~~ -1jcfh = q (x, t) 2, (la)

Transcription

1 1652 Progress of Theoretical Physics, Vol. 53, No. 6, June 1975 Simple Derivion of Backlund Transformion from Ricci Form of Inverse Methodt> Kimiaki KONNO and Miki W ADA TI* Department of Physics and Atomic Energy Research Institute, College of Science and Engineering, Nihon University, Tokyo *Institute for Optical Research, Tokyo University of Educion Shinjuku-ku, Tokyo (Received October 30, 1974) A simple method for deriving the Backlund transformion from the Ricci form of inverse method is presented for the nonlinear evolution equions. I) Recently it has been made clear th a class of the physically interesting nonlinear evolution equions can be solved by the inverse method and by the Backlund transformion. The purpose of this paper is to propose a simple method for deriving the Backlund transformion from the Ricci form of the inverse method. Similar problem has been considered independently by Chen. 1 > He derived the Backlund transformion by looking for the transformion which keeps the equion for the wave function invariant. However, the relion between his gauge-like invariance and the Backlund transformion is mhemically and physically unclear. In a previous paper. 2 > we have shown th the Backlund transformion is interpreted as the procedure to add one more soliton to the original solution of the same nonlinear evolution equion. Also we have shown th new soliton solution can be constructed from the old one and its wave function. Then, the Backlund transformion can be derived by elmining the wave function in the fundamental equions of the inverse method. By extending the previous results to the Ricci form of the inverse method, we have found a function for transforming a solution to another one where both solutions sisfy the same nonlinear evolution equion. II) We consider the following sctering problem : 3 > ~~ -1jcfh = q (x, t) 2, (la) in which eigenfunctions 1 ane 2 evolve in time according to (lb) 1' The result of this paper was reported the meeting on "The nonlinear dispersive system and soliton" held Plasma Institute of Nagoya University during September 26"-'29, 1974.

2 Simple Derivion of Backlund Transformion from Ricci Form 1653 The eigenvalues 1J arf-h =A(x, t; 1J)r/h +B(x, t; 11) 2, (2a) a 2 =C(x, t; 1J) 1-A(x, t; 1J) 2 (2b) are time invariant when a A =qc-rb, (3a) ab -21JB=qt-2Aq, ac +21JC=n+2Ar. Equions (3) are equivalent to a large class of the nonlinear evolution equions by the various choices of coefficients A, B and C: a) The Korteweg-de Vries (K-dV) equion: qt+6qq..,+q..,..,..,=o, r= -1, A= -41] 8-21jq-q..,, l B~ -q..-2w.-4n'q-2q', (4b) C=41J 2 +2q. b) The Modified Korteweg-de Vries (M. K-dV) equion: qt+6lq..,+q..,..,..,=o, (3b) (3c) (4a) (5a) c) The Sine-Gordon equion: Ut_,=sin u, u r- -q- "" - -2, r= -q, A= -41j 8-21Jq 2, B= -q..,..,-21jq_,-41j 2 q-2q 8, C=q..,..,-21Jq..,+4r/q+2l. 1 A=~ cos u, 41] d) The Nonlinear Schrodinger equion: iqt+q_,..,+2jqj 2 q=o, (5b) (6a) (6b) (7a) III) By introducing a function r= -q*, A=2ir/+iJqJ 2, B = iq.., + 2i1Jq, C = iq: - 2i1Jq*. (7b) (8)

