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Handbook of mathematics for engineers and scienteists part 114

Handbook of mathematics for engineers and scienteists part 114. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 15.12. Methods of the Inverse Scattering Problem Soliton Theory 759 Thus two linear systems 15.12.2.4 and 15.12.2.18 are compatible if the functions f and g satisfy the cubically nonlinear system 15.12.2.17 . For g -kf a 2i 15.12.2.19 where k is a real constant and the bar over a symbol stands for its complex conjugate both equations 15.12.2.18 turn into one and the same nonlinear Schrodinger equation ift fxx 2k f 2 f f f If 2 . 15.12.2.20 Likewise one can use other polynomials in A and determine the associated linear systems generating nonlinear evolution equations. 15.12.2.21 Example 1. Searching for solutions of the determining system 15.12.2.7 in the form third-order polynomials in À results in A a3À3 a2X2 2 a3 fg ai À 2a3 fg - jia2 fgx - gfx ao B ia-jfX2 ia2f - 2a2fx X iaif 2fa3f2g - 2a2fx - ja2fxx C ia3gX2 ia2g 2a3gx X iaig 2ia3fg2 2a2gx - jia3gxx where a0 a1 a2 and a3 are arbitrary constants. The evolution equations for f and g corresponding to 15.12.2.21 are ft jia3 fxxx - 6fgfx 2a2 fxx - 2f2g - iaf - 2aof 0 gt j ia3 gxxx - 6fggx - 2 a2 g-xx - 2fg - iag 2aog 0. Consider two important special cases leading to interesting nonlinear equations of mathematical physics. 15.12.2.22 1 . If a0 a1 a2 0 a3 -4i g 1 the second equation of 15.12.2.22 is satisfied identically and the first equation of 15.12.2.22 translates into the Korteweg-de Vries equation ft fxxx - 6ffx 0. 2 . If a0 a1 a2 0 a3 -4i g f both equations 15.12.2.22 translate into one and the same modified Korteweg-de Vries equation ft fxxx 6f2 fx 0. Example 2. Now we will look for a solution of the determining system 15.12.2.7 in the form of the simple one-term expansion in negative powers of X A a x t X-1 B b x t X 1 C c x t X 1. 15.12.2.23 This results in the relations ax 2 i fg t b - 2 ift c 2 igt. 15.12.2.24 The evolution equations for functions f and g corresponding to 15.12.2.24 are written as fxt -4iaf 15.12.2.25 gxt -4iag. If we set a ji cos w b c 2i sin w f -g -2wx 15.12.2.26 then the three .