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Ahmed111

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@sand15 Very well. But in my unders...

Ahmed111 130
November 21 2023
0 0

@sand15 Very well. But in my understanding V := (N, s) -> Vector[row](N, symbol=s) is a row vector (You wrote in CP_2 as column vector). And expressions for T[i] in (12) and (13) are different (see attached). Why do we need to perform the last step (i.e., (13))?

> 

restart
> 

with(LinearAlgebra):
> 

V := (N, s) -> Vector[row](N, symbol=s);
J := N -> IdentityMatrix(N);

proc (N, s) options operator, arrow; Vector[row](N, symbol = s) end proc

  

proc (N) options operator, arrow; LinearAlgebra:-IdentityMatrix(N) end proc

 (1)
> 

n   := 4;
k   := lambda+m__0;

M11 := I *~ ( DiagonalMatrix(  V(n, a) ) + k *~ IdentityMatrix(n) );
M12 := V(n, c);
M21 := -M12^+;
l1:=-2*I*lambda+I*k;

M  := `<,>`( `<|>`(l1, M12), `<|>`(M21, M11) );

n := 4

  

k := lambda+m__0

  

Matrix(4, 4, {(1, 1) = I*(a[1]+lambda+`#msub(mi("m"),mi("0"))`), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = I*(a[2]+lambda+`#msub(mi("m"),mi("0"))`), (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = I*(a[3]+lambda+`#msub(mi("m"),mi("0"))`), (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = I*(a[4]+lambda+`#msub(mi("m"),mi("0"))`)})

  

Vector[row](4, {(1) = c[1], (2) = c[2], (3) = c[3], (4) = c[4]})

  

Vector(4, {(1) = -c[1], (2) = -c[2], (3) = -c[3], (4) = -c[4]})

  

l1 := -(2*I)*lambda+I*(lambda+m__0)

  

Matrix(%id = 36893490902336311892)

 (2)
> 

 

> 


 


A pretty formula for the characteristic polynomial
 

> 

# Step 0: define X this way

X := eta *~ IdentityMatrix(n+1) - eval(M, lambda=theta-m__0)

Matrix(5, 5, {(1, 1) = eta+(2*I)*(theta-`#msub(mi("m"),mi("0"))`)-I*theta, (1, 2) = -c[1], (1, 3) = -c[2], (1, 4) = -c[3], (1, 5) = -c[4], (2, 1) = c[1], (2, 2) = eta-I*(a[1]+theta), (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (3, 1) = c[2], (3, 2) = 0, (3, 3) = eta-I*(a[2]+theta), (3, 4) = 0, (3, 5) = 0, (4, 1) = c[3], (4, 2) = 0, (4, 3) = 0, (4, 4) = eta-I*(a[3]+theta), (4, 5) = 0, (5, 1) = c[4], (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = eta-I*(a[4]+theta)})

 (3)
> 

# Step 1: write each diagonal element this way

for i from 1 to n+1 do
  X[i, i] := eta-u[i]
end do:

X;

Matrix(5, 5, {(1, 1) = eta-u[1], (1, 2) = -c[1], (1, 3) = -c[2], (1, 4) = -c[3], (1, 5) = -c[4], (2, 1) = c[1], (2, 2) = eta-u[2], (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (3, 1) = c[2], (3, 2) = 0, (3, 3) = eta-u[3], (3, 4) = 0, (3, 5) = 0, (4, 1) = c[3], (4, 2) = 0, (4, 3) = 0, (4, 4) = eta-u[4], (4, 5) = 0, (5, 1) = c[4], (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = eta-u[5]})

 (4)
> 

# Step 2: look to thexhape of the minots of W wrt to its rightmost column

for i from 1 to n+1 do
  Minor(X, i, n+1) * X[i, n+1] * (-1)^(i-1)
end do;

c[4]^2*(eta-u[2])*(eta-u[3])*(eta-u[4])

  

0

  

0

  

0

  

(eta^4-eta^3*u[1]-eta^3*u[2]-eta^3*u[3]-eta^3*u[4]+eta^2*c[1]^2+eta^2*c[2]^2+eta^2*c[3]^2+eta^2*u[1]*u[2]+eta^2*u[1]*u[3]+eta^2*u[1]*u[4]+eta^2*u[2]*u[3]+eta^2*u[2]*u[4]+eta^2*u[3]*u[4]-eta*c[1]^2*u[3]-eta*c[1]^2*u[4]-eta*c[2]^2*u[2]-eta*c[2]^2*u[4]-eta*c[3]^2*u[2]-eta*c[3]^2*u[3]-eta*u[1]*u[2]*u[3]-eta*u[1]*u[2]*u[4]-eta*u[1]*u[3]*u[4]-eta*u[2]*u[3]*u[4]+c[1]^2*u[3]*u[4]+c[2]^2*u[2]*u[4]+c[3]^2*u[2]*u[3]+u[1]*u[2]*u[3]*u[4])*(eta-u[5])

 (5)
> 

# Step 3: then the determinant of X writes

det := add(Minor(X, i, n+1) * X[i, n+1] * (-1)^(i-1), i=1..n+1);
 

c[4]^2*(eta-u[2])*(eta-u[3])*(eta-u[4])+(eta^4-eta^3*u[1]-eta^3*u[2]-eta^3*u[3]-eta^3*u[4]+eta^2*c[1]^2+eta^2*c[2]^2+eta^2*c[3]^2+eta^2*u[1]*u[2]+eta^2*u[1]*u[3]+eta^2*u[1]*u[4]+eta^2*u[2]*u[3]+eta^2*u[2]*u[4]+eta^2*u[3]*u[4]-eta*c[1]^2*u[3]-eta*c[1]^2*u[4]-eta*c[2]^2*u[2]-eta*c[2]^2*u[4]-eta*c[3]^2*u[2]-eta*c[3]^2*u[3]-eta*u[1]*u[2]*u[3]-eta*u[1]*u[2]*u[4]-eta*u[1]*u[3]*u[4]-eta*u[2]*u[3]*u[4]+c[1]^2*u[3]*u[4]+c[2]^2*u[2]*u[4]+c[3]^2*u[2]*u[3]+u[1]*u[2]*u[3]*u[4])*(eta-u[5])

 (6)
> 

Student:-Precalculus:-CompleteSquare(c[4]^2*(eta-u[2])*(eta-u[3])*(eta-u[4])+(eta^4-eta^3*u[1]-eta^3*u[2]-eta^3*u[3]-eta^3*u[4]+eta^2*c[1]^2+eta^2*c[2]^2+eta^2*c[3]^2+eta^2*u[1]*u[2]+eta^2*u[1]*u[3]+eta^2*u[1]*u[4]+eta^2*u[2]*u[3]+eta^2*u[2]*u[4]+eta^2*u[3]*u[4]-eta*c[1]^2*u[3]-eta*c[1]^2*u[4]-eta*c[2]^2*u[2]-eta*c[2]^2*u[4]-eta*c[3]^2*u[2]-eta*c[3]^2*u[3]-eta*u[1]*u[2]*u[3]-eta*u[1]*u[2]*u[4]-eta*u[1]*u[3]*u[4]-eta*u[2]*u[3]*u[4]+c[1]^2*u[3]*u[4]+c[2]^2*u[2]*u[4]+c[3]^2*u[2]*u[3]+u[1]*u[2]*u[3]*u[4])*(eta-u[5]), c[2])

(eta^2-eta*u[2]-eta*u[4]+u[2]*u[4])*(eta-u[5])*c[2]^2+c[4]^2*(eta-u[2])*(eta-u[3])*(eta-u[4])+(eta^4-eta^3*u[1]-eta^3*u[2]-eta^3*u[3]-eta^3*u[4]+eta^2*c[1]^2+eta^2*c[3]^2+eta^2*u[1]*u[2]+eta^2*u[1]*u[3]+eta^2*u[1]*u[4]+eta^2*u[2]*u[3]+eta^2*u[2]*u[4]+eta^2*u[3]*u[4]-eta*c[1]^2*u[3]-eta*c[1]^2*u[4]-eta*c[3]^2*u[2]-eta*c[3]^2*u[3]-eta*u[1]*u[2]*u[3]-eta*u[1]*u[2]*u[4]-eta*u[1]*u[3]*u[4]-eta*u[2]*u[3]*u[4]+c[1]^2*u[3]*u[4]+c[3]^2*u[2]*u[3]+u[1]*u[2]*u[3]*u[4])*(eta-u[5])

