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Tue Feb 10 12:26:55 2004 run on Linux %*******************************************************************% % % % L I E P D E . T S T % % ------------------- % % liepde.tst contains test examples for the program liepde.red. % % % % Author of this file: Thomas Wolf % % Date: 21. April 1998, 6. May 2003 % % % % Details about the syntax of liepde.red are given in liepde.tex. % % % % To run this demo you need to load liepde and crack through % % load crack,liepde$ % % and to read in this file as % % in "liepde.tst"; % % If you got the source code of a newer version of liepde then % % either read it in through % % in "crack.red","liepde.red"$ % % (with the appropriate directory name in front of liepde.red) % % or, to speed up the calculation, you compile before with % % faslout "crack"$ % % in "crack.red"$ % % faslend$ % % faslout "liepde"$ % % in "liepde.red"$ % % faslend$ % % and then load both it with % % load crack,liepde$ % % % %*******************************************************************% load crack; lisp(depl!*:=nil)$ % clearing of all dependences setcrackflags()$ lisp(print_:=nil)$ on dfprint$ comment ------------------------------------------------------- The following runs demonstrate the program LIEPDE for the computation of infinitesimal symmetries. Times given below refer to a 8 MB session under LINUX on a 133 MHz Pentium PC with the CRACK version of April 1998 running PSL Reduce. -------------------------------------------------------; lisp(prelim_:=nil)$ % not necessary as this is the default value lisp(individual_:=nil)$ % not necessary as this is the default value comment ------------------------------------------------------- The first example is a single ODE with a parametric function f=f(x) for which point symmetries are to be determined. (Time ~ 6 sec.); write"-------------------------------------------------------"; ------------------------------------------------------- lisp(freeint_:=nil)$ % This enables the solution of differential equ.s in % which unevaluated integrals remain. This becomes % necessary in this example through the parametric % function f=f(x) depend y,x$ depend f,x$ liepde({df(y,x,2)=-(y+3*f)*df(y,x)+y**3-f*y**2-(2*f**2+df(f,x))*y, {y}, {x}}, {"point"},{},{})$ The ODE under investigation is : 2 2 3 y = - f *y - 3*y *f - y *y - 2*f *y - f*y + y 2x x x x for the function(s) : y(x) The symmetries are: -------- 1. Symmetry: int(f,x) xi_x= - e int(f,x) eta_y=e *f*y -------- 2. Symmetry: int(f,x) 1 xi_x= - e *int(-----------,x) int(f,x) e int(f,x) 1 eta_y=e *int(-----------,x)*f*y + y int(f,x) e -------- {{}, int(f,x) 1 int(f,x) {xi_x= - e *int(-----------,x)*c_5 - e *c_6, int(f,x) e int(f,x) 1 int(f,x) eta_y=e *int(-----------,x)*c_5*f*y + e *c_6*f*y + c_5*y}, int(f,x) e {c_6,c_5}} nodepnd {y,f}$ lisp(freeint_:=t)$ % Because the simplification of differential % expressions which involve unevaluated integrals % may provide difficulties such solutions involving % unevaluated integrals are disabled. comment ------------------------------------------------------- The following example demonstrates a number of things. The Burgers equation is investigated concerning third order symmetries. The equation is used to substitute df(u,t) and all derivatives of df(u,t). This computation also shows that any equations that remain unsolved are returned, like in this case the heat quation. (Time ~ 15 sec.); write"-------------------------------------------------------"; ------------------------------------------------------- nodepnd {u}$ depend u,t,x$ liepde({df(u,t)=df(u,x,2)+df(u,x)**2,{u},{t,x}},{"general",3},{},{})$ The PDE under investigation is : 2 u =u + u t 2x x for the function(s) : u(t,x) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 2 eta_u=2*u *t + 2*u *t + u *x 2x x x -------- 2. Symmetry: xi_t=0 xi_x=0 2 2 2 2 eta_u=4*u *t + 4*u *t + 4*u *t*x + 