Polynomial Conserved Densities for the Sine-Gordon Equations

Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z, xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z,x, z, xx, etc. We prove that the generalized sine–Gordon equation z,xt = F(z) has an infinity of such polynomial conserved densities if, and only if, F(z) = A eαz + Be–αz for complex valued A, B and α ≠ 0. If F(z) does not take the form A eαz + Beβz there is no p. c. d. of rank greater than two. If α ≠ –β there is only a finite number of p. c. ds. If α = –β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β. Neither the odd rank p. c. ds in the case α = – β, nor the particular limited set of p. c. ds in the case when α ≠ –β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Bäcklund transformations and, via Noether’s theorem, with infinitesimal Bäcklund transformations. All equations with Bäcklund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Bäcklund transformations. We deduce that multiple sine–Gordon equations like z,xt = sin z + ½ sin ½z, which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Bäcklund transformations if, and only if Ḟ(z) – α2F(z) = 0.