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44) 54 3 The Benjamin-Ono Equation We shall now proceed to the periodic wave case. 9) and (3. 47) cx - x ; ( m Im x,@) > 0, j = 1, 2, . . j = 1, 2, . . 49) where x j and xi are complex functions o f t whose imaginary parts are positive or negative, respectively. 49). 1) yields the bilinearized BO equation iD, f ’ . 51) Dz f ’ . 17). 2). 55) where k , , a,, and xol are real constants and t\’) is an arbitrary phase constant. 49) it is necessary that 4lh ’0. 57) This form coincides with the periodic wave solution presented by Benjamin [l5] and Ono .
We shall now verify this relation by employing bilinear formalism. f2, withf,f,f, # 0. 204) p2 5 ( 0 2x -1 4Pl)fZ 2 'T12 = 0. 203)becomes a new solution. 189, (App. 203) to give . f l l f 2 - c(D: - iPp:)fo . f2)x + KP2 - P 3 f O fl f 2 = - 2c - Yo. Jl z + c - YOU0 f12)x + KPZ’ - P 3 f o fl f 2 =f0c-~-’~f0,xJ12 - f o J 1 2 , J + KP2 - P3flf21 0 = [(D2x - 1 4Pl)fO X@X + i ( P 2 - P:)flf21. )flf2. It follows that p1f0f2 = [(O: -id)fi = DXC(D,fl ‘f121fofz - f ~ . 208) where use has been made of (App.
155) = 0. 156) An introductorymonograph which treats the Bicklund transformations of nonlinear evolution equations that appear in physics and applied mathematics has been written by Rogers and Shadwick . 154), f ‘ also satisfies the same equation. 156) can be regarded as a relation that connects the pair of solutionsf and f‘, that is, the Backlund transformation of the KdV equation. Using formulas (App. 2) and (App. fl . f] . f’) = 0, where an arbitrary parameter 1 has been introduced. 159) These constitute the Backlund transformation of the KdV equation in the bilinear formalism.