 (7)
> 

# Step 4: simplify this expression bt setting Z[i] = eta-u[i]

expand( eval(det, [seq(eta-u[i]=Z[i], i=1..n+1)]) / mul(Z[i], i=1..n+1) );

eta^2*u[1]*u[2]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[1]^2*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])+c[2]^2*u[2]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*u[1]*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*u[3]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[3]^2*u[2]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*u[2]*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*u[2]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])+c[1]^2*u[3]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[1]^2*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[3]^2*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[2]^2*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*c[2]^2*u[2]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*u[1]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])+c[3]^2*u[2]*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])+u[1]*u[2]*u[3]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*u[1]*u[2]*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*u[1]*u[2]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*u[1]*u[3]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta*u[2]*u[3]*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])-eta^3*u[1]/(Z[1]*Z[2]*Z[3]*Z[4])-eta^3*u[2]/(Z[1]*Z[2]*Z[3]*Z[4])-eta^3*u[3]/(Z[1]*Z[2]*Z[3]*Z[4])-eta^3*u[4]/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*c[1]^2/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*c[2]^2/(Z[1]*Z[2]*Z[3]*Z[4])+eta^2*c[3]^2/(Z[1]*Z[2]*Z[3]*Z[4])+eta^4/(Z[1]*Z[2]*Z[3]*Z[4])+c[4]^2/(Z[1]*Z[5])

 (8)
> 

# Step 5: Thus this representation of the characteristic polynomial as a rational fraction

det := (-1)^'n' * Product(Z[k], k=1..'n'+1) * (1 + Sum(c[k]^2/Z[k]/Z[k+1], k=1..'n'));

print():
k := 'k':
reminder := Z[k] = eval(eta-u[k], u[k] = 'M[k, k]');

(-1)^n*(Product(Z[lambda+m__0], lambda+m__0 = 1 .. n+1))*(1+Sum(c[lambda+m__0]^2/(Z[lambda+m__0]*Z[lambda+m__0+1]), lambda+m__0 = 1 .. n))

  

 

Z[k] = eta-M[k, k]

 (9)


A direct use of  CharacteristicPolynomial

> 

# With CharacteristicPolynomial:

CP := CharacteristicPolynomial(eval(X, eta=0), eta);

eta^5-(-u[5]-u[4]-u[3]-u[2]-u[1])*eta^4-(-c[1]^2-c[2]^2-c[3]^2-c[4]^2-u[1]*u[2]-u[1]*u[3]-u[1]*u[4]-u[1]*u[5]-u[2]*u[3]-u[2]*u[4]-u[2]*u[5]-u[3]*u[4]-u[3]*u[5]-u[4]*u[5])*eta^3-(-c[1]^2*u[3]-c[1]^2*u[4]-c[1]^2*u[5]-c[2]^2*u[2]-c[2]^2*u[4]-c[2]^2*u[5]-c[3]^2*u[2]-c[3]^2*u[3]-c[3]^2*u[5]-c[4]^2*u[2]-c[4]^2*u[3]-c[4]^2*u[4]-u[1]*u[2]*u[3]-u[1]*u[2]*u[4]-u[1]*u[2]*u[5]-u[1]*u[3]*u[4]-u[1]*u[3]*u[5]-u[1]*u[4]*u[5]-u[2]*u[3]*u[4]-u[2]*u[3]*u[5]-u[2]*u[4]*u[5]-u[3]*u[4]*u[5])*eta^2-(-c[1]^2*u[3]*u[4]-c[1]^2*u[3]*u[5]-c[1]^2*u[4]*u[5]-c[2]^2*u[2]*u[4]-c[2]^2*u[2]*u[5]-c[2]^2*u[4]*u[5]-c[3]^2*u[2]*u[3]-c[3]^2*u[2]*u[5]-c[3]^2*u[3]*u[5]-c[4]^2*u[2]*u[3]-c[4]^2*u[2]*u[4]-c[4]^2*u[3]*u[4]-u[1]*u[2]*u[3]*u[4]-u[1]*u[2]*u[3]*u[5]-u[1]*u[2]*u[4]*u[5]-u[1]*u[3]*u[4]*u[5]-u[2]*u[3]*u[4]*u[5])*eta+c[1]^2*u[3]*u[4]*u[5]+c[2]^2*u[2]*u[4]*u[5]+c[3]^2*u[2]*u[3]*u[5]+c[4]^2*u[2]*u[3]*u[4]+u[1]*u[2]*u[3]*u[4]*u[5]

 (10)
> 

# A more synthetic form

with(LargeExpressions):

Synthetic_form := collect(CP, eta, Veil[T]);
 

eta^5+eta^4*T[1]+eta^3*T[2]+eta^2*T[3]+eta*T[4]+T[5]

 (11)
> 

# Where the coefficients are:

Coefficients_are := [ seq(T[i] = Unveil[T](T[i]), i=1..LastUsed[T]) ]:
print~(Coefficients_are):

T[1] = u[5]+u[4]+u[3]+u[2]+u[1]

  

T[2] = c[1]^2+c[2]^2+c[3]^2+c[4]^2+u[1]*u[2]+u[1]*u[3]+u[1]*u[4]+u[1]*u[5]+u[2]*u[3]+u[2]*u[4]+u[2]*u[5]+u[3]*u[4]+u[3]*u[5]+u[4]*u[5]

