Ahmed111

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5 years, 49 days

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@acer I double checked it. I didn't use lambda[1] in the entire worksheet. 

@dharr Thanks. But now I solved another determinant and this time Maple split 'lambda' into 'real' and 'imaginary' parts in some places and keep 'lambda' or 'conjugate(lambda)' remains same (look at (7)). How to fix it?

restart

with(LinearAlgebra)

assume(x::real); assume(t::real); assume(`α__1`::real); assume(`α__2`::real); assume(nu::real)

A2s := Matrix([[H__11*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__13*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`))+1, H__12*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__14*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__11*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__13*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__12*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__14*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))], [H__12*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__14*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), 1+H__11*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__13*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__12*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__14*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__11*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__13*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))], [H__13*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__33*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__14*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__34*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), 1+H__13*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__33*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__14*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__34*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))], [H__14*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__34*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__13*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__33*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__14*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__34*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__13*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__33*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))+1]])

vvalue := {v__11 = (conjugate(`λ__1`)-`λ__1`)*x+(4*`α__1`*(conjugate(`λ__1`)^3-`λ__1`^3)+2*`α__2`*(conjugate(`λ__1`)^2-`λ__1`^2)-8*nu*(conjugate(`λ__1`)^4-`λ__1`^4))*t, v__12 = (conjugate(`λ__1`)-`λ__2`)*x+(4*`α__1`*(conjugate(`λ__1`)^3-`λ__2`^3)+2*`α__2`*(conjugate(`λ__1`)^2-`λ__2`^2)-8*nu*(conjugate(`λ__1`)^4-`λ__2`^4))*t, v__21 = (conjugate(`λ__2`)-`λ__1`)*x+(4*`α__1`*(conjugate(`λ__2`)^3-`λ__1`^3)+2*`α__2`*(conjugate(`λ__2`)^2-`λ__1`^2)-8*nu*(conjugate(`λ__2`)^4-`λ__1`^4))*t, v__22 = (conjugate(`λ__2`)-`λ__2`)*x+(4*`α__1`*(conjugate(`λ__2`)^3-`λ__2`^3)+2*`α__2`*(conjugate(`λ__2`)^2-`λ__2`^2)-8*nu*(conjugate(`λ__2`)^4-`λ__2`^4))*t}

A2s2 := Determinant(A2s); dets22 := simplify(A2s2, size); length(%)

8949

(1)

dets22f := subs(vvalue, dets22)

dets22f2 := simplify(dets22f, size)

(((-(lambda__1-conjugate(lambda__1))*(-(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__1-conjugate(lambda__2)))*(lambda__2-conjugate(lambda__2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))-((lambda__2-conjugate(lambda__2))*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__1-conjugate(lambda__1))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(lambda__1-conjugate(lambda__2))*(((-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(H__11^2*H__33-H__11*H__13^2-H__11*H__14^2-H__12^2*H__33+2*H__12*H__13*H__14)*(lambda__2-conjugate(lambda__2)))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__1)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__1)^2-lambda__1*x+x*conjugate(lambda__1)))+(lambda__1-conjugate(lambda__1))*((-H__33*H__13^2+2*H__34*H__14*H__13-H__33*H__14^2+H__11*(H__33^2-H__34^2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(lambda__2-conjugate(lambda__2))*(H__11*H__33-H__13^2))))*(lambda__2-conjugate(lambda__1)))*exp(-I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(lambda__1-conjugate(lambda__2))*(((H__13^3+(-H__11*H__33-H__12*H__34-H__14^2)*H__13+H__14*(H__11*H__34+H__12*H__33))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(H__13-H__14)*(H__13+H__14)*(lambda__1-conjugate(lambda__2)))*(lambda__1-conjugate(lambda__1))*(lambda__2-conjugate(lambda__2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))+(-(H__12*H__34-H__13^2)*(lambda__2-conjugate(lambda__2))*(lambda__1-conjugate(lambda__1))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(((-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(lambda__2-conjugate(lambda__2))*(H__11*H__13-H__12*H__14))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__1)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__1)^2-lambda__1*x+x*conjugate(lambda__1)))+((H__13*H__33-H__14*H__34)*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+H__13*(lambda__2-conjugate(lambda__2)))*(lambda__1-conjugate(lambda__1)))*(lambda__1-conjugate(lambda__2)))*(lambda__2-conjugate(lambda__1))))*(lambda__1-conjugate(lambda__1))*(lambda__2-conjugate(lambda__2))*exp(-I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))+((lambda__2-conjugate(lambda__2))*(-(lambda__2-conjugate(lambda__2))*((-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(H__12*H__34-H__13^2)*(lambda__1-conjugate(lambda__2)))*(lambda__1-conjugate(lambda__1))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))+(lambda__2-conjugate(lambda__1))*((H__13-H__14)*(H__13+H__14)*(lambda__2-conjugate(lambda__2))*(lambda__1-conjugate(lambda__1))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(((-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(lambda__2-conjugate(lambda__2))*(H__11*H__13-H__12*H__14))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__1)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__1)^2-lambda__1*x+x*conjugate(lambda__1)))+((H__13*H__33-H__14*H__34)*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+H__13*(lambda__2-conjugate(lambda__2)))*(lambda__1-conjugate(lambda__1)))*(lambda__1-conjugate(lambda__2))))*(lambda__1-conjugate(lambda__1))*exp(-I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+((((lambda__1-conjugate(lambda__1))*(-(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__1-conjugate(lambda__2)))*(lambda__2-conjugate(lambda__2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))+((lambda__2-conjugate(lambda__2))*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__1-conjugate(lambda__1))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(lambda__1-conjugate(lambda__2))*(((-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(H__11^2*H__33-H__11*H__13^2-H__11*H__14^2-H__12^2*H__33+2*H__12*H__13*H__14)*(lambda__2-conjugate(lambda__2)))*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__1)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__1)^2-lambda__1*x+x*conjugate(lambda__1)))+(lambda__1-conjugate(lambda__1))*((-H__33*H__13^2+2*H__34*H__14*H__13-H__33*H__14^2+H__11*(H__33^2-H__34^2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(lambda__2-conjugate(lambda__2))*(H__11*H__33-H__13^2))))*(lambda__2-conjugate(lambda__1)))*exp(-I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))+(((-H__11^2*H__33+H__11*H__13^2+H__11*H__14^2+H__12^2*H__33-2*H__12*H__13*H__14)*exp(I*(8*t*nu*lambda__1^4-8*t*nu*conjugate(lambda__2)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*conjugate(lambda__2)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*conjugate(lambda__2)^2-lambda__1*x+x*conjugate(lambda__2)))+(lambda__1-conjugate(lambda__2))*(H__11*H__13-H__12*H__14))*(lambda__1-conjugate(lambda__1))*(lambda__2-conjugate(lambda__2))*exp(I*(8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-x*lambda__2+x*conjugate(lambda__1)))+((lambda__2-conjugate(lambda__2))*(lambda__1-conjugate(lambda__1))*(H__11*H__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*conjugate(lambda__2)^3-2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__2)^2-x*lambda__2+x*conjugate(lambda__2)))*H__33+lambda__2-conjugate(lambda__2))*(lambda__1-conjugate(lambda__1)))*(lambda__1-conjugate(lambda__2)))*(lambda__2-conjugate(lambda__1)))*(lambda__2-conjugate(lambda__2))))*(lambda__1-conjugate(lambda__2)))*(lambda__2-conjugate(lambda__1)))/((lambda__1-conjugate(lambda__1))^2*(lambda__1-conjugate(lambda__2))^2*(lambda__2-conjugate(lambda__1))^2*(lambda__2-conjugate(lambda__2))^2)

