Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications

1976 ◽  
Keyword(s):  
Inverse Scattering ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Backlund Transformations

Polynomial conserved densities for the sine-Gordon equations

1977 ◽  
Vol 352 (1671) ◽  
pp. 481-503 ◽  
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Sine Gordon Equation ◽  
Backlund Transformations

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 e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz 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 ½ z , 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 ) – α 2 F ( z ) = 0.

Bäcklund transformations for the sine–Gordon equations

1976 ◽  
Vol 351 (1667) ◽  
pp. 499-523 ◽  
Keyword(s):  
Pulse Propagation ◽  
Numerical Solutions ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Backlund Transformation ◽  
Optical Pulse Propagation ◽  
Backlund Transformations

The generalized sine–Gordon equations z ,xt = F ( z ) in two independent variables x , t include the sine–Gordon z ,xt = sin z and the multiple sine–Gordon’s like z ,xt = sin z + ½ sin ½ z . Among other physical applications all these sine–Gordon’s are significant to the theory of intense ultra-short optical pulse propagation. The sine–Gordon itself has analytical multi-soliton solutions. It also has an infinity of polynomial conserved densities and has auto-Bäcklund transformations which generate a second solution of the sine–Gordon from a first solution – particularly from the solution z ≡ 0. We prove first that the generalized multi-dimensional sine–Gordon in two or more space variables x 1 , x 2 , . . . has no auto-Bäcklund transformations. Next we prove that the generalized sine–Gordon’s z ,xt = F ( z ) and z ' ,xt = G ( z ') have an invertible Bäcklund transformation between solutions z and z ' if and only if F and G are solutions of F ¨ = α 2 F , G ¨ = β 2 G where, in general, β = αh -1 , α is a complex number and h 2 (≠ 0) is real. In case h = 1 and F and G are the same function z ,xt = F ( z ) has an auto-Bäcklund transformation if and only if F ¨ = α 2 F . We exhibit the B. ts and a. B. ts in these cases as well as the other B. ts for the generalized sine–Gordon. We conclude that the multiple sine–Gordon’s do not have a. B. ts and infer that, despite the soliton character of the numerical solutions, the multiple sine–Gordon’s are not soluble by present simplest formulations of the two by two inverse scattering method.

scholarly journals On the Extension of Inverse Scattering Method

1974 ◽  
Vol 52 (2) ◽  
pp. 397-414 ◽  
Author(s):  
M. Wadati ◽  
T. Kamijo
Keyword(s):  
Inverse Scattering ◽  
Inverse Scattering Method ◽  
Scattering Method
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