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. $\endgroup$systemnumerically 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 ofAbsin a differential equation is problematic. $\endgroup$