Streetj Chan and Fvomm 



[ A] W = 



Ct,, CX|2 c».,3 



^21 ^22 ^23 

 ^1 ^2 ^33 



U 



= 



(29) 



7:t* 



where the aj: = Rj: (ajP, y> JR»^ ^Ci »o-2»1Jl) . Because W ^ in 

 general, a Fourier component solution can exist and Eq, (29) can be 

 satisfied only if | A| = 0, viz. , only if a set of eigenvalues exists for 

 the matrix A. The condition JAJ = leads to a determination of jx} 

 linear stability depends on the amplitude of fx. If ||ji | ^ 1 the solu- 

 tion by Eqs. (16-21) would be termed linearly stable. 



Analysis of the coefficient matrix A in Eq, (29) is coinplex 

 and is not reproduced here, but the key results are as follows. 

 First, if 0| = 2TrA/\, and 62= 27r^/\2. then for finite At, X, and 

 X.2» 



lim ||i I = 1 



Similarly, for finite A, \, and \2, 



lim Ifi I = 1 

 At-^O 



The solution scheme is stable in these limit cases. 



Second, the case 62 = 0> namely, \2= 00, was investigated. 



Then, 



Ihl| = f(a,Y>0,) 



for assumed A and R. This approach leads to a complex, cubic 

 equation with a possible root 



|jx,| = 1 + O(a^) 



(30) 



and a possible root pair with ma-ximum modulus equal to or slightly 

 exceeding unity, i.e. , 



I M- 2,3 1 max — ^ 



(31) 



Specifically, in a solitary wave computation with A = 0. 5 and 



At = 0.25, we have for t|=0.1, y = 0.05, a<0.05, p = 0, R = 0.5, 



168 



