Q:

how to solve second order nonlinear coupled differential equations using NDSolve with hyperbolic function

i have to solve some solitons scattering through this coupled equations. i need to get two different graph, but still the graph did not come out. and also the equations quite complicated containing hyperbolic trigo . (maybe just for me). i dont know whether the problems come from the hyperbolic equations that i used, or becoz of initial condition.the coding as below:
u = 0.05;
g = 0.2;
s = NDSolve[{x''[t] == 4/(\[Pi]^2 x[t]^3) - 10/(\[Pi]^2 x[t]^2) - (80 g)/(3 
\[Pi]^2 x[t]^3) - ((6 u)/(\[Pi]^2 x[t]^2))[1/Cosh[y[t]/x[t]]^2 - (2 
y[t])/x[t] Sinh[y[t]/x[t]]/Cosh[y[t]/x[t]]^3], 
y''[t] == u (y[t]/(x[t]^3))[Sinh[y[t]/x[t]]/Cosh[y[t]/x[t]]^3], 
x[1] == -3, x'[1] == 0, y[0] == 1, y'[0] == 3}, {x, y}, {t, 0,100}]
Plot[Evaluate[{x[t], y[t]} /. %], {t, 0, 100}, PlotRange -> All, PlotPoints -> 200]

A:

First, your code contains simple mistake, you should distinguish [] from (), then your equations still can't be solved, it's a common problem for the boundary value problem (BVP) of nonlinear ODE(s), and the almost only solution as far as I know is "shooting method":
u = 5/100; 
g = 2/10; 
s = NDSolve[{x''[t] == 4/(π^2 x[t]^3) - 10/(π^2 x[t]^2) - 
             (80 g)/(3 π^2 x[t]^3) - ((6 u)/(π^2 x[t]^2))(1/Cosh[y[t]/x[t]]^2 - 
             (2 y[t])/x[t] Sinh[y[t]/x[t]]/Cosh[y[t]/x[t]]^3), 
             y''[t] == u (y[t]/(x[t]^3))(Sinh[y[t]/x[t]]/Cosh[y[t]/x[t]]^3), 
             x[1] == -3, x'[1] == 0, y[0] == 1, y'[0] == 3}, {x, y}, {t, 0, 100}, 
    Method -> {"Shooting", "StartingInitialConditions" -> 
                           {x[0] == -3, x'[0] == 0, y[0] == 1, y'[0] == 3}}]

Plot[{x[t], y[t]} /. s, {t, 0, 100}, PlotRange -> All, Evaluated -> True]

