By Robert M. Miura (auth.), Robert M. Miura (eds.)
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Additional info for Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications: NSF Research Workshop on Contact Transformations
After several trials we find (8) is transformed into (9) (GtF - GF t + GxxxF - 3GxxFx + 3Gx xxF - GFxxx)/F2 + 3(GxF - GFx)[=G2 - 2 ( F F x x - F~)]/F 4 = 0 . --l'r)+ 3 3t (~- 8 ~) 3 , , ]gl(x,t)f2(x ,t )l~x, 2 ~--~f432+ (~3 _~_fr>2f2(x,t>f2(~,t,>l~=~,t=t ' and so on. t=t, = 0 , = 2=glg 3 We start with a solution gl = ex~POll) + ex'p(r~2) where ni = pix - ~it - 71, ~i " p3i' 7i " constant, for i = 1,2. Note that the above equations are linear differential equations with known inhomogeneous terms.
PRELIMINARIES In order to illustrate the present method we first consider the modified Korteweg-deVries equation [i],  (1) v t + 0~(V3)x + Vxx x = 0 where subscripts indicate partial differentiations. We solve the equation by the usual perturbation method. as a power series in a parameter (2) We expand c v = ~v I + ~3v 3 + c5v 5 + "'" D Substituting (2) into (1) and collecting terms with the same powers of ob tain v ~, we 41 ~3 (~ + ~)v I ffi0 , 83 (~+~)v3 (3) =-~(v3) x , 83 ( ~ + _ ~ ) v 5 = _ 3c~(vlv3) 2 x and so on.
Eigenvalues, the solution V will interact uonlinearly, position of N formula for lVl 2 Y b(1,0) = 0, will consist of but, asymptotically solitons in the form when If (22). N and there are such solitons, V N discrete these solitons will consist of the super- From Zakharov and Shabat , the is an N-soliton solution is IV[ 2 = 4 %n[det(l+ZZ*)]xx where Z is an NxN matrix with elements defined by Zjs = e-i(Xj-Is (lj-xg) 0ne interesting type of solution is the "breather". These are real solutions of (16) formed by two solitons whose associated elgenvalues by I I = -I 2 E I ~ I r + iIi .