2*t + x 2x x x -------- 3. Symmetry: xi_t=0 xi_x=0 2 eta_u=u + u 2x x -------- 4. Symmetry: xi_t=0 xi_x=0 3 2 eta_u=2*u *t + 6*u *u *t + u *x + 2*u *t + u *x 3x 2x x 2x x x -------- 5. Symmetry: xi_t=0 xi_x=0 3 3 2 3 3 2 2 2 eta_u=8*u *t + 24*u *u *t + 12*u *t *x + 8*u *t + 12*u *t *x + 12*u *t 3x 2x x 2x x x x 2 3 + 6*u *t*x + 6*t*x + x x -------- 6. Symmetry: xi_t=0 xi_x=0 2 2 3 2 2 2 eta_u=4*u *t + 12*u *u *t + 4*u *t*x + 4*u *t + 4*u *t*x + u *x - x 3x 2x x 2x x x x -------- 7. Symmetry: xi_t=0 xi_x=0 3 eta_u=u + 3*u *u + u 3x 2x x x -------- 8. Symmetry: xi_t=0 xi_x=0 eta_u=2*u *t + x x -------- 9. Symmetry: xi_t=0 xi_x=0 eta_u=u x -------- 10. Symmetry: xi_t=0 xi_x=0 eta_u=1 -------- Further symmetries: xi_t=0 xi_x=0 c_25 eta_u=------ u e with c_25(t,x) which still have to satisfy: 0=c_25 - c_25 t 2x {{c_25 - c_25 }, t 2x {xi_t=0, xi_x=0, u u 2 u 3 u eta_u=(e *u *c_31 + 4*e *u *c_32*t + 8*e *u *c_33*t + 2*e *u *c_34*t 3x 3x 3x 3x u u 2 u 3 + 3*e *u *u *c_31 + 12*e *u *u *c_32*t + 24*e *u *u *c_33*t 2x x 2x x 2x x u u u 2 + 6*e *u *u *c_34*t + e *u *c_28 + 4*e *u *c_29*t 2x x 2x 2x u u u 2 + 2*e *u *c_30*t + 4*e *u *c_32*t*x + 12*e *u *c_33*t *x 2x 2x 2x u u 3 u 3 2 u 3 3 + e *u *c_34*x + e *u *c_31 + 4*e *u *c_32*t + 8*e *u *c_33*t 2x x x x u 3 u 2 u 2 2 u 2 + 2*e *u *c_34*t + e *u *c_28 + 4*e *u *c_29*t + 2*e *u *c_30*t x x x x u 2 u 2 2 u 2 + 4*e *u *c_32*t*x + 12*e *u *c_33*t *x + e *u *c_34*x x x x u u u 2 u 2 + 4*e *u *c_29*t*x + e *u *c_30*x + e *u *c_32*x + 12*e *u *c_33*t x x x x u 2 u u u + 6*e *u *c_33*t*x + e *u *c_35 + 2*e *u *c_36*t + 2*e *c_29*t x x x u 2 u u u 3 u + e *c_29*x - e *c_32*x + 6*e *c_33*t*x + e *c_33*x + e *c_36*x u u + e *c_37 + c_25)/e }, {c_25, c_30, c_29, c_28, c_34, c_33, c_32, c_31, c_36, c_35, c_37}} comment ------------------------------------------------------- Now the same equation is investigated, this time only df(u,x,2) and its derivatives are substituted. As a consequence less jet-variables (u-derivatives of lower order) are generated in the process of formulating the symmetry conditions. Less jet-variables in which the conditions have to be fulfilled identically means less overdetermined conditions and more solutions which to compute takes longer than before. (Time ~ 85 sec.); write"-------------------------------------------------------"; ------------------------------------------------------- liepde({df(u,x,2)=df(u,t)-df(u,x)**2,{u},{t,x}},{"general",3},{},{})$ The PDE under investigation is : 2 u =u - u 2x t x for the function(s) : u(t,x) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 2 eta_u= - 8*u *u *t + u - 4*u *t - 4*u *u *t - 2*u *x - 4*u *u *t t,x t t,x 2t,x 2t x 2t t x 2 - 2*u *x + u *u t t x -------- 2. Symmetry: xi_t=0 xi_x=0 3 2 2 3 3 eta_u= - 32*u *u *t + 12*u *t - 12*u *t*x - 16*u *t - 16*u *u *t t,x t t,x t,x 2t,x 2t x 2 2 3 2 2 2 2 - 24*u *t *x - 16*u *u *t - 24*u *t *x + 12*u *u *t - 12*u *u *t*x 2t t x t t x t x 3 2 + 24*u *t*x - 2*u *x + 9*u *x - 9*x t t x -------- 3. Symmetry: xi_t=0 xi_x=0 5 4 3 2 5 eta_u= - 64*u *u *t - 160*u *t - 80*u *t *x - 32*u *t t,x t t,x t,x 2t,x 5 4 2 5 2 4 4 - 32*u *u *t - 80*u *t *x - 32*u *u *t - 80*u *t *x - 160*u *u *t 2t x 2t t x t t x 3 2 3 2 3 3 2 2 - 80*u *u *t *x - 240*u *t *x - 40*u *t *x - 120*u *t - 120*u *t *x t x t t x x 4 2 3 5 - 10*u *t*x - 60*t *x - 20*t*x - x x -------- 4. Symmetry: xi_t=0 xi_x=0 4 3 2 2 4 4 eta_u= - 32*u *u *t + 16*u *t - 24*u *t *x - 16*u *t - 16*u *u *t t,x t t,x t,x 2t,x 2t x 3 2 4 2 3 3 2 2 - 32*u *t *x - 16*u *u *t - 32*u *t *x + 16*u *u *t - 24*u *u *t *x 2t t x t t x t x 2 3 2 2 4 3 + 48*u *t *x - 8*u *t*x + 84*u *t + 36*u *t*x - u *x + 48*t*x + 8*x t t x x x -------- 5. Symmetry: xi_t=0 xi_x=0 2 2 2 2 eta_u= - 8*u *u *t + 2*u *t - u *x - 4*u *t - 4*u *u *t t,x t t,x t,x 2t,x 2t x 2 2 2 2 - 4*u *t*x - 4*u *u *t - 4*u *t*x + 2*u *u *t - u *u *x + 2*u *x 2t t x t t x t x t -------- 6. Symmetry: