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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.

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    $\begingroup$ Maple also fails with it, returning NULL. $\endgroup$ – user64494 Sep 14 at 5:32
  • $\begingroup$ Did you check system numerically with NDSolve? What are approbriate parameter values (examplary)? $\endgroup$ – Ulrich Neumann Sep 14 at 10:10
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    $\begingroup$ My perspective on "simple": (1) I'd say the only simple nonlinear ODE is one that is separable, or can be transformed into separable. (2) I wouldn't say 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. (Compare NIntegrate[1/Abs[z], {z, 1, I}] and NIntegrate[1/Abs[z], {z, 1, 2 + I, I}].) $\endgroup$ – Michael E2 Sep 14 at 13:47
  • $\begingroup$ From the "Possible Issues" section of the documentation for 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 of Abs in a differential equation is problematic. $\endgroup$ – Bob Hanlon Sep 14 at 17:28
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    $\begingroup$ The answer to 229656 that I derived for you last week may be of value here. $\endgroup$ – bbgodfrey Sep 14 at 22:53
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Based on my answer to 229656, the solution is

s = {A1 -> Function[{x}, Sqrt[P10] Exp[I α11 x (P10/2 + P20)]], 
     A2 -> Function[{x}, Sqrt[P20] Exp[I α22 x (P20/2 + P10)]]}

as can be demonstrated by

Simplify[system /. s, P10 > 0 && P20 > 0 && (α11 | α22 | x) ∈ Reals]
(* {True, True, True, True} *)
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