These boundary conditions are equivalent to the fact, that this function at infinity tend to zero. But however I changed 500 to 100 (-500 to -100) and get the result!

I do not know if I can change this, but still thank you very much for your help. I am very grateful to you! 

@Carl Love  Thank you very much! You are absolutely right, I mage a mistake, Thank you!
But unfortunately , I still get an error , as You.

> plots[display]([seq(sol:-plot(U, t = 0))]);

Error, (in pdsolve/numeric/plot) unable to compute solution for t<HFloat(0.0):
matrix is singular

What does it mean, can you help me?

Thank you!!! You were absolutly right. I added new nitial conditions and get 

> sol := pdsolve({l1, l2}, IBC, funcs, numeric, time = t, range = -500 .. 500);
print(`output redirected...`); # input placeholder
module () local INFO; export plot, plot3d, animate, value,

I'm new to working with the program, can I use a plot after this message, and what kind of operation I should use for building the plot (not 3d) of ui(e, t=0) and ur(e,t=0) ?

Please help.

Thank you!!! You were absolutly right. I added new nitial conditions and get 

> sol := pdsolve({l1, l2}, IBC, funcs, numeric, time = t, range = -500 .. 500);
print(`output redirected...`); # input placeholder
module () local INFO; export plot, plot3d, animate, value,

I'm new to working with the program, can I use a plot after this message, and what kind of operation I should use for building the plot (not 3d) of ui(e, t=0) and ur(e,t=0) ?

Please help.

it's my program

> restart; with(PDEtools); a := 1;
> m := 1/64;

> l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0;

> l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0;
> sys := {l1, l2};

> IBC := {ui(-500, t) = 0, ui(0, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0};

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs

a and m have numerical values. And as for e and t its are dependent variables with respect to which it is necessary to solve the system. For the Plot (according my task I wil take t=0) And finally the result should contain the Plot of dependence ui from e and ur from e

it's my program

> restart; with(PDEtools); a := 1;
> m := 1/64;

> l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0;

> l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0;
> sys := {l1, l2};

> IBC := {ui(-500, t) = 0, ui(0, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0};

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs

a and m have numerical values. And as for e and t its are dependent variables with respect to which it is necessary to solve the system. For the Plot (according my task I wil take t=0) And finally the result should contain the Plot of dependence ui from e and ur from e

@Preben Alsholm 

The program in my case doesn't require the initial conditions. As I understand, I should set boundary or initial conditions, as initial I have only one.
so

> restart; with(PDEtools); a := 1;
> m := 1/64;

> l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0;

> l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0;
> sys := {l1, l2};

> IBC := {ui(-500, t) = 0, ui(0, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0};

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs
>

@Preben Alsholm 

The program in my case doesn't require the initial conditions. As I understand, I should set boundary or initial conditions, as initial I have only one.
so

> restart; with(PDEtools); a := 1;
> m := 1/64;

> l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0;

> l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0;
> sys := {l1, l2};

> IBC := {ui(-500, t) = 0, ui(0, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0};

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs
>

I change some steps, and finally get

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs

So, as I understand right, Maple can't solve such kind of PDE?

This is my system:
l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0

l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0

sys:={l1,l2}

I change some steps, and finally get

> sol := pdsolve(sys, IBC, [ur(e, t), ui(e, t)], numeric);

Error, (in pdsolve/numeric) unable to handle elliptic PDEs

So, as I understand right, Maple can't solve such kind of PDE?

This is my system:
l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0

l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0

sys:={l1,l2}

Thank you! I fixed it, but now I get another error

IBC := {ui(-500, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0}

funcs:={ui(e, t), ur(e, t)};

sol := pdsolve(sys, IBC, funcs, numeric)

Error, (in pdsolve/numeric/process_PDEs) specified dependent variable(s) {funcs} not present in input PDE


Can you help me, why its not present?
 This is my system:
l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0

l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0

sys:={l1,l2}

I will be very grateful for any help

Thank you! I fixed it, but now I get another error

IBC := {ui(-500, t) = 0, ui(500, t) = 0, ur(-500, t) = 0, ur(500, t) = 0, (D[1](ui))(-500, t) = 0, (D[1](ui))(500, t) = 0, (D[1](ur))(-500, t) = 0, (D[1](ur))(500, t) = 0}

funcs:={ui(e, t), ur(e, t)};

sol := pdsolve(sys, IBC, funcs, numeric)

Error, (in pdsolve/numeric/process_PDEs) specified dependent variable(s) {funcs} not present in input PDE


Can you help me, why its not present?
 This is my system:
l1 := 2*(diff(ur(e, t), t))+(1+tanh((1/20)*e))*(diff(ui(e, t), `$`(e, 2)))+2*a*(ui(e, t)*ur(e, t)*ur(e, t)+(ui(e, t)*ui(e, t))*ui(e, t))+m*ui(e, t)*(diff(ur(e, t)^2, e))+m*ui(e, t)*(diff(ui(e, t)*ui(e, t), e)) = 0

l2 := -2*(diff(ui(e, t), t))+(1+tanh((1/20)*e))*(diff(ur(e, t), `$`(e, 2)))+2*a*(ur(e, t)^3+ur(e, t)*ui(e, t)^2)+m*ur(e, t)*(diff(ur(e, t)^2, e))+m*ur(e, t)*(diff(ui(e, t)^2, e)) = 0

sys:={l1,l2}

I will be very grateful for any help