Solution 1 to problem N2f1b0o35w2

Expressions | Parameters | Inequalities | Relevance | Back to problem N2f1b0o35w2

Expressions

The solution is given through the following expressions:


q25=0


q24=0


q23=0


q22=0


        1
q21= - ---*i*q9
        2


q20= - q9


     1
q19=---*i*q9
     2


        1
q18= - ---*q9
        2


q17=i*q9


     1
q16=---*q9
     2


q15= - q9


q14= - i*q9


q13=i*q9


q12= - q9


q11=i*q9


q10= - q9


q8=i*q9


q7=0


q6=0


q5=0


q4=0


q3=0


q2=0


q1=0


p5= - i*p4


p3=0


p2=0


p1=0

Parameters

Apart from the condition that they must not vanish to give a non-trivial solution and a non-singular solution with non-vanishing denominators, the following parameters are free:

 q27, q26, q9, p4

Inequalities

In the following not identically vanishing expressions are shown. Any auxiliary variables g00?? are used to express that at least one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3 means that either p4 or p3 or both are non-vanishing.

 
{p4,

 q9,

 2*g0003*q26 - 2*g0007*q9 + g0008*i*q9 - g0009*q9 + 2*g0010*i*q9 + g0011*q9

  - 2*g0012*q9 + 2*g0013*i*q9 - 2*g0014*q9 + 2*g0015*i*q9 - 2*g0016*q9

  + 2*g0017*q9 + 2*g0018*i*q9 + 2*g0024*p4,

 p5,

 2*g0079*q27 + 2*g0080*q26 - g0085*i*q9 - 2*g0086*q9 + g0087*i*q9 - g0088*q9

  + 2*g0089*i*q9 + g0090*q9 - 2*g0091*q9 - 2*g0092*i*q9 + 2*g0093*i*q9

  - 2*g0094*q9 + 2*g0095*i*q9 - 2*g0096*q9 + 2*g0097*q9 + 2*g0098*i*q9,

 2*g0027*q26 - g0031*i*q9 - 2*g0032*q9 - g0033*q9 + 2*g0034*i*q9 + g0035*q9

  - 2*g0036*q9 - 2*g0037*i*q9 - 2*g0038*q9 + 2*g0039*i*q9 - 2*g0040*q9

  + 2*g0041*q9 + 2*g0042*i*q9 + 2*g0048*p4}

Relevance for the application:

The equation:


f =D D f  *p4 - f  *i*p4
 t  1 2 2x       3x

The symmetry:

f =D f *D f*D D f*q9 + D f *D f*f *i*q9 + D f *D D f*D f*i*q9 - D f *D f*f *q9
 s  2 x  2   1 2        2 x  2   x         2 x  1 2   1          2 x  1   x

    1       2              1       2
 + ---*(D f) *D D f *q9 + ---*(D f) *f  *i*q9 + D f*D D f *D f*i*q9
    2    2     1 2 x       2    2     2x         2   1 2 x  1

 + D f*D D f*D f *i*q9 - D f*D f *f *q9 - D f*D f*f  *q9 + D D f  *q26
    2   1 2   1 x         2   1 x  x       2   1   2x       1 2 4x

    1              2
 - ---*D D f *(D f) *q9 - D D f*D f *D f*q9 - D f *D f*f *i*q9
    2   1 2 x   1          1 2   1 x  1        1 x  1   x

    1       2
 - ---*(D f) *f  *i*q9 + f  *q27
    2    1     2x         5x

And now in machine readable form:

The system:

df(f(1),t)=d(1,d(2,df(f(1),x,2)))*p4 - df(f(1),x,3)*i*p4$

The symmetry:

df(f(1),s)=d(2,df(f(1),x))*d(2,f(1))*d(1,d(2,f(1)))*q9 + d(2,df(f(1),x))*d(2,f(1
))*df(f(1),x)*i*q9 + d(2,df(f(1),x))*d(1,d(2,f(1)))*d(1,f(1))*i*q9 - d(2,df(f(1)
,x))*d(1,f(1))*df(f(1),x)*q9 + 1/2*d(2,f(1))**2*d(1,d(2,df(f(1),x)))*q9 + 1/2*d(
2,f(1))**2*df(f(1),x,2)*i*q9 + d(2,f(1))*d(1,d(2,df(f(1),x)))*d(1,f(1))*i*q9 + d
(2,f(1))*d(1,d(2,f(1)))*d(1,df(f(1),x))*i*q9 - d(2,f(1))*d(1,df(f(1),x))*df(f(1)
,x)*q9 - d(2,f(1))*d(1,f(1))*df(f(1),x,2)*q9 + d(1,d(2,df(f(1),x,4)))*q26 - 1/2*
d(1,d(2,df(f(1),x)))*d(1,f(1))**2*q9 - d(1,d(2,f(1)))*d(1,df(f(1),x))*d(1,f(1))*
q9 - d(1,df(f(1),x))*d(1,f(1))*df(f(1),x)*i*q9 - 1/2*d(1,f(1))**2*df(f(1),x,2)*i
*q9 + df(f(1),x,5)*q27$