(2)

A2snum := Matrix([[H__11*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__13*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`))+1, H__12*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__14*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__11*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__13*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__12*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__14*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__12*conjugate(psi1)+H__14*conjugate(psi2)], [H__12*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__14*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), 1+H__11*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__13*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__12*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__14*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__11*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__13*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__11*conjugate(psi1)+H__13*conjugate(psi2)], [H__13*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__33*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__14*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__34*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), 1+H__13*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__33*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__14*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__34*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__14*conjugate(psi1)+H__34*conjugate(psi2)], [H__14*exp(I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__34*exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__13*exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`))+H__33*exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), H__14*exp(I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__34*exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), H__13*exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))+H__33*exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))+1, H__13*conjugate(psi1)+H__33*conjugate(psi2)], [0, phi1, 0, phi2, 0]])

Matrix(%id = 36893490104711191356)

(3)

valphipsi12 := {phi1 = exp(-I*`λ__1`*x-((4*I)*`λ__1`^3*`α__1`+(2*I)*`λ__1`^2*`α__2`-(8*I)*nu*`λ__1`^4)*t), phi2 = exp(-I*`λ__2`*x-((4*I)*`λ__2`^3*`α__1`+(2*I)*`λ__2`^2*`α__2`-(8*I)*nu*`λ__2`^4)*t), conjugate(psi1) = exp(-I*conjugate(`λ__1`)*x-((4*I)*conjugate(`λ__1`)^3*`α__1`+(2*I)*conjugate(`λ__1`)^2*`α__2`-(8*I)*nu*conjugate(`λ__1`)^4)*t), conjugate(psi2) = exp(-I*conjugate(`λ__2`)*x-((4*I)*conjugate(`λ__2`)^3*`α__1`+(2*I)*conjugate(`λ__2`)^2*`α__2`-(8*I)*nu*conjugate(`λ__2`)^4)*t)}

{phi1 = exp(-I*lambda__1*x-((4*I)*lambda__1^3*alpha__1+(2*I)*lambda__1^2*alpha__2-(8*I)*nu*lambda__1^4)*t), phi2 = exp(-I*lambda__2*x-((4*I)*lambda__2^3*alpha__1+(2*I)*lambda__2^2*alpha__2-(8*I)*nu*lambda__2^4)*t), conjugate(psi1) = exp(-I*conjugate(lambda__1)*x-((4*I)*conjugate(lambda__1)^3*alpha__1+(2*I)*conjugate(lambda__1)^2*alpha__2-(8*I)*nu*conjugate(lambda__1)^4)*t), conjugate(psi2) = exp(-I*conjugate(lambda__2)*x-((4*I)*conjugate(lambda__2)^3*alpha__1+(2*I)*conjugate(lambda__2)^2*alpha__2-(8*I)*nu*conjugate(lambda__2)^4)*t)}