xi_t=0 xi_x=0 2 eta_u= - 2*u *u - u - u *u - u *u t,x t 2t,x 2t x t x -------- 7. Symmetry: xi_t=0 xi_x=0 eta_u 3 2 = - 2*u *u *x - 2*u *t - u *x - 6*u *u *t - u *u *x - 2*u *t - u *u *x t,x t 3t 2t,x 2t t 2t x t t x -------- 8. Symmetry: xi_t=0 xi_x=0 2 3 3 2 eta_u= - 48*u *u *t *x + 12*u *t*x - 2*u *x - 16*u *t - 24*u *t *x t,x t t,x t,x 3t 2t,x 3 2 2 3 3 2 2 - 48*u *u *t - 24*u *u *t *x - 12*u *t*x - 16*u *t - 24*u *u *t *x 2t t 2t x 2t t t x 2 2 3 2 - 12*u *t*x + 12*u *u *t*x - 2*u *u *x + 6*u *x - 6*u *x + 3 t t x t x t x -------- 9. Symmetry: xi_t=0 xi_x=0 4 3 2 3 5 eta_u= - 160*u *u *t *x + 80*u *t *x - 40*u *t *x - 32*u *t t,x t t,x t,x 3t 4 5 4 3 2 3 5 - 80*u *t *x - 96*u *u *t - 80*u *u *t *x - 80*u *t *x - 32*u *t 2t,x 2t t 2t x 2t t 2 4 2 3 2 3 2 3 3 - 80*u *u *t *x - 80*u *t *x + 80*u *u *t *x - 40*u *u *t *x + 360*u *t t x t t x t x t 2 2 4 2 3 5 2 + 120*u *t *x - 10*u *t*x + 420*u *t *x + 60*u *t*x - u *x + 120*t t t x x x 2 4 + 120*t*x + 10*x -------- 10. Symmetry: xi_t=0 xi_x=0 5 4 3 3 6 eta_u= - 384*u *u *t *x - 960*u *t *x - 160*u *t *x - 64*u *t t,x t t,x t,x 3t 5 6 5 5 4 2 - 192*u *t *x - 192*u *u *t - 192*u *u *t *x - 480*u *t - 240*u *t *x 2t,x 2t t 2t x 2t 2t 3 6 2 5 2 5 2 4 2 4 - 64*u *t - 192*u *u *t *x - 480*u *t - 240*u *t *x - 960*u *u *t *x t t x t t t x 3 3 4 3 2 2 4 3 - 160*u *u *t *x - 720*u *t - 720*u *t *x - 60*u *t *x - 720*u *t *x t x t t t x 2 3 5 3 2 2 4 6 - 240*u *t *x - 12*u *t*x - 120*t - 180*t *x - 30*t*x - x x x -------- 11. Symmetry: xi_t=0 xi_x=0 3 2 3 4 eta_u= - 64*u *u *t *x + 24*u *t *x - 8*u *t*x - 16*u *t t,x t t,x t,x 3t 3 4 3 2 2 3 4 - 32*u *t *x - 48*u *u *t - 32*u *u *t *x - 24*u *t *x - 16*u *t 2t,x 2t t 2t x 2t t 2 3 2 2 2 2 3 2 - 32*u *u *t *x - 24*u *t *x + 24*u *u *t *x - 8*u *u *t*x + 24*u *t*x t x t t x t x t 4 3 2 - u *x - 24*u *t*x + 6*u *x - 30*t - 15*x t x x -------- 12. Symmetry: xi_t=0 xi_x=0 2 2 eta_u= - 8*u *u *t*x + u *x - 4*u *t - 4*u *t*x - 12*u *u *t t,x t t,x 3t 2t,x 2t t 2 3 2 2 2 2 - 4*u *u *t*x - u *x - 4*u *t - 4*u *u *t*x - u *x + u *u *x 2t x 2t t t x t t x -------- 13. Symmetry: xi_t=0 xi_x=0 3 eta_u= - u - 3*u *u - u 3t 2t t t -------- 14. Symmetry: xi_t=0 xi_x=0 2 eta_u= - u *x - 2*u *t - 2*u *t - u *u *x t,x 2t t t x -------- 15. Symmetry: xi_t=0 xi_x=0 2 3 2 3 2 2 eta_u= - 12*u *t *x - 8*u *t - 8*u *t - 12*u *u *t *x - 6*u *t*x t,x 2t t t x t 3 2 + 6*u *t*x - u *x + 6*t + 3*x x x -------- 16. Symmetry: xi_t=0 xi_x=0 3 4 2 4 3 3 eta_u= - 32*u *t *x - 16*u *t - 16*u *t - 32*u *u *t *x - 48*u *t t,x 2t t t x t 2 2 2 3 2 2 4 - 24*u *t *x - 48*u *t *x - 8*u *t*x - 12*t - 12*t*x - x t x x -------- 17. Symmetry: xi_t=0 xi_x=0 2 2 2 2 eta_u= - 8*u *t*x - 8*u *t - 8*u *t - 8*u *u *t*x - 2*u *x + 2*u *x - 1 t,x 2t t t x t x -------- 18. Symmetry: xi_t=0 xi_x=0 2 eta_u=u + u 2t t -------- 19. Symmetry: xi_t=0 xi_x=0 eta_u= - 2*u *t - 2*u *u *t - u *x t,x t x t -------- 20. Symmetry: xi_t=0 xi_x=0 3 3 2 2 2 3 eta_u= - 8*u *t - 8*u *u *t - 12*u *t *x - 12*u *t - 6*u *t*x - 6*t*x - x t,x t x t x x -------- 21. Symmetry: xi_t=0 xi_x=0 2 2 2 eta_u= - 4*u *t - 4*u *u *t - 4*u *t*x - u *x + x t,x t x t x -------- 22. Symmetry: xi_t=0 xi_x=0 eta_u=u + u *u t,x t x -------- 23. Symmetry: xi_t=0 xi_x=0 eta_u= - 2*u *t - x x -------- 24. Symmetry: xi_t=0 xi_x=0 eta_u=u x -------- 25. Symmetry: xi_t=0 xi_x=0 eta_u= - 4*u *t - 2*u *x + 1 t x -------- 26. Symmetry: xi_t=0 xi_x=0 2 2 eta_u= - 4*u *t - 4*u *t*x - 2*t - x t x -------- 27. Symmetry: xi_t=0 xi_x=0 eta_u=u t -------- 28. Symmetry: xi_t=0 xi_x=0 eta_u=1 -------- Further symmetries: xi_t=0 xi_x=0 c_94 eta_u=------ u e with c_94(t,x) which still have to satisfy: 0= - c_94 + c_94 t 2x {{ - c_94 + c_94 }, t 2x {xi_t=0, xi_x=0, u u 2 u 4 eta_u=( - 2*e *u *u *c_100 - 8*e *u *u *c_101*t - 32*e *u *u *c_102*t t,x t t,x t t,x t u 5 u 3 - 64*e *u *u *c_103*t - 32*e *u *u *c_104*t t,x t t,x t u u - 8*e *u *u *c_105*t - 8*e *u *u *c_107*t*x t,x t t,x t u 3 u 5 - 64*e *u *u *c_108*t *x - 384*e *u *u *c_109*t *x t,x t t,x t u 4 u 2 - 160*e *u *u *c_110*t *x - 48*e *u *u *c_111*t *x t,x t t,x t u u u 2 - 2*e *u *u *c_112*x + 2*e *u *c_101*t - e *u *c_101*x t,x t t,x t,x u 3 u 2 2 u 4 + 16*e *u *c_102*t - 24*e *u *c_102*t *x - 160*e *u *c_103*t t,x t,x t,x u 3 2 u 2 u 2 - 80*e *u *c_103*t *x + 12*e *u *c_104*t - 12*e *u *c_104*t*x t,x t,x t,x u u u 2 + e *u *c_105 + e *u *c_107*x + 24*e *u *c_108*t *x t,x t,x t,x u 3 u 4 - 8*e *u *c_108*t*x - 960*e *u *c_109*t *x t,x t,x u 3 3 u 3 - 160*e *u *c_109*t *x + 80*e *u *c_110*t *x t,x t,x u 2 3 u u 3 - 40*e *u *c_110*t *x + 12*e *u *c_111*t*x - 2*e *u *c_111*x t,x t,x t,x u u 3 u 2 - 8*e *u *c_115*t*x - 32*e *u *c_116*t *x - 12*e *u *c_117*t *x t,x t,x t,x u u u 2 - e *u *c_118*x + e *u *c_119 - 4*e *u *c_120*t t,x t,x t,x u 3 u u - 8*e *u *c_121*t - 2*e *u *c_122*t - e *u *c_106 t,x t,x 3t u 2 u 4 u 6 - 4*e *u *c_107*t - 16*e *u *c_108*t - 64*e *u *c_109*t 3t 3t 3t u 5 u 3 u - 32*e *u *c_110*t - 16*e *u *c_111*t - 2*e *u *c_112*t 3t 3t 3t u u 2 u 4 - e *u *c_100 - 4*e *u *c_101*t - 16*e *u *c_102*t 2t,x 2t,x 2t,x u 5 u 3 u - 32*e *u *c_103*t - 16*e *u *c_104*t - 4*e *u *c_105*t 2t,x 2t,x 2t,x u u 3 - 4*e *u *c_107*t*x - 32*e *u *c_108*t *x 2t,x 2t,x u 5 u 4 - 192*e *u *c_109*t *x - 80*e *u *c_110*t *x 2t,x 2t,x u 2 u u - 24*e *u *c_111*t *x - e *u *c_112*x - 3*e *u *u *c_106 2t,x 2t,x 2t t u 2 u 4 - 12*e *u *u *c_107*t - 48*e *u *u *c_108*t 2t t 2t t u 6 u 5 - 192*e *u *u *c_109*t - 96*e *u *u *c_110*t 2t t 2t t u 3 u u - 48*e *u *u *c_111*t - 6*e *u *u *c_112*t - e *u *u *c_100 2t t 2t t 2t x u 2 u 4 u 5 - 4*e *u *u *c_101*t - 16*e *u *u *c_102*t - 32*e *u *u *c_103*t 2t x 2t x 2t x u 3 u u - 16*e *u *u *c_104*t - 4*e *u *u *c_105*t - 4*e *u *u *c_107*t*x 2t x 2t x 2t x u 3 u 5 - 32*e *u *u *c_108*t *x - 192*e *u *u *c_109*t *x 2t x 2t x u 4 u 2 - 80*e *u *u *c_110*t *x - 24*e *u *u *c_111*t *x 2t x 2t x u u u 3 - e *u *u *c_112*x - 4*e *u *c_101*t*x - 32*e *u *c_102*t *x 2t x 2t 2t u 4 u 2 u - 80*e *u *c_103*t *x - 24*e *u *c_104*t *x - 2*e *u *c_105*x 2t 2t 2t u 2 u 2 2 u 5 - e *u *c_107*x - 24*e *u *c_108*t *x - 480*e *u *c_109*t 2t 2t 2t u 4 2 u 3 2 - 240*e *u *c_109*t *x - 80*e *u *c_110*t *x 2t 2t u 2 u u 2 - 12*e *u *c_111*t*x + e *u *c_114 - 8*e *u *c_115*t 2t 2t 2t u 4 u 3 u - 16*e *u *c_116*t - 8*e *u *c_117*t - 2*e *u *c_118*t 2t 2t 2t u 3 u 3 2 u 3 4 - e *u *c_106 - 4*e *u *c_107*t - 16*e *u *c_108*t t t t u 3 6 u 3 5 u 3 3 - 64*e *u *c_109*t - 32*e *u *c_110*t - 16*e *u *c_111*t t t t u 3 u 2 u 2 2 - 2*e *u *c_112*t - e *u *u *c_100 - 4*e *u *u *c_101*t t t x t x u 2 4 u 2 5 - 16*e *u *u *c_102*t - 32*e *u *u *c_103*t t x t x u 2 3 u 2 u 2 - 16*e *u *u *c_104*t - 4*e *u *u *c_105*t - 4*e *u *u *c_107*t*x t x t x t x u 2 3 u 2 5 - 32*e *u *u *c_108*t *x - 192*e *u *u *c_109*t *x t x t x u 2 4 u 2 2 - 80*e *u *u *c_110*t *x - 24*e *u *u *c_111*t *x t x t x u 2 u 2 u 2 3 - e *u *u *c_112*x - 4*e *u *c_101*t*x - 32*e *u *c_102*t *x t x t t u 2 4 u 2 2 u 2 - 80*e *u *c_103*t *x - 24*e *u *c_104*t *x - 2*e *u *c_105*x t t t u 2 2 u 2 2 2 u 2 5 - e *u *c_107*x - 24*e *u *c_108*t *x - 480*e *u *c_109*t t t t u 2 4 2 u 2 3 2 - 240*e *u *c_109*t *x - 80*e *u *c_110*t *x t t u 2 2 u 2 u 2 2 - 12*e *u *c_111*t*x + e *u *c_114 - 8*e *u *c_115*t t t t u 2 4 u 2 3 u 2 - 16*e *u *c_116*t - 8*e *u *c_117*t - 2*e *u *c_118*t t t t u u 2 u 3 + 2*e *u *u *c_101*t - e *u *u *c_101*x + 16*e *u *u *c_102*t t x t x t x u 2 2 u 4 - 24*e *u *u *c_102*t *x - 160*e *u *u *c_103*t t x t x u 3 2 u 2 - 80*e *u *u *c_103*t *x + 12*e *u *u *c_104*t t x t x u 2 u u - 12*e *u *u *c_104*t*x + e *u *u *c_105 + e *u *u *c_107*x t x t x t x u 2 u 3 + 24*e *u *u *c_108*t *x - 8*e *u *u *c_108*t*x t x t x u 4 u 3 3 - 960*e *u *u *c_109*t *x - 160*e *u *u *c_109*t *x t x t x u 3 u 2 3 + 80*e *u *u *c_110*t *x - 40*e *u *u *c_110*t *x t x t x u u 3 