I have problem with coupled nonlinear reaction diffusion equations, but adding more initial and boundary conditions (as mentioned in this post: here) did not help :
B=10;
c=1;
k=10;
V=1.;
A1=2.;
A0=(B+2*A1^2)/(2*A1);
pde={
D[uu[x,t],t]==k*D[uu[x,t],x,x]+uu[x,t]*uu[x,t]*vv[x,t]-B*uu[x,t],
D[vv[x,t],t]==k*D[vv[x,t],x,x]-uu[x,t]*uu[x,t]*vv[x,t]+B*uu[x,t]
};
ics={
uu[x,0]==A1+A1*Tanh[c*x],
vv[x,0]==A0-A1*Tanh[c*x],
Derivative[0,1][uu][x,0]==0,
Derivative[0,1][vv][x,0]==0
};
bcs={
DirichletCondition[uu[x,t]==A1*(1+Tanh[c*(x-V*t)]),x==0],
DirichletCondition[uu[x,t]==A1*(1+Tanh[c*(x-V*t)]),x==10],
DirichletCondition[vv[x,t]==A0-A1*Tanh[c*(x-V*t)],x==0],
DirichletCondition[vv[x,t]==A0-A1*Tanh[c*(x-V*t)],x==10],
(D[uu[x,t],x]/.x->0)==0,
(D[uu[x,t],x]/.x->10)==0,
(D[vv[x,t],x]/.x->0)==0,
(D[vv[x,t],x]/.x->10)==0
};
NDSolveValue[{pde,ics,bcs},{uu,vv},{x,0,10},{t,0,1}];
and I got the same error:
NDSolveValue::femnonlinear: Nonlinear coefficients are not supported in this version of NDSolve.