  

T[3] = c[1]^2*u[3]+c[1]^2*u[4]+c[1]^2*u[5]+c[2]^2*u[2]+c[2]^2*u[4]+c[2]^2*u[5]+c[3]^2*u[2]+c[3]^2*u[3]+c[3]^2*u[5]+c[4]^2*u[2]+c[4]^2*u[3]+c[4]^2*u[4]+u[1]*u[2]*u[3]+u[1]*u[2]*u[4]+u[1]*u[2]*u[5]+u[1]*u[3]*u[4]+u[1]*u[3]*u[5]+u[1]*u[4]*u[5]+u[2]*u[3]*u[4]+u[2]*u[3]*u[5]+u[2]*u[4]*u[5]+u[3]*u[4]*u[5]

  

T[4] = c[1]^2*u[3]*u[4]+c[1]^2*u[3]*u[5]+c[1]^2*u[4]*u[5]+c[2]^2*u[2]*u[4]+c[2]^2*u[2]*u[5]+c[2]^2*u[4]*u[5]+c[3]^2*u[2]*u[3]+c[3]^2*u[2]*u[5]+c[3]^2*u[3]*u[5]+c[4]^2*u[2]*u[3]+c[4]^2*u[2]*u[4]+c[4]^2*u[3]*u[4]+u[1]*u[2]*u[3]*u[4]+u[1]*u[2]*u[3]*u[5]+u[1]*u[2]*u[4]*u[5]+u[1]*u[3]*u[4]*u[5]+u[2]*u[3]*u[4]*u[5]

  

T[5] = c[1]^2*u[3]*u[4]*u[5]+c[2]^2*u[2]*u[4]*u[5]+c[3]^2*u[2]*u[3]*u[5]+c[4]^2*u[2]*u[3]*u[4]+u[1]*u[2]*u[3]*u[4]*u[5]

 (12)

 

> 

# In these expressions of T[i] we recognize:

T[1] = Determinant(eval(X, eta=0));
T[2] = -Trace(eval(X, eta=0));;
T[3] = - expand( 1/2*(Trace(eval(X, eta=0)^2) - Trace(eval(X, eta=0))^2) )

T[1] = -c[1]^2*u[3]*u[4]*u[5]-c[2]^2*u[2]*u[4]*u[5]-c[3]^2*u[2]*u[3]*u[5]-c[4]^2*u[2]*u[3]*u[4]-u[1]*u[2]*u[3]*u[4]*u[5]

  

T[2] = u[5]+u[4]+u[3]+u[2]+u[1]

  

T[3] = c[1]^2+c[2]^2+c[3]^2+c[4]^2+u[1]*u[2]+u[1]*u[3]+u[1]*u[4]+u[1]*u[5]+u[2]*u[3]+u[2]*u[4]+u[2]*u[5]+u[3]*u[4]+u[3]*u[5]+u[4]*u[5]

 (13)
> 

# and so on, those are classical formulas.
> 

NULL

Download CP_3.mw

@sand15 zero on the 2,2 is a vector...

Ahmed111 130
November 21 2023
0 0

@sand15 zero on the 2,2 is a vector i.e., Matrix([-Transpose(c), 0]). Also, see M_0 in the attached pic 

diag(-2*lambda,a)...

Ahmed111 130
November 20 2023
0 0

@sand15 Many thanks. It would be helpful for me.  But I have a question (see attached) about diag(-2*lambda,a). And M12 is a (1x3) row vector for n=3. 

> 

restart
> 

with(LinearAlgebra)
> 

NULL
> 

diag(-2*lambda, a1) := Matrix([[-2*lambda, 0], [0, a1]])

Matrix(%id = 36893490769549446068)

 (1)
> 

NULL
> 

diag(-2*lambda, a1) := Matrix([[-2*lambda*a1, 0], [0, 0]])

Matrix(%id = 36893490769553617132)

 (2)

NULL

Download CP1.mw

@acer I double checked it. I didn&#...

Ahmed111 130
February 04 2023
0 0

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

another determinant...

