# 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]

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There's a bigger problem at the moment NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.. You should review your equations. – Sektor Jun 18 '14 at 10:47
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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]


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at first, the initial condition used were x[1] == -3, x'[1] == 0, y[0] == 1, y'[0] == 3 then why at the shooting command, we must change t value to 0 for both x and x'? "Shooting", "StartingInitialConditions" -> {x[0] == -3, x'[0] == 0, y[0] == 1, y'[0] == 3} – ameera Jun 23 '14 at 4:31
@ameera "StartingInitialConditions" is just a guess, here I just choose it casually. For your equations there're many available "StartingInitialConditions", for example {x[0] == -1, x'[0] == -1, y[0] == 0, y'[0] == 0}, but it's also worth to mention that it can be really hard to choose proper initial conditions for some equations, for example this one. – xzczd Jun 25 '14 at 3:03