u + 12*e *u *u *c_111*t*x - 2*e *u *u *c_111*x - 8*e *u *u *c_115*t*x t x t x t x u 3 u 2 u - 32*e *u *u *c_116*t *x - 12*e *u *u *c_117*t *x - e *u *u *c_118*x t x t x t x u u 2 u 3 + e *u *u *c_119 - 4*e *u *u *c_120*t - 8*e *u *u *c_121*t t x t x t x u u u 2 - 2*e *u *u *c_122*t + 2*e *u *c_101*x + 48*e *u *c_102*t *x t x t t u 3 u 3 u 2 3 - 8*e *u *c_102*t*x - 240*e *u *c_103*t *x - 40*e *u *c_103*t *x t t t u u 3 u 2 + 24*e *u *c_104*t*x - 2*e *u *c_104*x + 24*e *u *c_108*t*x t t t u 4 u 4 u 3 2 - e *u *c_108*x - 720*e *u *c_109*t - 720*e *u *c_109*t *x t t t u 2 4 u 3 u 2 2 - 60*e *u *c_109*t *x + 360*e *u *c_110*t + 120*e *u *c_110*t *x t t t u 4 u 2 u 2 - 10*e *u *c_110*t*x + 6*e *u *c_111*x - 2*e *u *c_115*x t t t u 3 u 2 2 u 2 - 48*e *u *c_116*t - 24*e *u *c_116*t *x - 6*e *u *c_117*t*x t t t u u 2 u - 4*e *u *c_120*t*x - 12*e *u *c_121*t *x - e *u *c_122*x t t t u u 2 u u 2 + e *u *c_125 - 4*e *u *c_126*t - 4*e *u *c_127*t + 84*e *u *c_102*t t t t x u 2 u 4 u 3 + 36*e *u *c_102*t*x - e *u *c_102*x - 120*e *u *c_103*t x x x u 2 2 u 4 u 2 - 120*e *u *c_103*t *x - 10*e *u *c_103*t*x + 9*e *u *c_104*x x x x u u 3 u 3 - 24*e *u *c_108*t*x + 6*e *u *c_108*x - 720*e *u *c_109*t *x x x x u 2 3 u 5 u 2 - 240*e *u *c_109*t *x - 12*e *u *c_109*t*x + 420*e *u *c_110*t *x x x x u 3 u 5 u + 60*e *u *c_110*t*x - e *u *c_110*x - 6*e *u *c_111*x x x x u u 2 u 3 + 2*e *u *c_115*x - 48*e *u *c_116*t *x - 8*e *u *c_116*t*x x x x u u 3 u 2 + 6*e *u *c_117*t*x - e *u *c_117*x - e *u *c_120*x x x x u 2 u 2 u - 12*e *u *c_121*t - 6*e *u *c_121*t*x + e *u *c_123 x x x u u u - 2*e *u *c_124*t - 4*e *u *c_126*t*x - 2*e *u *c_127*x x x x u u 3 u 2 + 48*e *c_102*t*x + 8*e *c_102*x - 60*e *c_103*t *x u 3 u 5 u u - 20*e *c_103*t*x - e *c_103*x - 9*e *c_104*x - 30*e *c_108*t u 2 u 3 u 2 2 - 15*e *c_108*x - 120*e *c_109*t - 180*e *c_109*t *x u 4 u 6 u 2 u 2 - 30*e *c_109*t*x - e *c_109*x + 120*e *c_110*t + 120*e *c_110*t*x u 4 u u u 2 + 10*e *c_110*x + 3*e *c_111 - e *c_115 - 12*e *c_116*t u 2 u 4 u u 2 - 12*e *c_116*t*x - e *c_116*x + 6*e *c_117*t + 3*e *c_117*x u u u 3 u + e *c_120*x - 6*e *c_121*t*x - e *c_121*x - e *c_124*x u u 2 u u u - 2*e *c_126*t - e *c_126*x + e *c_127 + e *c_128 + c_94)/e }, {c_94, c_105, c_104, c_103, c_102, c_101, c_100, c_112, c_111, c_110, c_109, c_108, c_107, c_106, c_118, c_117, c_116, c_115, c_114, c_122, c_121, c_120, c_119, c_124, c_123, c_127, c_126, c_125, c_128}} nodepnd {u}$ comment ------------------------------------------------------- The following example includes the Karpman equations for three unknown functions in 4 variables. If point symmetries are to be computed for a single equation or a system of equations of higher than first order then there is the option to formulate at first preliminary conditions for each equation, have CRACK solving these conditions before the full set of conditions is formulated and solved. This strategy is adopted if a lisp flag prelim_ has the value t. The default value is nil. Similarly, if a system of equations is to be investigated and a flag individual_ has the value t then symmetry conditions are formulated and investigated for each individual equation successively. The default value is nil. It is advantageous to split a large set of conditions into smaller sets to be investigated successively if each set is sufficiently overdetermined to be solvable quickly. Then any substitutions are done in the smaller set and the next set of conditions is shorter. For example, for the Karpman equations below the speedup for prelim_:=t and individual_:=t is a factor of 10. (Time ~ 1 min.); write"-------------------------------------------------------"; ------------------------------------------------------- lisp(prelim_:=t)$ lisp(individual_:=t)$ depend r,x,y,z,t; depend f,x,y,z,t; depend v,x,y,z,t; on time$ Time: 8610 ms plus GC time: 150 ms liepde({ first