u ˆ x (1 + 4 ²t ) 2 2 - γ , t 1 + 4 ²t ! , where F 0 ( x ) = f ( x ) /x γ and u is a solution of the PDE. That is, for ² sufficiently small, U ² is a solution of (2.2) whenever u is. If u ( x, t ) = u 0 ( x ) with u 0 an analytic, stationary solution then there is a fundamental solution p ( x, y, t ) of (2.2) such that Z 0 e - λy 2 - γ u 0 ( y ) p ( x, y, t ) dy = U λ ( x, t ) . (2.4) Here U λ ( x, t ) = U 1 4 σ (2 - γ ) 2 λ . Further, if u 0 = 1 , then R 0 p ( x, y, t ) dy = 1 . Thus we can find fundamental solutions by inverting a generalized Laplace transform. These generalized Laplace transforms can always be explicitly inverted, see [10]. Inversion of the transforms which arise from Theorem 2.3 will frequently involve the use of distributions. The following result is useful for the purpose of inverting the Laplace trans- forms that we will encounter. Proposition 2.4. When n is a non-negative integer we have L - 1 [ λ n e k λ ] = n X l =0 k l l ! δ ( n - l ) ( y ) + k y n +1 2 I n +1 2 p ky · , (2.5) where L is the Laplace transform, δ ( y ) is the Dirac delta and I n is a modified Bessel function of the first kind. For a proof, together with more general Laplace transforms of dis- tributions, see [10]. 2.0.1. Reduction to Canonical Form Revisited. Using Theorem 2.3 we reconsider the method of reduction to canonical form. Example 2.1. Consider the PDE u t = u xx - A x 2 u, x > 0 , (2.6) and for simplicity let A > - 3 4 . Stationary solutions of the PDE are u 0 ( x ) = x 1+ 1+4 A 2 and u 1 ( x ) = x 1 - 1+4 A 2 . Using Theorem 2.3 and u 0 we
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8 MARK CRADDOCK know that there is a fundamental solution Q with Z 0 e - λy 2 u 0 ( y ) Q ( x, y, t ) dy = u 0 ( x ) (1 + 4 λt ) 1+ 1+4 A 2 exp - λx 2 1 + 4 λt .
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