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Extra resources for Communications In Mathematical Physics - Volume 285
Without any loss of generality one can set α = −1, which gives f (s) = 1 ln(s − 1) + 3 2 ln(s − ) + ln(s − 2 ) , 3 = 1. Hamiltonian Systems of Hydrodynamic Type in 2 + 1 Dimensions 41 Potential (19). The corresponding system (14) takes the form vT + 2(vw 2 )Y = 0, w X + 2(v 2 w)Y = 0. (30) It possesses the Lax pair ψT = w 2 a(ψY ), ψ X = −v 2 b(ψY ), where the dependence of a and b on ψ y ≡ ξ is governed by the ODEs a = −4 a b − 2, b = 4 + 2. b a To solve these equations we proceed as follows. Expressing b from the first equation, b = −4a/(a + 2), and substituting into the second one arrives at a second order ODE 2aa − 3(a )2 + 12 = 0.
Keeping in mind that F3 = G 1 = K 2 = 0, we can rewrite these equations in the form F GK = 0, 12 G FK = 0, 23 K FG = 0, F3 = G 1 = K 2 = 0. (53) 13 The system (53) possesses obvious symmetries F → f 1 (u 1 ) f 2 (u 2 )F, G → f 2 (u 2 ) f 3 (u 3 )G, K → f 1 (u 1 ) f 3 (u 3 )K , u 1 → g1 (u 1 ), u 2 → g2 (u 2 ), u 3 → g3 (u 3 ); (54) here f i and gi are six arbitrary functions of the indicated arguments. As a first step, we introduce the new variables p= K1 F1 F2 G2 G3 K3 − , q= − , r= − , K F F G G K which are nothing but the invariants of the first ‘half’ of the symmetry group (54).
To do so one has to demonstrate the consistency of the relations (8), (10) where the characteristic speeds ν i and µi satisfy the dispersion relation det (ν I3 + µA + B) = 0, and ∂i u is the right eigenvector of the matrix ν i I3 + µi A + B — see Sect. 2. Theorem 5. The diagonalizability conditions (32) are necessary and sufficient for the existence of an infinity of n-component hydrodynamic reductions parametrized by n arbitrary functions of a single variable. Proof. The necessity follows from the general result of  which states that, for a quasilinear system (4), the diagonalizability is a necessary condition for the existence of an infinity of hydrodynamic reductions.