Consider the single soliton 2 sech2x and its associated
Chapter 14, Problem 9(choose chapter or problem)
Consider the single soliton 2 sech2x and its associated eigenvalue problem xx+(2sech2 x +) =0. (a) Substitute s=tanh x to convert it to [(1s2)]+2+ 1s2 =0 where = d ds . ThisistheassociatedLegendreequationofdegreeone(seeSection 10.6). (b) Showthatasolutionthatvanishesat x =(s =1)is = 1 and = 1 2 1s2 = 1 2 sechx. This is the unique bound state of this Schrodinger equation! (c) (difcult) By analyzing the solutions of the associated Legendre equation for =k2 > 0, show that R(k)0 and T(k)= (ik1)/(ik+1).
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