(4)

A2s2num := Determinant(A2snum); dets22num := simplify(A2s2num, size); length(%)

10115

(5)

dets22fnum := subs(vvalue, dets22num)

Iprint*valphipsi12

Iprint*{phi1 = exp(-I*lambda__1*x-((4*I)*lambda__1^3*alpha__1+(2*I)*lambda__1^2*alpha__2-(8*I)*nu*lambda__1^4)*t), phi2 = exp(-I*lambda__2*x-((4*I)*lambda__2^3*alpha__1+(2*I)*lambda__2^2*alpha__2-(8*I)*nu*lambda__2^4)*t), conjugate(psi1) = exp(-I*conjugate(lambda__1)*x-((4*I)*conjugate(lambda__1)^3*alpha__1+(2*I)*conjugate(lambda__1)^2*alpha__2-(8*I)*nu*conjugate(lambda__1)^4)*t), conjugate(psi2) = exp(-I*conjugate(lambda__2)*x-((4*I)*conjugate(lambda__2)^3*alpha__1+(2*I)*conjugate(lambda__2)^2*alpha__2-(8*I)*nu*conjugate(lambda__2)^4)*t)}

(6)

dets22f2num := simplify(subs(valphipsi12, dets22fnum))

(1/8)*(-(4*I)*(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(64*Re(lambda__2)^3*nu*t-24*Re(lambda__2)^2*t*alpha__1-8*Re(lambda__2)*t*alpha__2-2*x)*Im(lambda__2)-(8*I)*conjugate(lambda__2)^4*nu*t+(4*I)*conjugate(lambda__2)^3*t*alpha__1+(2*I)*conjugate(lambda__2)^2*t*alpha__2+I*conjugate(lambda__2)*x+((-(8*I)*lambda__1^3-(4*I)*lambda__2^3)*alpha__1+((16*I)*lambda__1^4+(8*I)*lambda__2^4)*nu-(4*I)*(lambda__1^2+(1/2)*lambda__2^2)*alpha__2)*t-(2*I)*x*(lambda__1+(1/2)*lambda__2))+8*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp((8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x-I*(8*nu*lambda__1^4-16*nu*lambda__2^4-4*alpha__1*lambda__1^3+8*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2+4*alpha__2*lambda__2^2)*t+I*(lambda__1-2*lambda__2)*x)+8*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2*(H__11*H__33-H__13^2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp((8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x-I*(8*t*nu*lambda__1^4-8*t*nu*lambda__2^4-8*t*nu*conjugate(lambda__1)^4-4*t*alpha__1*lambda__1^3+4*t*alpha__1*lambda__2^3+4*t*alpha__1*conjugate(lambda__1)^3-2*t*alpha__2*lambda__1^2+2*alpha__2*t*lambda__2^2+2*alpha__2*t*conjugate(lambda__1)^2-lambda__1*x+x*lambda__2+x*conjugate(lambda__1)))+2*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*(-H__33*H__13^2+2*H__34*H__14*H__13-H__33*H__14^2+H__11*(H__33^2-H__34^2))*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(-64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(64*Re(lambda__1)^3*nu*t-24*Re(lambda__1)^2*t*alpha__1-8*Re(lambda__1)*t*alpha__2-2*x)*Im(lambda__1)+64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*lambda__2*(lambda__2*(nu*lambda__2^2-(1/2)*alpha__1*lambda__2-(1/4)*alpha__2)*t-(1/8)*x))+2*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*lambda__2*(lambda__2*(nu*lambda__2^2-(1/2)*alpha__1*lambda__2-(1/4)*alpha__2)*t-(1/8)*x))-8*(lambda__2-conjugate(lambda__1))*(H__13*H__33-H__14*H__34)*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(I*(8*t*nu*conjugate(lambda__2)^4-8*t*nu*conjugate(lambda__1)^4+16*t*nu*lambda__2^4-4*t*alpha__1*conjugate(lambda__2)^3+4*t*alpha__1*conjugate(lambda__1)^3-8*t*alpha__1*lambda__2^3-2*alpha__2*t*conjugate(lambda__2)^2+2*alpha__2*t*conjugate(lambda__1)^2-4*alpha__2*t*lambda__2^2-x*conjugate(lambda__2)+x*conjugate(lambda__1)-2*x*lambda__2))+8*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2*(H__12*H__34-H__13^2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp((8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x-(8*I)*conjugate(lambda__2)^4*nu*t+(4*I)*conjugate(lambda__2)^3*t*alpha__1+(2*I)*conjugate(lambda__2)^2*t*alpha__2+I*conjugate(lambda__2)*x+I*(8*nu*lambda__1^4+8*nu*lambda__2^4-4*alpha__1*lambda__1^3-4*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2-2*alpha__2*lambda__2^2)*t-I*x*(lambda__1+lambda__2))+8*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(I*(8*t*nu*conjugate(lambda__1)^4+16*t*nu*lambda__1^4-8*t*nu*lambda__2^4-4*t*alpha__1*conjugate(lambda__1)^3-8*t*alpha__1*lambda__1^3+4*t*alpha__1*lambda__2^3-2*alpha__2*t*conjugate(lambda__1)^2-4*t*alpha__2*lambda__1^2+2*alpha__2*t*lambda__2^2-x*conjugate(lambda__1)-2*lambda__1*x+x*lambda__2))+8*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-(8*I)*conjugate(lambda__2)^4*nu*t+(4*I)*conjugate(lambda__2)^3*t*alpha__1+(2*I)*conjugate(lambda__2)^2*t*alpha__2+I*conjugate(lambda__2)*x+I*(8*nu*lambda__1^4+16*nu*lambda__2^4-4*alpha__1*lambda__1^3-8*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2-4*alpha__2*lambda__2^2)*t-I*(lambda__1+2*lambda__2)*x)-8*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*((-H__13*H__33+H__14*H__34)*conjugate(lambda__1)+(-H__11*H__13+H__12*H__14)*conjugate(lambda__2)+(H__11*H__13-H__12*H__14)*lambda__1+(H__13*H__33-H__14*H__34)*lambda__2)*exp(I*(8*nu*lambda__1^4+8*nu*lambda__2^4-4*alpha__1*lambda__1^3-4*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2-2*alpha__2*lambda__2^2)*t-I*x*(lambda__1+lambda__2))-8*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*(H__11*H__13-H__12*H__14)*exp(I*(-8*t*nu*conjugate(lambda__2)^4+8*t*nu*conjugate(lambda__1)^4+16*t*nu*lambda__1^4+4*t*alpha__1*conjugate(lambda__2)^3-4*t*alpha__1*conjugate(lambda__1)^3-8*t*alpha__1*lambda__1^3+2*alpha__2*t*conjugate(lambda__2)^2-2*alpha__2*t*conjugate(lambda__1)^2-4*t*alpha__2*lambda__1^2+x*conjugate(lambda__2)-x*conjugate(lambda__1)-2*lambda__1*x))+(4*I)*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*((H__33^2-H__34^2)*Im(lambda__1)^2-(1/4)*(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12)))*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*lambda__2*(lambda__2*(nu*lambda__2^2-(1/2)*alpha__1*lambda__2-(1/4)*alpha__2)*t-(1/8)*x))+8*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*(H__12*H__34-H__13^2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(-(8*I)*conjugate(lambda__1)^4*nu*t+(4*I)*conjugate(lambda__1)^3*t*alpha__1+(2*I)*conjugate(lambda__1)^2*t*alpha__2+I*conjugate(lambda__1)*x+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+I*(8*nu*lambda__1^4+8*nu*lambda__2^4-4*alpha__1*lambda__1^3-4*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2-2*alpha__2*lambda__2^2)*t-I*x*(lambda__1+lambda__2))+8*(lambda__2-conjugate(lambda__1))*(H__11^2*H__33-H__11*H__13^2-H__11*H__14^2-H__12^2*H__33+2*H__12*H__13*H__14)*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-(8*I)*conjugate(lambda__2)^4*nu*t+(4*I)*conjugate(lambda__2)^3*t*alpha__1+(2*I)*conjugate(lambda__2)^2*t*alpha__2+I*conjugate(lambda__2)*x+I*(16*nu*lambda__1^4+8*nu*lambda__2^4-8*alpha__1*lambda__1^3-4*alpha__1*lambda__2^3-4*alpha__2*lambda__1^2-2*alpha__2*lambda__2^2)*t-(2*I)*x*(lambda__1+(1/2)*lambda__2))+2*(lambda__2-conjugate(lambda__1))^2*(H__11^2*H__33-H__11*H__13^2-H__11*H__14^2-H__12^2*H__33+2*H__12*H__13*H__14)*Im(lambda__2)*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)-64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(64*Re(lambda__2)^3*nu*t-24*Re(lambda__2)^2*t*alpha__1-8*Re(lambda__2)*t*alpha__2-2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*(lambda__1*(nu*lambda__1^2-(1/2)*alpha__1*lambda__1-(1/4)*alpha__2)*t-(1/8)*x)*lambda__1)+8*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*(-H__33*H__13^2+2*H__34*H__14*H__13-H__33*H__14^2+H__11*(H__33^2-H__34^2))*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-(8*I)*conjugate(lambda__1)^4*nu*t+(4*I)*conjugate(lambda__1)^3*t*alpha__1+(2*I)*conjugate(lambda__1)^2*t*alpha__2+I*conjugate(lambda__1)*x+I*(8*nu*lambda__1^4+16*nu*lambda__2^4-4*alpha__1*lambda__1^3-8*alpha__1*lambda__2^3-2*alpha__2*lambda__1^2-4*alpha__2*lambda__2^2)*t-I*(lambda__1+2*lambda__2)*x)+8*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)^2*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-(8*I)*conjugate(lambda__1)^4*nu*t+(4*I)*conjugate(lambda__1)^3*t*alpha__1+(2*I)*conjugate(lambda__1)^2*t*alpha__2+I*conjugate(lambda__1)*x+I*(16*nu*lambda__1^4+8*nu*lambda__2^4-8*alpha__1*lambda__1^3-4*alpha__1*lambda__2^3-4*alpha__2*lambda__1^2-2*alpha__2*lambda__2^2)*t-(2*I)*x*(lambda__1+(1/2)*lambda__2))+2*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha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3*H__13^2+2*H__34*H__14*H__13-H__33*H__14^2+H__11*(H__33^2-H__34^2))*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*(((-(1/2)*lambda__1^2-(1/2)*lambda__1*lambda__2-(1/2)*lambda__2^2)*alpha__1+(-(1/4)*alpha__2+nu*(lambda__1^2+lambda__2^2))*(lambda__1+lambda__2))*t-(1/8)*x)*(-lambda__2+lambda__1))+(4*I)*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2*(H__11*H__33-H__14^2)*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*lambda__2*(lambda__2*(nu*lambda__2^2-(1/2)*alpha__1*lambda__2-(1/4)*alpha__2)*t-(1/8)*x))+(4*I)*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*(H__11*H__33-H__13^2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(-64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(64*Re(lambda__2)^3*nu*t-24*Re(lambda__2)^2*t*alpha__1-8*Re(lambda__2)*t*alpha__2-2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*(lambda__1*(nu*lambda__1^2-(1/2)*alpha__1*lambda__1-(1/4)*alpha__2)*t-(1/8)*x)*lambda__1)+(4*I)*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*(H__11*H__13-H__12*H__14)*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*(lambda__1*(nu*lambda__1^2-(1/2)*alpha__1*lambda__1-(1/4)*alpha__2)*t-(1/8)*x)*lambda__1)-2*(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(lambda__2-conjugate(lambda__1))*Im(lambda__2)*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*Im(lambda__1)*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*(((-(1/2)*lambda__1^2-(1/2)*lambda__1*lambda__2-(1/2)*lambda__2^2)*alpha__1+(-(1/4)*alpha__2+nu*(lambda__1^2+lambda__2^2))*(lambda__1+lambda__2))*t-(1/8)*x)*(-lambda__2+lambda__1))-2*(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12))*Im(lambda__1)*(lambda__1-conjugate(lambda__2))*exp(64*(nu*Re(lambda__1)-(1/8)*alpha__1)*t*Im(lambda__1)^3+(-64*Re(lambda__1)^3*nu*t+24*Re(lambda__1)^2*t*alpha__1+8*Re(lambda__1)*t*alpha__2+2*x)*Im(lambda__1)+64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x-(8*I)*(((-(1/2)*lambda__1^2-(1/2)*lambda__1*lambda__2-(1/2)*lambda__2^2)*alpha__1+(-(1/4)*alpha__2+nu*(lambda__1^2+lambda__2^2))*(lambda__1+lambda__2))*t-(1/8)*x)*(-lambda__2+lambda__1))+(4*I)*(-H__13^3+(H__11*H__33+H__12*H__34+H__14^2)*H__13-H__14*(H__11*H__34+H__12*H__33))*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))*exp(-64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(64*Re(lambda__2)^3*nu*t-24*Re(lambda__2)^2*t*alpha__1-8*Re(lambda__2)*t*alpha__2-2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__1)^4*nu*t-(4*I)*conjugate(lambda__1)^3*t*alpha__1-(2*I)*conjugate(lambda__1)^2*t*alpha__2-I*conjugate(lambda__1)*x-(8*I)*conjugate(lambda__2)^4*nu*t+(4*I)*conjugate(lambda__2)^3*t*alpha__1+(2*I)*conjugate(lambda__2)^2*t*alpha__2+I*conjugate(lambda__2)*x+(16*I)*(lambda__1*(nu*lambda__1^2-(1/2)*alpha__1*lambda__1-(1/4)*alpha__2)*t-(1/8)*x)*lambda__1)+(4*I)*(lambda__2-conjugate(lambda__1))^2*(H__13*H__33-H__14*H__34)*Im(lambda__2)*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(64*(nu*Re(lambda__2)-(1/8)*alpha__1)*t*Im(lambda__2)^3+(-64*Re(lambda__2)^3*nu*t+24*Re(lambda__2)^2*t*alpha__1+8*Re(lambda__2)*t*alpha__2+2*x)*Im(lambda__2)+(8*I)*conjugate(lambda__2)^4*nu*t-(4*I)*conjugate(lambda__2)^3*t*alpha__1-(2*I)*conjugate(lambda__2)^2*t*alpha__2-I*conjugate(lambda__2)*x+(8*I)*(lambda__1*(nu*lambda__1^2-(1/2)*alpha__1*lambda__1-(1/4)*alpha__2)*t-(1/8)*x)*lambda__1)-8*Im(lambda__2)^2*((-(1/2)*H__12*H__13*H__14+H__33*Im(lambda__1)^2-(1/4)*H__11^2*H__33+(1/4)*H__11*H__13^2+(1/4)*H__11*H__14^2+(1/4)*H__12^2*H__33)*(lambda__2-conjugate(lambda__1))^2*(lambda__1-conjugate(lambda__2))^2*exp(I*(8*t*nu*conjugate(lambda__2)^4+8*t*nu*lambda__2^4-4*t*alpha__1*conjugate(lambda__2)^3-4*t*alpha__1*lambda__2^3-2*alpha__2*t*conjugate(lambda__2)^2-2*alpha__2*t*lambda__2^2-x*conjugate(lambda__2)-x*lambda__2))+((lambda__2-conjugate(lambda__1))^2*(H__13*conjugate(lambda__2)^2-2*lambda__1*H__13*conjugate(lambda__2)+lambda__1^2*H__13+H__13^3+(-H__11*H__33-H__12*H__34-H__14^2)*H__13+H__14*(H__11*H__34+H__12*H__33))*exp(I*(8*t*nu*conjugate(lambda__1)^4+8*t*nu*lambda__2^4-4*t*alpha__1*conjugate(lambda__1)^3-4*t*alpha__1*lambda__2^3-2*alpha__2*t*conjugate(lambda__1)^2-2*alpha__2*t*lambda__2^2-x*conjugate(lambda__1)-x*lambda__2))+(conjugate(lambda__1)^2*H__13-2*lambda__2*conjugate(lambda__1)*H__13+lambda__2^2*H__13+H__13^3+(-H__11*H__33-H__12*H__34-H__14^2)*H__13+H__14*(H__11*H__34+H__12*H__33))*(lambda__1-conjugate(lambda__2))^2*exp(I*(8*t*nu*conjugate(lambda__2)^4+8*t*nu*lambda__1^4-4*t*alpha__1*conjugate(lambda__2)^3-4*t*alpha__1*lambda__1^3-2*alpha__2*t*conjugate(lambda__2)^2-2*t*alpha__2*lambda__1^2-x*conjugate(lambda__2)-lambda__1*x))+((H__13^2-H__14^2)*conjugate(lambda__1)^2+(-2*H__13^2+2*H__14^2)*lambda__2*conjugate(lambda__1)+(H__13^2-H__14^2)*lambda__2^2+(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12)))*(lambda__1-conjugate(lambda__2))*exp((2*I)*lambda__1*(8*nu*t*lambda__1^3-4*t*alpha__1*lambda__1^2-2*t*alpha__2*lambda__1-x))+exp((2*I)*lambda__2*(8*nu*t*lambda__2^3-4*t*alpha__1*lambda__2^2-2*t*alpha__2*lambda__2-x))*((H__13^2-H__14^2)*conjugate(lambda__2)^2+(-2*H__13^2+2*H__14^2)*lambda__1*conjugate(lambda__2)+(H__13^2-H__14^2)*lambda__1^2+(-H__13^2-2*H__13*H__14-H__14^2+(H__33+H__34)*(H__11+H__12))*(-H__13^2+2*H__13*H__14-H__14^2+(H__33-H__34)*(H__11-H__12)))*(lambda__2-conjugate(lambda__1)))*Im(lambda__1)^2)-8*(lambda__2-conjugate(lambda__1))^2*(Im(lambda__2)^2*H__11+(1/4)*H__33*H__13^2-(1/2)*H__34*H__14*H__13+(1/4)*H__33*H__14^2+H__11*(-(1/4)*H__33^2+(1/4)*H__34^2))*Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*exp(I*(8*t*nu*conjugate(lambda__1)^4+8*t*nu*lambda__1^4-4*t*alpha__1*conjugate(lambda__1)^3-4*t*alpha__1*lambda__1^3-2*alpha__2*t*conjugate(lambda__1)^2-2*t*alpha__2*lambda__1^2-x*conjugate(lambda__1)-lambda__1*x)))/(Im(lambda__1)^2*(lambda__1-conjugate(lambda__2))^2*(lambda__2-conjugate(lambda__1))^2*Im(lambda__2)^2)