3 1654 K. Konno and M. Wadi Eqs. (1). and (2) are reduced to the Ricci equions: ar. 2 =21JT+q-rT, (9a) ar =B+2Ar-cr. (9b). Our procedure in the following is th we construct a transformion F 1 sisfying the same equion as (9a) with a potential q 1 (x) where q 1 (x) =q(x) + f(t, "IJ). (10) Thus, elimining T in Eqs. (9) and. (10), we have a Backlund transformion to a desired nonlinear equion.. We consider the following thr~e classes: 1) r= -1, 2) r= -q and 3) r= -q*. Class 1) r= -1 The Ricci equion (9a) becomes If we choose T 1 and q 1 as ar =21JT+q+r. (11) T 1 = -T-21J, (12a) q 1 (x) =q(x) +2_1_( -T-21J), (12b) r' with q 1 sisfies Eq. (11). If we elimine T in Eqs. (11) and (9b) with (12b), we get the Backlund transformion ~ + _ (:w - W 1 Y W:c W:c.-- "IJ, Wt -Wt=2B+4A 2 1 w -w where we put q= -w:c and q 1 = -w/. Class 2) r=...:.q Equi'on (9a) becomes [ I 2 (13a) (13b) If we choose T 1 and q 1 as ar =21JT+q+qr. r' =.!.. r' q 1 (x) =q(x)-2_1_ tan- 1 T, (14) (15a) (15b)

4 Simple Derivion of Backlund Transformion from Ricci Form 1655 T' with q' sisfies Eq. (14). If we elimine. T in Eqs. (14) and (9b) with (15b), we get the Backlund transformion for this class as w..,+w,' = -21} sin(w-w'), (16a) Wt-wt' = (C-B)- (B+C)cos(w-w') +2A sin(w-w'), (16b) where we.put q= -w.., and q'= -w,'. Equions (16) are the Backlund transformion for the M. K-dV equion with A, B and C given in (5b) and the one for the Sine-Gordon equion putting w = u/2 with A, B and C given in (6b). Class III) r = - q*, Equion (9a) becomes If we choose T' and q' as fjt =21JT+q+q*r. f).x (17) r' =_l T* ' I (x) = (x) + 2 r (fjt* /fjx) - (fjt jf)x) q q 1-ITI4 ' (18a) (18b) then T' with q'(x) sisfies Eq. (17) for real 'l/ Equion (18b) reduces to a simpler form Solving Eq. (18c). for T, we find q'(x) +q(x) = -41} r 1+ITI 2 (18c) 21J+.. J:41j 2 -Jq' +ql 2 (19) T=- (q'*+q*) After the eliminion of T in (17) and (9b) with (19) and (7b), we get the Backlund transformion for the nonlinear Schrodinger equion as q.r'+qx' = (q- q')..j 41J~ -Jq + q'j 2, qt+q/ =i(q..,-q,')..j41j 2 -Jq + q'j 2 (20a) + ~ (q+4) (Jq+q'J2+ Jq-q'J2). (20b) The nonlin~ar Schrodinger equion is invariant under transformion x~x-2kt, t~t and q~eikx-wtq. Then, under the transformions, Eqs. (20) are found to be the same as the result by Lamb. 4 l Another solution of (18c) (21)

5 1656 K. Konno and M. Wadi derives the Backlund transformion (20) from the Ricci'form by putting 1/F* instead of r in Eqs. (17) and (9b). IV) We have previously discussed the relionships among the inverse method, the Backlund transformion and an infinite number of the conservion.laws. 2 > There, we have restricted our discussion to the nonlinear evolution equions which can be reduced to the Schrodinger form of the inverse method such as the K-dV equion, the M. K-dV equion and the Sine-Gordon equion. In this paper, we start discussion from the Ricci form of the inverse method. Since the lter is more general, then, we can cover wider.classes of inverse problem for the nonlinear evolution.equion. We wul extend our discussion to equions which have the second derivives with respect to time such as the Boussinesq equion and equion of motion for Toda ltice (one-dimensional exponential ltice). The authors would like to thank Professor M. Toda and Professor Y. H. Ichikawa for their encouragement. The authors also would like to thank Professor N. Yajima for helpful discussions. One of the authors (M.W.) is partially supported by the Sakkokai Foundion. References 1) H. H. Chen, Phys. Rev. Letters 33 (1974), 925~ 2) M. Wadi, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975), 41_9. 3) M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys. Rev. Letters 31 (1973), ) G. L. Lamb. Jr., Phys. Letters 48A (1974), 73 and preprint.