Ahmed111 130
February 04 2023
0 0

@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(`&alpha;__1`::real); assume(`&alpha;__2`::real); assume(nu::real)
> 

A2s := Matrix([[H__11*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__13*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`))+1, H__12*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__14*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__11*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__13*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__12*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__14*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`))], [H__12*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__14*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), 1+H__11*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__13*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__12*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__14*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__11*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__13*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`))], [H__13*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__33*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__14*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__34*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), 1+H__13*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__33*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__14*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__34*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`))], [H__14*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__34*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__13*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__33*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__14*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__34*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__13*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__33*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`))+1]])
> 

vvalue := {v__11 = (conjugate(`&lambda;__1`)-`&lambda;__1`)*x+(4*`&alpha;__1`*(conjugate(`&lambda;__1`)^3-`&lambda;__1`^3)+2*`&alpha;__2`*(conjugate(`&lambda;__1`)^2-`&lambda;__1`^2)-8*nu*(conjugate(`&lambda;__1`)^4-`&lambda;__1`^4))*t, v__12 = (conjugate(`&lambda;__1`)-`&lambda;__2`)*x+(4*`&alpha;__1`*(conjugate(`&lambda;__1`)^3-`&lambda;__2`^3)+2*`&alpha;__2`*(conjugate(`&lambda;__1`)^2-`&lambda;__2`^2)-8*nu*(conjugate(`&lambda;__1`)^4-`&lambda;__2`^4))*t, v__21 = (conjugate(`&lambda;__2`)-`&lambda;__1`)*x+(4*`&alpha;__1`*(conjugate(`&lambda;__2`)^3-`&lambda;__1`^3)+2*`&alpha;__2`*(conjugate(`&lambda;__2`)^2-`&lambda;__1`^2)-8*nu*(conjugate(`&lambda;__2`)^4-`&lambda;__1`^4))*t, v__22 = (conjugate(`&lambda;__2`)-`&lambda;__2`)*x+(4*`&alpha;__1`*(conjugate(`&lambda;__2`)^3-`&lambda;__2`^3)+2*`&alpha;__2`*(conjugate(`&lambda;__2`)^2-`&lambda;__2`^2)-8*nu*(conjugate(`&lambda;__2`)^4-`&lambda;__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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 (2)
> 

A2snum := Matrix([[H__11*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__13*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`))+1, H__12*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__14*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__11*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__13*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__12*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__14*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__12*conjugate(psi1)+H__14*conjugate(psi2)], [H__12*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__14*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), 1+H__11*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__13*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__12*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__14*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__11*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__13*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__11*conjugate(psi1)+H__13*conjugate(psi2)], [H__13*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__33*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__14*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__34*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), 1+H__13*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__33*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__14*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__34*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__14*conjugate(psi1)+H__34*conjugate(psi2)], [H__14*exp(I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__34*exp(I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__13*exp(-I*v__11)/(`&lambda;__1`-conjugate(`&lambda;__1`))+H__33*exp(-I*v__21)/(`&lambda;__1`-conjugate(`&lambda;__2`)), H__14*exp(I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__34*exp(I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`)), H__13*exp(-I*v__12)/(`&lambda;__2`-conjugate(`&lambda;__1`))+H__33*exp(-I*v__22)/(`&lambda;__2`-conjugate(`&lambda;__2`))+1, H__13*conjugate(psi1)+H__33*conjugate(psi2)], [0, phi1, 0, phi2, 0]])

Matrix(%id = 36893490104711191356)

 (3)
> 

valphipsi12 := {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)}

{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 de...

Ahmed111 130
January 16 2023
0 0

@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.

tried it...

Ahmed111 130
January 16 2023
0 0

@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("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))` end do

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

  

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

  

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

  

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

  

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

  

`#mrow(msup(mi("&lambda;",fontstyle = "normal"),mi("m")),mo("&Assign;"),mi("coeff"),mfenced(mrow(mi("lhs"),mfenced(mi("z1")),mo("&comma;"),mi("lambda",fontstyle = "normal"),mo("&comma;"),mi("m"))),mo("&equals;"),mn("0"),mo("&semi;"))`

 (3)
> 

NULL

Download compare.mw

@mmcdara when we change "y1:=-...

Ahmed111 130
November 25 2022
0 0

@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 n...

Ahmed111 130
November 16 2022
0 0

@ecterrab I removed it. But still not working. 

@tomleslie Can we plot (4)? I tried...

Ahmed111 130
October 06 2022
0 0

@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 L...

Ahmed111 130
September 25 2022
0 0

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

Error, (in collect) cannot collect 0

  
> 

 

``

Download Lineaize.mw

@Rouben Rostamian  I got it. But wh...

Ahmed111 130
September 25 2022
0 0

@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 ...

Ahmed111 130
September 24 2022
0 0

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

@Carl Love After doing mtaylor...

Ahmed111 130
September 23 2022
0 0

@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, ...

Ahmed111 130
September 23 2022
0 0

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

Calculation.mw  

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