(7)

NULL

 

Download sol1det.mw

@rlopez because u, ub, and their derivatives are non-abelian i.e. their multiplication are non-commutative. That's why I used bold black dots.

@rlopez I try it as you suggested but got a '0' answer for all the coefficients.

restart

with(LinearAlgebra)

alias(u = u(x, t), ub = ub(x, t))

u, ub

(1)

NULL

z1 := I*ub*gamma*(diff(u, x, x, x))+(2*I)*u*(2*gamma*lambda^2+alpha*lambda-1/4)*(diff(ub, x))+(2*I)*ub*(-2*gamma*lambda^2+alpha*lambda-1/4)*(diff(u, x))+Typesetting[delayDotProduct](4*ub*gamma, lambda*u.ub.u, true)+Typesetting[delayDotProduct]((3*I)*gamma, ub.u.ub.(diff(u, x)), true)+Typesetting[delayDotProduct]((3*I)*gamma, ub.u.(diff(ub, x)).u, true)+Typesetting[delayDotProduct]((3*I)*gamma, ub.(diff(u, x)).ub.u, true)+Typesetting[delayDotProduct]((3*I)*gamma, (diff(ub, x)).u.ub.u, true)+Typesetting[delayDotProduct](I*gamma, (diff(u, x)).(diff(ub, x, x)), true)-Typesetting[delayDotProduct](I*gamma, (diff(ub, x, x)).(diff(u, x)), true)+Typesetting[delayDotProduct](I*gamma, u.(diff(ub, x, x, x)), true)+Typesetting[delayDotProduct](I*gamma, ub.(diff(u, x, x, x)), true)-Typesetting[delayDotProduct]((2*I)*u*gamma, ub.(3*u.(diff(ub, x))-(4*I)*lambda^3), true)+Typesetting[delayDotProduct]((2*I)*ub*gamma, u.(3*ub.(diff(u, x))-(4*I)*lambda^3), true)+(1/2)*Typesetting[delayDotProduct](-4*gamma*lambda-2*alpha, u.(diff(ub, x, x)), true)+(1/2)*Typesetting[delayDotProduct](4*gamma*lambda+2*alpha, ub.(diff(u, x, x)), true)+(1/2)*Typesetting[delayDotProduct](-4*gamma*lambda-2*alpha, (diff(u, x)).(diff(ub, x)), true)+(1/2)*Typesetting[delayDotProduct](4*gamma*lambda+2*alpha, (diff(ub, x)).(diff(u, x)), true)+(1/2)*Typesetting[delayDotProduct](-(8*I)*gamma*lambda^2-(4*I)*alpha*lambda+I, (diff(ub, x)).u, true)+(1/2)*Typesetting[delayDotProduct](-(8*I)*gamma*lambda^2-(4*I)*alpha*lambda+I, ub.(diff(u, x)), true)+Typesetting[delayDotProduct](4*u*gamma, lambda*ub.u.ub, true)+Typesetting[delayDotProduct](2*u, u.ub.u, true)*alpha-Typesetting[delayDotProduct](2*ub, ub.u.ub, true)*alpha-I*u*gamma*(diff(ub, x, x, x))+u*(2*gamma*lambda+alpha)*(diff(ub, x, x))-ub*(-2*gamma*lambda+alpha)*(diff(u, x, x)) = 0

