Stuck on a complex ODE (conjugate issues )

In summary, the conversation revolves around a time-evolution problem in complex space, where the state of the system is described by a type of ODE. There is a concern about integrating the expression to obtain the time evolution of the system for any given starting point. One approach is to treat the equation as a system of ODEs, while another is to use standard methods to solve a simple harmonic oscillator equation.
  • #1
Simplexed
2
0
Hi everyone.

I'm stuck on a problem, banging of head etc. Basically, I've got a time-evolution problem restricted to the positive quadrant of complex space, where the state of the system [tex]\psi \in \mathbb{C}[/tex] is described by the following type of ODE:

[tex]
\frac{{d\psi \left( t \right)}}{{dt}} = i\overline {\psi \left( t \right)}
[/tex]

i.e, the time evolution of the state depends on the conjugate, not the state itself (this is determined by the first principles of my system).

Numerically, I can easily generate the quiver diagrams of this first difference, and then use it to simulate the time evolution of the system. The norm seems to grow roughly exponentially, while the argument exponentially approaches [tex]\pi /4 [/tex] , which is what I expect.

My problem is that I am at a loss of ideas for how to actually integrate the expression, in order to obtain the expression for the time evolution of [tex]\psi \left( 0 \right)[/tex] for any given starting point in the space.

Any assistance on this would be very much appreciated.
 
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  • #2
This is just a pair of coupled DE's. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Your equation can be written

[tex]\frac{d}{dt}(\psi_R +i\psi_I) = -\psi_I + i\psi_R[/tex]

which splits up into

[tex]\frac{d \psi_R}{dt}= -\psi_I[/tex]

and

[tex]\frac{d \psi_I}{dt} = \psi_R[/tex]

which you should hopefully be able to solve, yes?
 
  • #3
:redface:

Well, that figures... I was trying to work it while keeping the complex number whole, but completely overlooked treating it as a system of ODEs (and not a complicated one at that).

Thanks a lot for that wake-up call!
 
  • #4
In this case, this is likely the easier way to solve it: You could also note that if you take the conjugate of your original DE you get

[tex]\frac{d \overline{\psi}}{dt} = -i\psi[/tex]

so taking another derivative of your original equation gives

[tex]\frac{d^2\psi}{dt^2} = i\frac{d \overline{\psi}}{dt} = \psi[/tex]
 
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1. What is a complex ODE?

A complex ODE is a type of ordinary differential equation that involves complex-valued functions and their derivatives. It can be written in the form f(z, z') = 0, where z is a complex variable and z' is its derivative. Complex ODEs are often used to model systems in physics and engineering that involve complex quantities.

2. What are conjugate issues in a complex ODE?

Conjugate issues in a complex ODE refer to cases where the complex conjugate of a solution is also a solution. This can lead to difficulty in finding a unique solution, as there may be multiple solutions that satisfy the equation. It is important to carefully consider conjugate issues when solving complex ODEs.

3. How do you solve a complex ODE with conjugate issues?

There are various methods for solving complex ODEs with conjugate issues. One approach is to convert the complex ODE into a system of two real-valued ODEs, which can then be solved using standard techniques. Another method is to use the method of Frobenius, which involves expanding the solution as a power series and solving for the coefficients.

4. Are there any software packages for solving complex ODEs with conjugate issues?

Yes, there are several software packages available for solving complex ODEs with conjugate issues. Some popular options include MATLAB, Mathematica, and Maple. These packages have built-in functions and algorithms specifically designed for solving complex ODEs, including those with conjugate issues.

5. Can complex ODEs with conjugate issues be applied to real-world problems?

Yes, complex ODEs with conjugate issues have many applications in physics, engineering, and other fields. They can be used to model a wide range of systems, including electrical circuits, fluid dynamics, and quantum mechanics. By properly considering conjugate issues, complex ODEs can provide accurate and insightful solutions to complex real-world problems.

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