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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 . 57) in the long-wave limit. To show this, keep a, and xol finite, choose tio)= n, and take the long-wave limit k, -+ 0. 59) [Eq. 3). 66) denotes the summation over all possible combinations where lr=o, of p1 = 0, 1, p 2 = O , l , . ,pN = O,1, and 1 i T ) k means the summation under the condition j c k.
F - - iD, f * - f = D: f*-f, which is the bilinearized BO equation. 18) which can be verified by direct substitution. 19) where cl, c, c 3 ,and b are unknown constants. 19) may be arbitrarily chosen. 20) + cle, + c,e, + b. 21) It follows by direct calculation that + i(c, - c:)a18t - ia,(c,ct - c:cz - b + b*)dl - ia,(c,cf - clcf - b* + b)& - i(a,cf + a,cf)b + i(alc, + azc2)b* i D t f t . f 2 = i(c, - cf)a,8: and ~ f f rf,. = + e: + (c, + c;)e, + (cl + c:)e, + b + b* + (cl + cz)(c: + c t ) ] . 17), the equations required to determine the unknown constants c,, c,, and b may be derived.
5) and (App. fz) *flfz = D x fl fz . 215) that Qifofz = W x f i f z . + + ( 3 / 2 c N f O-Jl1. fZlfl + 3 w : - *PP:)fO - f 1 l x f z - 3 C E - iPp:)fo *fZlXf1. 221) Using formulas (App. 1), (App. 3), and (App. 221) is converted to 0 = -f0[0,+ %P? fz + (3/C)f0(Dxf0 . ~ I z ) , + (3/2c)fO(fo J; dXx + 3P: + P W X - +D;31fl * f Z - ( 6 / ~ ) f o . Tlz}. 224) =O, since fo fi # 0. 211) has been verified. 212, and therefore we have completed the proof of the commutability relation for the time part of the Backlund transformation.