(1/2)*(-(8*I)*gamma*lambda^2-(4*I)*alpha*lambda+I)*(ub.(diff(u, x)))-ub*(-2*gamma*lambda+alpha)*(diff(diff(u, x), x))+u*(2*gamma*lambda+alpha)*(diff(diff(ub, x), x))+(1/2)*(4*gamma*lambda+2*alpha)*((diff(ub, x)).(diff(u, x)))+(1/2)*(-4*gamma*lambda-2*alpha)*((diff(u, x)).(diff(ub, x)))+(1/2)*(4*gamma*lambda+2*alpha)*(ub.(diff(diff(u, x), x)))-2*ub*(`.`(ub, u, ub))*alpha+2*u*(`.`(u, ub, u))*alpha-I*gamma*((diff(diff(ub, x), x)).(diff(u, x)))+(1/2)*(-4*gamma*lambda-2*alpha)*(u.(diff(diff(ub, x), x)))+(2*I)*u*(2*gamma*lambda^2+alpha*lambda-1/4)*(diff(ub, x))+(2*I)*ub*(-2*gamma*lambda^2+alpha*lambda-1/4)*(diff(u, x))+I*ub*gamma*(diff(diff(diff(u, x), x), x))-I*u*gamma*(diff(diff(diff(ub, x), x), x))+(2*I)*ub*gamma*(u.(3*(ub.(diff(u, x)))-(4*I)*lambda^3))-(2*I)*u*gamma*(ub.(3*(u.(diff(ub, x)))-(4*I)*lambda^3))+(1/2)*(-(8*I)*gamma*lambda^2-(4*I)*alpha*lambda+I)*((diff(ub, x)).u)+(3*I)*gamma*(`.`(ub, u, diff(ub, x), u))+(3*I)*gamma*(`.`(ub, u, ub, diff(u, x)))+I*gamma*((diff(u, x)).(diff(diff(ub, x), x)))+4*u*gamma*(`.`(lambda*ub, u, ub))+(3*I)*gamma*(`.`(diff(ub, x), u, ub, u))+(3*I)*gamma*(`.`(ub, diff(u, x), ub, u))+4*ub*gamma*(`.`(lambda*u, ub, u))+I*gamma*(ub.(diff(diff(diff(u, x), x), x)))+I*gamma*(u.(diff(diff(diff(ub, x), x), x))) = 0

(2)

for m from 0 to 5 do `#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))` end do

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

 

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

 

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

 

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

 

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

 

`#mrow(msup(mi("λ",fontstyle = "normal"),mi("m")),mo("≔"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo(","),mi("lambda",fontstyle = "normal"),mo(","),mi("m"))),mo("="),mn("0"),mo(";"))`

(3)

NULL

Download compare.mw

@mmcdara when we change "y1:=-1*v^2+1/8*v^4" to "y1:=-1*v^2+1/4*v^4", it shows an error. How to remove it?

pot_mmcdara1.mw 

@ecterrab I removed it. But still not working. 

@tomleslie Can we plot (4)? I tried it by putting "
plot(rhs(%), {_C1 = 1, c = 1, g = 1.5}, x = -10 .. 10);" just after (4). But encountered with "Error, invalid input: rhs received [w(x) = 2*JacobiSN((1/2)*(-2*c+2*(c^2+2*g)^(1/2))^(1/2)*x+_C1, ((c*(c^2+2*g)^(1/2)-c^2-g)*g)^(1/2)/(c*(c^2+2*g)^(1/2)-c^2-g))*g/(g*(-c+(c^2+2*g)^(1/2)))^(1/2)], which is not valid for its 1st argument, expr"

@Rouben Rostamian  When I execute Lineaize.mw on my computer. Maple returns "Error, (in collect) cannot collect 0"

restart

with(LinearAlgebra):

with(PDEtools):

with(Physics):

with(plots):

Setup(mathematicalnotation = true);

[mathematicalnotation = true]

