# Coupled Nonlinear Differential Equations Problem

I have a big problem solving a set of coupled nonlinear differential equations using NDSolve.

Solving the equations by themselves works quite well but if I want to couple them I get

"NDSolve::conarg: The arguments should be ordered consistently. >>"

but I can not see my mistake.

Maybe you guys can help me with this.

The first equation is:

D[Int[z, t], z] == -(c Int[z, t]^k), Int[0, t] == a (Exp[-4 Log[2] (t/f)^2])


This is easy to solve using Parametric NDsolve.

The second equation is (not coupled):

D[nn[t], t] == b Int2[t]^k, nn[t <= tmin1] == 0


with

Int2[t]= a (Exp[-4 Log[2] (t/f)^2])


This is also solved using ParametricNDSolve with Method -> {StiffnessSwitching, Method -> {ExplicitRungeKutta, Automatic}}

The coupling is simply givien by the fact that Int2[t] should now be Int[z,t] thus my code should be:

ParametricNDSolve[
{
D[Int[z, t], z] == -(c Int[z, t]^k)
, Int[0, t] == a (Exp[-4 Log[2] (t/f)^2])
, D[nn[z, t], t] == σ Int[z, t]^k
, nn[z, t /; t <= tmin1] == 0
}
, {Int, nn}
, {z, 0, 100 10^-9}
, {t, tmin1, tmax1}
, {a}
, MaxSteps -> 100000
, MaxStepSize -> Automatic
, StartingStepSize -> Automatic
, Method -> {StiffnessSwitching,
Method -> {ExplicitRungeKutta, Automatic}}
, AccuracyGoal -> 2
, PrecisionGoal -> 2
, EvaluationMonitor -> Automatic
];


b,c,f,k are just constants, a is a parameter of interest.

I would be glad if you could help me with this (maybe not so big) problem.

Thank you so much in advance.

Best greetings

nn[z, t /; t <= tmin1] == 0

nn[z, tmin1] == 0