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Handbook of mathematics for engineers and scienteists part 113. 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. | 752 Nonlinear Partial Differential Equations where it has been taken into account that ix 0. If follows from the first equation of 15.11.2.5 that wo -2vix. 15.11.2.6 Substituting this into the second and third equations of 15.11.2.5 after some rearrangements we obtain W1 x - v xx 0 15.11.2.7 it W1 ix - v xx x 0. It is obvious that if the first equation of 15.11.2.7 is valid then the second is satisfied identically. The last equation in 15.11.2.5 is the Burgers equation. Hence formula 15.11.2.4 in view of 15.11.2.6 can be rewritten as d . . . w -2v ln i w1 15.11.2.8 dx where the functions w and w1 satisfy the Burgers equation and the function i is described by the first equation of 15.11.2.7 . Given a solution w1 of the Burgers equation formula 15.11.2.8 allows obtaining other solutions of it by solving the first equation in 15.11.2.7 which is linear in . Taking into account that w1 0 is a particular solution of the Burgers equation let us substitute it into 15.11.2.7 and 15.11.2.8 . This results in the Cole-Hopf transformation n ix W -2v . This transformation allows constructing solutions of the nonlinear Burgers equation 15.11.2.1 via solutions of the linear heat equation it vixx. Example 2. Consider the Korteweg-de Vries equation w dW 0. 15.11.2.9 dt dx dx3 First step. Let us substitute the leading term of the expansion 15.11.1.1 into equation 15.11.2.9 and then multiply the resulting relation by ip 3 the product ip 3wxxx gives a zeroth order quantity to obtain w0i3 pw0x0i2 - pw0i2 p -p p 1 p 2 w0 0 where i x - x0 x0 x0 t and w0 w0 t . From the balance of the highest-order terms only the last two terms are taken into account it follows that p 2 w0 -12 m 0 . Since p is a positive integer the first necessary condition of the Painleve test is satisfied. Second step. To fine the Fuchs indices resonances we substitute the binomial w - 12i 2 wmim-2 into the leading terms of equation 15.11.2.9 where the second and the third term are taken into account. Isolating the .