solve( {df(r,t) + w1*df(r,z) + s1*(df(r,x)*df(f,x)+df(r,y)*df(f,y)+r*df(f,x,2)/2+r*df(f,y,2)/2) + s2*(df(r,z)*df(f,z)+r*df(f,z,2)/2), df(f,t) + w1*df(f,z) - (s1*(df(r,x,2)/r+df(r,y,2)/r-df(f,x)**2-df(f,y)**2) + s2*(df(r,z,2)/r-df(f,z)**2))/2 + a1*v, df(v,t,2) - w2**2*(df(v,x,2)+df(v,y,2)+df(v,z,2)) - 2*a2*r*(df(r,x,2)+df(r,y,2)+df(r,z,2)) - 2*a2*(df(r,x)**2+df(r,y)**2+df(r,z)**2)}, {df(v,x,2), df(r,x,2), df(f,x,2)} ) , {r,f,v}, {x,y,z,t}}, {"point"}, {},{})$ The PDE-system under investigation is : 2 2 2 2 2 2 2 v =( - 4*f *a2*r - 2*f *a2*r *s1 - 2*f *a2*r *s1 - 2*f *a2*r *s2 2x t x y z 2 2 2 - 4*f *a2*r *w1 - 2*r *a2*s1 - 2*r *a2*s1 - 2*r *a2*r*s1 z x y 2z 2 2 2 + 2*r *a2*r*s2 - 2*r *a2*s1 + v *s1 - v *s1*w2 - v *s1*w2 2z z 2t 2y 2z 2 2 - 4*a1*a2*r *v)/(s1*w2 ) 2 2 2 r =(2*f *r + f *r*s1 + f *r*s1 + f *r*s2 + 2*f *r*w1 - r *s1 - r *s2 2x t x y z z 2y 2z + 2*a1*r*v)/s1 f = 2x - 2*f *r *s1 - f *r*s1 - 2*f *r *s1 - f *r*s2 - 2*f *r *s2 - 2*r - 2*r *w1 x x 2y y y 2z z z t z -------------------------------------------------------------------------------- r*s1 for the function(s) : r(z,y,t,x), f(z,y,t,x), v(z,y,t,x) =============== Initializations time for initializations: 30 ms GC time : 0 ms =============== Preconditions for the 1. equation time to formulate conditions: 260 ms GC time : 20 ms CRACK needed : 721 ms GC time : 10 ms =============== Preconditions for the 2. equation =============== Preconditions for the 3. equation time to formulate conditions: 130 ms GC time : 0 ms CRACK needed : 230 ms GC time : 0 ms =============== Full conditions for the 1. equation time to formulate conditions: 50 ms GC time : 0 ms CRACK needed : 1150 ms GC time : 20 ms =============== Full conditions for the 2. equation time to formulate conditions: 20 ms GC time : 0 ms CRACK needed : 70 ms GC time : 0 ms =============== Full conditions for the 3. equation time to formulate conditions: 40 ms GC time : 0 ms CRACK needed : 50 ms GC time : 0 ms The symmetries are: -------- 1. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 eta_f=1 eta_v=0 -------- 2. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 2 eta_f= - t 2*t eta_v=----- a1 -------- 3. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=0 eta_r=0 eta_f= - t 1 eta_v=---- a1 -------- 4. Symmetry: xi_x=0 1 xi_y=---- s2 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 5. Symmetry: 1 xi_x=---- s2 xi_y=0 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 6. Symmetry: y xi_x=---- s2 - x xi_y=------ s2 xi_z=0 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 7. Symmetry: xi_x=0 xi_y=0 1 xi_z=---- s1 xi_t=0 eta_r=0 eta_f=0 eta_v=0 -------- 8. Symmetry: xi_x=0 xi_y=0 xi_z=0 xi_t=1 eta_r=0 eta_f=0 eta_v=0 -------- {{}, c_211 + c_212*y {xi_x=-----------------, s2 - c_212*x + c_219 xi_y=--------------------, s2 c_208 xi_z=-------, s1 xi_t=c_221, eta_r=0, 2 eta_f=c_222 - c_223*t - c_224*t, 2*c_223*t + c_224 eta_v=-------------------}, a1 {c_222, c_223, c_224, c_219, c_211, c_212, c_208, c_221}} Time: 2931 ms plus GC time: 50 ms off time$ nodepnd {r,f,v}$ comment ------------------------------------------------------- In the following example a system of two equations (by V.Sokolov) is investigated concerning a special ansatz for 4th order symmetries. The ansatz for the symmetries includes two unknown functions f,g. Because x is the second variable in the list of variables {t,x}, the name u!