Dear Erik, I thank you very much for your assistance. Indeed, your help gave me some solution. But now I can not understand something again.

So, I need to solve the following system of two equations

Re(ikV+α(U))=0,        iV+d α(U)/dU = 0,                                                                             (1)

with  V the real variable, U the imaginary variable, and the known functions

α(U)=U^2(a-(1-U^2)^2),                   d α(U)/dU=2U(a-(1-U^2)^2+2U^2(1-U^2)),            (2)

where a is the constant.

For the system of equations (1)-(2), I write the following (very short) program on MAPLE:

> restart;

#                                    WILLIES PROGRAM CODE No. 1

> Digits:=6;

# VARIABLES:        U the imagimary variable, V the real variable

Initial data :                  a the constant

> a:=0.5;

> ####################################################

> # BEGINNING OF COMPUTATIONS                        #

> #      COMPUTATIONS FOR alpha(U)                   #

> alph:= (U) -> U^2*(a-(1-U^2)^2);

> # Derivative of alph(U) with respect to U #

> dadU:= (U) -> 2*U*(a-(1-U^2)^2+2*U^2*(1.0-U^2));

> solve({ Re(I*U*V+alph(U))=0, -V+I*dadU(U)=0 })  assuming V :: real;                                        

> solve({ Re(I*U*V+alph(U))=0, I*V+dadU(U)=0 })  assuming V :: real;

Could you please answer the following questions:

1. pre-last and last operators (of the above program code) describe, respectively,  absolutely the same second equation from the system (1) written in a different manner, i.e.:

-V+I*dadU(U)=0 gives -V+id α(U)/dU = 0 ,               (3)

and

I*V+dadU(U)=0 gives iV+d α(U)/dU = 0.                     (4)

Obviously, from the algebraic view point, Eqs. (3) and (4) are equivalent to the second equation from the system (1). But not for the MAPLE: I have got different roots in solution of this system of equations.

2. We define above V as the real variable but the obtained solution for V looks like imaginary ...
3. After finding values for V and U, I need to calculate the following value Kf=Im(iUV+ α(U))/V using the ontained values for V and U. How, one could do this using MAPLE?

Thanking in advance.

Best regards,

Willie