(1)

assume(x::real);

assume(t::real);

assume(xi::real);

assume(tau::real);

interface(showassumed = 0);

0

(2)

alias(phi = phi(x, t), epsilon[1] = epsilon[1](x, t), epsilon[2] = epsilon[2](x, t), rho = rho(x, t), r = r(x, t), q = q(x, t), conjugate(q) = (conjugate(q))(x, t));

phi, epsilon[1], epsilon[2], rho, r, q, conjugate(q)

(3)

Eq := Physics:-`*`(diff(r, x, t), diff(r, x, t))-Physics:-`*`(Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q), Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), diff(r, x, t)), Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q))-Physics:-`*`(Physics:-`*`(4, r^2), 4-Physics:-`*`(diff(r, t), diff(r, t))-Physics:-`*`(Physics:-`*`(4, diff(q, t)), diff(conjugate(q), t))) = 0;

(diff(diff(r, t), x))^2-((diff(q, t))*conjugate(q)-(diff(conjugate(q), t))*q)^2-(2*I)*(diff(diff(r, t), x))*((diff(q, t))*conjugate(q)-(diff(conjugate(q), t))*q)-4*r^2*(4-(diff(r, t))^2-4*(diff(q, t))*(diff(conjugate(q), t))) = 0

(4)

va := {q = Physics:-`*`(Physics:-`*`(Physics:-`*`(sqrt(2), c), exp(Physics:-`*`(Physics:-`*`(I, -1/sqrt(3)), t))), 1+epsilon[2]), r = Physics:-`*`(c, 1+epsilon[1]), conjugate(q) = Physics:-`*`(Physics:-`*`(Physics:-`*`(sqrt(2), c), exp(-Physics:-`*`(Physics:-`*`(I, -1/sqrt(3)), t))), 1+epsilon[2])};

{q = 2^(1/2)*c*exp(-((1/3)*I)*3^(1/2)*t)*(1+epsilon[2]), r = c*(1+epsilon[1]), conjugate(q) = 2^(1/2)*c*exp(((1/3)*I)*3^(1/2)*t)*(1+epsilon[2])}

(5)

Eq1 := simplify(eval(Eq, va), size);

4*c^2*((4/3)*c^2*(exp(((1/3)*I)*3^(1/2)*t))^2*(1+epsilon[2])^4*(exp(-((1/3)*I)*3^(1/2)*t))^2+8*exp(((1/3)*I)*3^(1/2)*t)*c*(-(1/12)*3^(1/2)*(1+epsilon[2])^2*(diff(diff(epsilon[1], t), x))+c*(1+epsilon[1])^2*((diff(epsilon[2], t))^2+(1/3)*(1+epsilon[2])^2))*exp(-((1/3)*I)*3^(1/2)*t)+(1/4)*(diff(diff(epsilon[1], t), x))^2+(1+epsilon[1])^2*(c*(diff(epsilon[1], t))-2)*(c*(diff(epsilon[1], t))+2)) = 0

(6)

indets(Eq1, function);

{epsilon[1], epsilon[2], diff(epsilon[1], t), diff(epsilon[2], t), diff(diff(epsilon[1], t), x), exp(-((1/3)*I)*3^(1/2)*t), exp(((1/3)*I)*3^(1/2)*t)}

(7)

In Eq1 replace diff(`ε`[1], t, x) by the symbol e__1tx, etc., linearize with respect to those symbols,

and then restore the symbols to their original meanings:

subs(diff(`ε`[1], t, x) = e__1tx, diff(`ε`[1], t) = e__1t, `ε`[1] = e__1, diff(`ε`[2], t) = e__2t, `ε`[2] = e__2, lhs(Eq1)); mtaylor(%, [e__1, e__1t, e__1tx, e__2, e__2t], 2); subs(e__1 = `ε`[1], e__2 = `ε`[2], e__1tx = diff(`ε`[1], t, x), %); simplify(%); collect(%, [`ε`[1], `ε`[2], diff(`ε`[1], t, x)]) = 0

Error, (in collect) cannot collect 0

 

 

``

Download Lineaize.mw

@Rouben Rostamian  I got it. But when we use eq (5) into eq (4), maple gives eq (6) after simplification. How can we know the result (eq (6)) is correct?

@Carl Love epsilon derivative raised to power 2 or higher can be removed.

@Carl Love After doing mtaylor, no derivative involve in the final equation. See attachment (where I didn't use mtaylor), there are also derivative terms like diff(epsilon[2], t), etc in equation (6). But I am not clear whether the calculations are going in the right direction or not?

Calculation.mw

@Carl Love as per your suggestion, I got linearize equation. But epsilon[1] and epsilon[2] are functions of and t. How to do it with epsilon[1]=epsilon[1](x,t) and epsilon[2]=epsilon[2](x,t)?

Calculation.mw  

@Kitonum how to get continuous system from dicrete system by sending h (lattice parameter along x-axis) approaches to zero in maple? i.e., in the attached worksheet, if we send h --> 0, then eq (5) reduces to q_{xt} + (2r)r_{x}=0. How to do that in maple?

 

 

CD_new.mw

@Kitonum D2 is the derivative?

@Carl Love very nice explanation. To understand completely I need more details on "order", "digits" and "degree". 

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