By C. Rogers and W.F. Shadwick (Eds.)

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**Extra resources for Bäcklund Transformations and Their Applications**

**Sample text**

Once ¢ has been determined, the free surface is given by z = h(x,t) = h* - g-l¢t(x,h*,t). 161) that are appropriate to horizontal wave propagation are now sought. 162) 48 1. 157) shows that Z(z) = Z(O) cosh(kz). 164) + h*. 166) connecting the wave frequency w = 2n/T and wave number k = w/Vp = 2n/A (namely, the number of cycles per unit distance). Here, in the usual notation, T is the wave period, Vp = w/k is the wave speed with which the disturbance propagates to the right, while A = Vp T is the wave length (that is, the distance between successive wave crests); H/A is called the wave steepness, while e = kx - wt is known as the phase of the motion.

0-PI'~-G Fig. 1 A Bianchi diagram. 2 There exist cPn+ 1 and I/In+ 1 such that cPn+1 = I/In+1 = cP, where In order to establish the above result, we follow the argument developed by McLaughlin and Scott [170]' Thus, on appropriate use of the Backlund 20 1. 19), the Bianchi diagram (Fig. 1) indicates that t A.. A.. {¢n +2¢n-l} ' 'f'n,x' = 'f'nI,x' + 2/3' 1 SIn A.. 24) 2 . {¢n - ¢n-l} 2 ' 'f'n-1,x 2 + /31 SIn A.. 25) A.. A.. 26) A.. A.. 2. t, A.. 28) ' 2 . {ljIn - ¢n-l} 2 ' A.. 30) 2. {ljIn+ 1 -ljIn} n,x 2 + /31 SIn 2 .

12. 4 LONG JOSEPHSON JUNCTION: ELLIPTIC SINE-GORDON EQUATION ! • 4>2 4>2 1 i {13(2)+v 2} a +v l 4>1 I Fig. 1l 2 The Bianchi diagram for second-generation ! C( solutions. °2 °2 a(2)+ vI I 37 • . UI 1 i{l3+v2} 2 Fig. 12 The Bianchi diagram for second-generation fJ solutions. 5 may be used iteratively to generate multiple o: and f3 solutions according to the sequence of relations ( r:t. t an { n 2n + vt ) - ( 4 = 1,2,3, ... , r:t. 2n- 2 + VI)} ¢2n # ¢2n-1 ± Nn, N = 0, 1,2, ... 107) 38 1. BACKLUND TRANSFORMATIONS AND NONLINEAR EQUATIONS and = cot ( (J 2n - 1 - 2 (J 2n) n = 1,2,3, ...