I have the following two first order non-linear DEs that I denote as system
system = {A1'[x] == I*α11/2*A1[x]*Abs[A1[x]]^2 + I*α11*A1[x]*Abs[A2[x]]^2, A2'[x] == I*α22*A2[x]*Abs[A1[x]]^2 + I*α22/2*A2[x]*Abs[A2[x]]^2, A1[0] == Sqrt[P10], A2[0] == Sqrt[P20]};
where P10, P20
are the initial conditions (real and independent of x
) and α11, α22
are real parameters that are independent of x
. I proceed to solve them with
DSolve[system, {A1, A2}, {x}]
but I'm returned with
(*DSolve[{Derivative[1][A1][x] == 1/2 I α11 A1[x] Abs[A1[x]]^2 + I α11 A1[x] Abs[A2[x]]^2, Derivative[1][A2][x] == I α22 A2[x] Abs[A1[x]]^2 + 1/2 I α22 A2[x] Abs[A2[x]]^2, A1[0] == Sqrt[P10], A2[0] == Sqrt[P20]}, {A1, A2}, {x}]*)
which is pretty much the same thing as my system
. In other words, Mathematica isn't solving it at all and is regurgitating system
, yet it doesn't return as an error or warning. Furthermore I'm surprise that DSolve is unable to solve relatively simple systems of ODEs. I am confident that an analytical solution exists as these corresponds to pump amplitudes equations for travelling waves.
NULL
. $\endgroup$system
numerically withNDSolve
? What are approbriate parameter values (examplary)? $\endgroup$Abs[z]
is a particularly simple function. Here's a separable ODE that doesn't have a closed-form solution:DSolve[z'[t] == I*Abs[z[t]], z, t]
. The returned solution is problematic because the integral is not path-independent. (CompareNIntegrate[1/Abs[z], {z, 1, I}]
andNIntegrate[1/Abs[z], {z, 1, 2 + I, I}]
.) $\endgroup$Abs
, "Abs is a function of a complex variable and is therefore not differentiable" and "In particular, the limit that defines the derivative is direction dependent and therefore does not exist" Consequently, the presence ofAbs
in a differential equation is problematic. $\endgroup$