This problem may be considered as a separatrix crossing in confluent point. The passing through a separatrix of the second order equations in a general position was considered by A.V. Timofeev in 1979, A.I. Neishtadt in 1986. The sepapraprix crossing for the second order equations in the confluent point was considered in a preliminary fashion by R.Haberman in 1979 and D.C.Diminnie and R.Haberman in 2000 in more detail. The asymptotic solution crossing the separatrix in the confluent point was constructed by O.M.Kiselev in 1999,2001 for the Painleve-2 equation.

Using our approach one can construct the uniform asymptotic solution crossing the separatrix in the confluent point for second order equation in general case.

If we suppose that the derivative in the equation (2) is bounded then we obtain a nonlinear algebraic equation for the main term of asymptotics U(t): 

what happened with the asymptotic solution of the equation (2) when two roots of the equation (3) coalesces?

To obtain an answer on the precision question one can consider an autonomous equation with a "frozen" coefficient T:

	iV'+(|V|^2-T)V=1. This equation has three equilibrium positions T > t*. There are U1 < U2 < U3, where U1 is a saddle, U2 and U3 are centers.

 

At T = t* the saddle-node bifurcation takes place and there exist center U3 and confluent saddle-center point U*.

 

	When T < t* there exists along center U3.

 

We will construct the formal asymptotic solution of the equation (1) in the interval t Î [t_*-C,t_*+C] where C = const > 0 uniform on e . We suppose that the solution in the domain t > t_* has the form

The qualitative analysis shows that this asymptotic solution oscillates when t < t_*. Our problem is to study the transition layer between the nonoscillating asymptotics when t > t_* and the oscillating asymptotics when t < t_*.

The numeric evaluations for the special solution of the equation (2) give the picture:

In the domain (t-t*)e-4/5 >> 1   the asymptotics has the form:  y(t,e) = exp([(it2)/(2e)])ån = 0¥enU(t).   Here U(t) = U2(t) and corrections   U(t) are algebraic functions of t.

where t = (t-t*)e-4/5. The leader term a(t) is a special solution of the Painlevé-1 equation :

In the neighborhood of t = t₀ the coefficients of the asymptotic expansion depend on one more fast time scale q = (t-t₀)e^-1/5. Denote by

y(t,e) = (U_*+e^1/3å_{n = 0}^¥e^i/30(A_k+ie^1/6B_k))exp([(it²)/(2e)]). The leader term satisfies to the equation :

A_k = -2Ã(T_k;0,g₃(k)),  g₃(k) = [1/56](g₃(k-1)+p/2).
 

Here g3(0) = [(a4)/56], a4 is the   coefficient as (t-t0)4 in the   Laurent expansion of a(t).   The intermediate expansion   with the leader term   is valid in the domain   between two poles Tk = 0 and   Tk = Wk of theÃ-function:    -e-1/6Tk >> 1,  e-2/15(Tk+Wk) >> 1.   At the large values of k the intermediate asymptotics are constructed in the form

y(t,e) = (U_*+e^1/3å_{n = 0}^¥e^n/6(A_k + ie^1/6B_k))exp([(it²)/(2e)]).

g₃(k,e) = g₃(k)+å_{n = 1}^¥e^n/30g₃(k).
The intermediate expansion with the leader term is valid in the domain between the poles of the Weierstrass function as

The separatrix expansions are valid in a small neighborhood of the Weierstrass function poles. Denote:

q_k = (T_k+W_k-¹/₄å_{n = 1}^¥e^n/30x_k⁺)e^-1/6,    k = 1,2,.... When |q_k|e^1/6 << 1 the formal asymptotic solution of equation (1) has the form :

The leader term of the  asymptotics W(qk) is a  separatrix solution of the  autonomous equation (5):   W(qk) =  (-2)/((qk-iU*)2)].  The sequence of the   alternating intermediate  expansions and separatrix   asymptotics is valid as e-1/6(t*-t) << 1.

In the domain (t_*-t)e^-2/3 >> 1 the asymptotic solution becomes two-phase. The amplitude of the stimulated oscillations in the solution of (1) oscillates fast. The form of the solution is:

y = (U(t₁,t,e) + eU(t₁,t,e) + e² U(t₁,t,e))exp([(it²)/(e)]),

where t1 is a new fast variable t1 = S(t)/e+f(t). The   main term of the symptotics   U lies on the curve G(t):    1/2|y|4 -t |y|2 - (y + [`y]) = E(t),   and satisfies to the Cauchy   problem for the equation

 

with an initial condition U|_{t1 = 0} = u₀, such, that Á(u₀) = 0,    Â(u₀) = min_{y Î G(t)}(Â(y)). The function S(t) is a solution for the Cauchy problem

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Kiselev O.M., Hard loss of stability in Painlevé-2 equation. J. of Nonlin. Math. Phys., 2001, v.8, n1, pp.65-95.

Kiselev O.M., Glebov S.G., Asymptotic description of nonlinear resonance. http://arxiv.org/abs/math/0105011