`2 stands for df(u,x). Because higher order symmetries are investigated we have to set prelim_:=nil. The symmetries to be calculated are lengthy and therefore conditions are not very overdetermined. In that case CRACK can take long to solve a single subset of conditions. The complete set of conditions would have been more overdetermined and easier to solve. Therefore the advantage of first formulating all conditions and then solving them together with one CRACK call is that having more equations, the chance of finding short integrable equations among then is higher, i.e. CRACK has more freedom in optimizing the computation. Therefore individual_:=nil is more appropriate in this example. Because 4th order conditions are to be computed the `binding stack size' is increased. (Time ~ 5 min.); write"-------------------------------------------------------"; ------------------------------------------------------- lisp(prelim_:=nil)$ lisp(individual_:=nil)$ lisp(if getd 'set_bndstk_size then set_bndstk_size(7000))$ nodepnd {u,v}$ depend u,x,t; depend v,x,t; des:={df(u,t)=+df(u,x,2) + (u + v)*df(u,x) + u*df(v,x), df(v,t)=-df(v,x,2) + (u + v)*df(v,x) + v*df(u,x) }$ nodepnd {f,g}$ depend f,t,x,u,v,u!`2,v!`2,u!`2!`2,v!`2!`2,u!`2!`2!`2,v!`2!`2!`2$ depend g,t,x,u,v,u!`2,v!`2,u!`2!`2,v!`2!`2,u!`2!`2!`2,v!`2!`2!`2$ liepde({des,{u,v},{t,x}}, {xi_t=0, xi_x=0, eta_u=+df(u,x,4)+f, eta_v=-df(v,x,4)+g }, {f,g},{} )$ The PDE-system under investigation is : u =u + u *u + u *v + v *u t 2x x x x v =u *v - v + v *u + v *v t x 2x x x for the function(s) : u(x,t), v(x,t) The symmetries are: -------- 1. Symmetry: xi_t=0 xi_x=0 eta_u=u + 4*u *t + 2*u *u + 2*u *v + 5*u *u + 3*u *v + 6*u *t*u 4x 3x 3x 3x 2x x 2x x 2x 3 2 9 3 2 2 + 6*u *t*v + ---*u *u + ---*u *u*v + ---*u *v + 2*u *x + 6*u *t 2x 2 2x 2 2x 2 2x 2x x 2 9 2 9 + 3*u *u + ---*u *v + 2*u *v + 6*u *v *t + ---*u *v *u + 3*u *v *v x 2 x x 2x x x 2 x x x x 2 2 1 3 9 2 9 2 + 3*u *t*u + 12*u *t*u*v + 3*u *t*v + ---*u *u + ---*u *u *v + ---*u *u*v x x x 2 x 2 x 2 x 1 3 2 + 2*u *u*x + ---*u *v + 2*u *v*x + 8*u + v *u + 6*v *t*u + 6*v *t*u*v x 2 x x x 3x x x 3 3 9 2 3 2 2 + ---*v *u + ---*v *u *v + ---*v *u*v + 2*v *u*x + u + 3*u*v 2 x 2 x 2 x x 9 eta_v=u *v + 2*u *v + 3*u *v - 6*u *v *t - 3*u *v *u - ---*u *v *v 3x 2x x x 2x x x x x 2 x x 2 3 2 9 2 3 3 + 6*u *t*u*v + 6*u *t*v + ---*u *u *v + ---*u *u*v + ---*u *v + 2*u *v*x x x 2 x 2 x 2 x x - v + 4*v *t + 2*v *u + 2*v *v + 5*v *v - 6*v *t*u - 6*v *t*v 4x 3x 3x 3x 2x x 2x 2x 3 2 9 3 2 2 9 2 - ---*v *u - ---*v *u*v - ---*v *v - 2*v *x - 6*v *t - ---*v *u 2 2x 2 2x 2 2x 2x x 2 x 2 2 2 1 3 9 2 - 3*v *v + 3*v *t*u + 12*v *t*u*v + 3*v *t*v + ---*v *u + ---*v *u *v x x x x 2 x 2 x 9 2 1 3 2 + ---*v *u*v + 2*v *u*x + ---*v *v + 2*v *v*x + 3*u*v + v 2 x x 2 x x -------- 2. Symmetry: xi_t=0 xi_x=0 3 2 eta_u=u + 2*u *u + 2*u *v + 4*u + 5*u *u + 3*u *v + ---*u *u 4x 3x 3x 3x 2x x 2x x 2 2x 9 3 2 2 9 2 2 + ---*u *u*v + 6*u *u + ---*u *v + 6*u *v + 3*u *u + ---*u *v + 6*u 2 2x 2x 2 2x 2x x 2 x x 9 1 3 9 2 + 2*u *v + ---*u *v *u + 3*u *v *v + 6*u *v + ---*u *u + ---*u *u *v x 2x 2 x x x x x x 2 x 2 x 2 9 2 1 3 2 3 3 + 3*u *u + ---*u *u*v + 12*u *u*v + ---*u *v + 3*u *v + v *u + ---*v *u x 2 x x 2 x x 3x 2 x 9 2 2 3 2 + ---*v *u *v + 6*v *u + ---*v *u*v + 6*v *u*v 2 x x 2 x x 9 eta_v=u *v + 2*u *v + 3*u *v - 3*u *v *u - ---*u *v *v - 6*u *v 3x 2x x x 2x x x 2 x x x x 3 2 9 2 3 3 2 + ---*u *u *v + ---*u *u*v + 6*u *u*v + ---*u *v + 6*u *v - v + 2*v *u 2 x 2 x x 2 x x 4x 3x 3 2 9 3 2 + 2*v *v + 4*v + 5*v *v - ---*v *u - ---*v *u*v - 6*v *u - ---*v *v 3x 3x 2x x 2 2x 2 2x 2x 2 2x 9 2 2 2 1 3 9 2 2 - 6*v *v - ---*v *u - 3*v *v - 6*v + ---*v *u + ---*v *u *v + 3*v *u 2x 2 x x x 2 x 2 x x 9 2 1 3 2 + ---*v *u*v + 12*v *u*v + ---*v *v + 3*v *v 2 x x 2 x x -------- 3. Symmetry: xi_t=0 xi_x=0 3 2 eta_u=u + 2*u *u + 2*u *v + 5*u *u + 3*u *v + 2*u *t + ---*u *u 4x 3x 3x 2x x 2x x 2x 2 2x 9 3 2 2 9 2 9 + ---*u *u*v + ---*u *v + 3*u *u + ---*u *v + 2*u *v + ---*u *v *u 2 2x 2 2x x 2 x x 2x 2 x x 1 3 9 2 9 2 + 3*u *v *v + 2*u *t*u + 2*u *t*v + ---*u *u + ---*u *u *v + ---*u *u*v x x x x 2 x 2 x 2 x 1 3 3 3 9 2 3 2 + ---*u *v + u *x + v *u + 2*v *t*u + ---*v *u + ---*v *u *v + ---*v *u*v 2 x x 3x x 2 x 2 x 2 x + u 9 eta_v=u *v + 2*u *v + 3*u *v - 3*u *v *u - ---*u *v *v + 2*u *t*v 3x 2x x x 2x x x 2 x x x 3 2 9 2 3 3 + ---*u *u *v + ---*u *u*v + ---*u *v - v + 2*v *u + 2*v *v + 5*v *v 2 x 2 x 2 x 4x 3x 3x 2x x 3 2 9 3 2 9 2 2 - 2*v *t - ---*v *u - ---*v *u*v - ---*v *v - ---*v *u - 3*v *v 2x 2 2x 2 2x 2 2x 2 x x 1 3 9 2 9 2 1 3 + 2*v *t*u + 2*v *t*v + ---*v *u + ---*v *u *v + ---*v *u*v + ---*v *v x x 2 x 2 x 2 x 2 x + v *x + v x -------- 4. Symmetry: xi_t=0 xi_x=0 3 2 9 eta_u=u + 2*u *u + 2*u *v + 5*u *u + 3*u *v + ---*u *u + ---*u *u*v 4x 3x 3x 2x x 2x x 2 2x 2 2x 3 2 2 9 2 9 + ---*u *v + u + 3*u *u + ---*u *v + 2*u *v + ---*u *v *u + 3*u *v *v 2 2x 2x x 2 x x 2x 2 x x x x 1 3 9 2 9 2 1 3 + ---*u *u + ---*u *u *v + ---*u *u*v + u *u + ---*u *v + u *v + v *u 2 x 2 x 2 x x 2 x x 3x 3 3 9 2 3 2 + ---*v *u + ---*v *u *v + ---*v *u*v + v *u 2 x 2 x 2 x x 9 3 2 eta_v=u *v + 2*u *v + 3*u *v - 3*u *v *u - ---*u *v *v + ---*u *u *v 3x 2x x x 2x x x 2 x x 2 x 9 2 3 3 + ---*u *u*v + ---*u *v + u *v - v + 2*v *u + 2*v *v + 5*v *v 2 x 2 x x 4x 3x 3x 2x x 3 2 9 3 2 9 2 2 1 3 - ---*v *u - ---*v *u*v - ---*v *v - v - ---*v *u - 3*v *v + ---*v *u 2 2x 2 2x 2 2x 2x 2 x x 2 x 9 2 9 2 1 3 + ---*v *u *v + ---*v *u*v + v *u + ---*v *v + v *v 2 x 2 x x 2 x x -------- 5. Symmetry: xi_t=0 xi_x=0 3 2 9 eta_u=u + 2*u *u + 2*u *v + 5*u *u + 3*u *v + ---*u *u + ---*u *u*v 4x 3x 3x 2x x 2x x 2 2x 2 2x 3 2 2 9 2 9 + ---*u *v + 3*u *u + ---*u *v + 2*u *v + ---*u *v *u + 3*u *v *v 2 2x x 2 x x 2x 2 x x x x 1 3 9 2 9 2 1 3 3 3 + ---*u *u + ---*u *u *v + ---*u *u*v + ---*u *v + u + v *u + ---*v *u 2 x 2 x 2 x 2 x x 3x 2 x 9 2 3 2 + ---*v *u *v + ---*v *u*v 2 x 2 x 9 3 2 eta_v=u *v + 2*u *v + 3*u *v - 3*u *v *u - ---*u *v *v + ---*u *u *v 3x 2x x x 2x x x 2 x x 2 x 9 2 3 3 3 2 + ---*u *u*v + ---*u *v - v + 2*v *u + 2*v *v + 5*v *v - ---*v *u 2 x 2 x 4x 3x 3x 2x x 2 2x 9 3 2 9 2 2 1 3 9 2 - ---*v *u*v - ---*v *v - ---*v *u - 3*v *v + ---*v *u + ---*v *u *v 2 2x 2 2x 2 x x 2 x 2 x 9 2 1 3 + ---*v *u*v + ---*v *v + v 2 x 2 x x -------- {{}, {xi_t=0, xi_x=0, eta_u=(2*u + 8*u *c_326 + 8*u *c_328*t + 4*u *u + 4*u *v + 10*u *u 4x 3x 3x 3x 3x 2x x + 6*u *v + 12*u *c_326*u + 12*u *c_326*v + 12*u *c_328*t*u 2x x 2x 2x 2x + 12*u *c_328*t*v + 4*u *c_328*x + 2*u *c_332 + 4*u *c_333*t 2x 2x 2x 2x 2 2 2 2 + 3*u *u + 9*u *u*v + 3*u *v + 12*u *c_326 + 12*u *c_328*t 2x 2x 2x x x 2 2 + 6*u *u + 9*u *v + 4*u *v + 12*u *v *c_326 + 12*u *v *c_328*t x x x 2x x x x x 2 + 9*u *v *u + 6*u *v *v + 6*u *c_326*u + 24*u *c_326*u*v x x x x x x 2 2 + 6*u *c_326*v + 6*u *c_328*t*u + 24*u *c_328*t*u*v x x x 2 + 6*u *c_328*t*v + 4*u *c_328*u*x + 4*u *c_328*v*x + 16*u *c_328 x x x x + 2*u *c_332*u + 2*u *c_332*v + 4*u *c_333*t*u + 4*u *c_333*t*v x x x x 3 2 2 3 + 2*u *c_333*x + 2*u *c_334 + u *u + 9*u *u *v + 9*u *u*v + u *v x x x x x x 2 2 + 2*v *u + 12*v *c_326*u + 12*v *c_326*u*v + 12*v *c_328*t*u 3x x x x + 12*v *c_328*t*u*v + 4*v *c_328*u*x + 2*v *c_332*u + 4*v *c_333*t*u x x x x 3 2 2 2 + 3*v *u + 9*v *u *v + 3*v *u*v + 2*c_328*u + 6*c_328*u*v x x x + 2*c_333*u)/2, eta_v=(2*u *v + 4*u *v + 6*u *v - 12*u *v *c_326 - 12*u *v *c_328*t 3x 2x x x 2x x x x x 2 - 6*u *v *u - 9*u *v *v + 12*u *c_326*u*v + 12*u *c_326*v x x x x x x 2 + 12*u *c_328*t*u*v + 12*u *c_328*t*v + 4*u *c_328*v*x + 2*u *c_332*v x x x x 2 2 3 + 4*u *c_333*t*v + 3*u *u *v + 9*u *u*v + 3*u *v - 2*v x x x x 4x + 8*v *c_326 + 8*v *c_328*t + 4*v *u + 4*v *v + 10*v *v 3x 3x 3x 3x 2x x - 12*v *c_326*u - 12*v *c_326*v - 12*v *c_328*t*u 2x 2x 2x - 12*v *c_328*t*v - 4*v *c_328*x - 2*v *c_332 - 4*v *c_333*t 2x 2x 2x 2x 2 2 2 2 - 3*v *u - 9*v *u*v - 3*v *v - 12*v *c_326 - 12*v *c_328*t 2x 2x 2x x x 2 2 2 2 - 9*v *u - 6*v *v + 6*v *c_326*u + 24*v *c_326*u*v + 6*v *c_326*v x x x x x 2 2 + 6*v *c_328*t*u + 24*v *c_328*t*u*v + 6*v *c_328*t*v x x x + 4*v *c_328*u*x + 4*v *c_328*v*x + 2*v *c_332*u + 2*v *c_332*v x x x x 3 + 4*v *c_333*t*u + 4*v *c_333*t*v + 2*v *c_333*x + 2*v *c_334 + v *u x x x x x 2 2 3 2 + 9*v *u *v + 9*v *u*v + v *v + 6*c_328*u*v + 2*c_328*v + 2*c_333*v x x x )/2}, {c_328, c_326, c_333, c_332, c_334}} nodepnd {f,g}$ end$ Time for test: 20401 ms, plus GC time: 360 ms