Equilibrium states of a variational formulation for the Ginzburg-Landau equation

Periodic boundary value problem for one of the versions of the complex Ginzburg- Landau equation, which is commonly called the variational Ginzburg-Landau equation are studied. Questions of existence and stability in the sense of Lyapunov, and also the local bifurcations problem of spatially nonhomogeneous equilibrium states are investigated. Three types of such solutions for the given problem are indicated. The exact formulas of the solutions for the first two types are suggested. Equilibrium states of the second type are expressed through elliptic functions. The third type of equilibrium states appears as a result of bifurcations of automodel equilibrium states, i.e., solutions of the first type in the case when the stability changes. It is shown that equilibrium states of the second and third types are unstable.


Introduction
The cubic complex equation can be written in the form u t = u − (1 + ic)u|u| 2 + (d 1 + id 2 )u xx , (1.1) where u = u 1 (t, x) + iu 2 (t, x), c, d 1 , d 2 ∈ R, d 1 > 0 is commonly called a quantic time-dependent Ginzburg-Landau equation (QTDGL) [1][2][3]. More general forms of this equation and also particular variants of this equation are considered. For example, in applications to nonlinear optics and to some model problems of oscillations theory one considers its weakly dissipative form when d 1 = 0 [3][4].
In [1] the case when equation (1.1) has the form (1.1) is singled out in a separate case For equation (1.2) in [1] it was suggested a special name -variational QTDGL (see [5]). In the given work it will be considered the following boundary value problem with c = 0 u(t, x + 2π) = u(t, x), (1.4) which was obtained in studying of phenomena in condensed media (Ψ 4 -Ginzburg-Landau's model). The questions of existence and stability of nontrivial equilibrium states will be further considered. For such solutions it will be obtained explicit or asymptotic formulas. We will i.e., belongs to the space of periodic functions with period 2π, which have distributional derivatives. They and this function belong to L 2 (−π, π) (see [6][7]), i.e., π −π |f (k) (x)| 2 dx < ∞, k = 0, 1, 2.

Automodel equilibrium states
The boundary value problem (1.3), (1.4) has the solutions of the form where a natural number n 0 will be defined later. Together with solutions (2.1) there are exist the solutions u n (x + h) = η n exp(inh) exp(inx), h ∈ R. Therefore, considering the solutions (2.1) we suppose that η n ∈ R + (η n > 0), and a positive constant η n should be found as the corresponding root of the equation η − η 3 − dηn 3 = 0. Except a zero root the last equation has a root η n = (1 − dn 2 ) 1/2 if 1 − dn 2 > 0. Therefore, the given equilibrium states S n exist, if We remind that as [a] we denote an integer part of the number a.
To analyze the stability of S n (u n (x) = η n exp(inx)) we set u n (x) = η n exp(inx)(1 + w(t, x)). Then, for w(t, x) = w 1 (t, x) + iw 2 (t, x) we obtain the boundary value problem in the form The boundary value problem (2.2), (2.3) for a complex-value function w(t, x) = w 1 (t, x) + iw 2 (t, x) may be rewritten in a real form for a vector-function w(t, x) = colon(w 1 , w 2 )
A linearized form of the boundary value problem (2.4), (2.5) supposes the analysis of the equation w t = A n w with the boundary conditions (2.5). We should found the eigenfunctions of a linear differential operator A n in the following form for each n (|n| ≤ n 0 ). Standard constructions allow reducing the question about finding the eigenvalues A n to the analysis of the characteristic equation Lemma 1. Let n = ±1, . . . , ±n 0 and d < d n = 2/(6n 2 − 1). Then for the given d, the eigenvalues A n , except λ 0 = 0, are located in the half-lane Reλ ≤ −γ 0 (n) < 0.
The proof is reduced to the verification of the inequalities Q n (m) > 0. The analysis of characteristic equation shows that eigenvalues λ n (m) ∈ R and lim m→∞ λ n (m) = −∞. Theorem 1. The boundary value problem (1.3), (1.4) has 2n 0 + 1 the one-dimensional invariant manifolds S n (α) which are formed by solutions u n (x + α), α ∈ R, n = 0, ±1, . . . , n 0 . Let For n = 0 the manifold S 0 (α) is a local attractor for all d > 0. We underline that each of equilibrium states of the family S n (α), which are existing with d ∈ (0; 1/n 2 ) are stable with d ∈ (0; d n ) and unstable with d ∈ (d n ; 1/n 2 ), d n < 1/n 2 for all considering n (n = 0).

Equilibrium states of the second kind
Let u(t, x) = u 1 (t, x) + iu 2 (t, x) and u 2 (t, x) = 0. Then for the function v(t, x) = u 1 (t, x) we derived the following boundary value problem The boundary value problem (3.1),(3.2) has a zero equilibrium state, but may have the nonzero equilibrium states. They should be found as nontrivial solutions v(x) of the boundary value problem dv The ordinary differential equation (3.3) has the solutions, which can be express through the elliptic functions v(x) = psn(δx, k), (3.5) where k ∈ (0; 1) is a real parameter (a module of an elliptic sine). We can assume that p, δ > 0. The function (3.5) satisfies the equation (3.3), if dδ 2 (1 + k 2 ) = 1, and 2dδ 2 k 2 = p 2 , i.e., p = √ 2(k/ √ 1 + k 2 ), δ = (d(1 + k 2 )) −1/2 . We should further choose k ∈ (0; 1) so that the function (3.5) has a period 2π. It has a period 2π/m, where m = 1, 2, . . . , if It is clear that G(0) = 1, G (k) > 0 for all k ∈ (0; 1), lim for this problem it should be considered the question of stability of a zero equilibrium state. For that, we consider a linear variant of the boundary value problem (3.7), (3.8). We write the boundary value problem in a real form. For this, we set w = colon(w 1 , w 2 ). And as a result, we obtain a linear differential equation in R 2 where w(t, x) satisfies the periodic boundary conditions, and B m w = The given linear operator B m has the eigenvalue λ 0m = 1 − 1 1 + k 2 m > 0, corresponding to the eigenvector function colon(0, cn(δ m x, k m )), where cn(y, k) is an elliptic cosine, and the periodic boundary value problem for the partial differential equation (3.9) has the exponentially growing solution w 1 = 0, w 2 (t, x) = cn(δ m x, k m ) exp(λ 0m t). The main boundary value problem (1.3), (1.4) with the solution v m (x) has a two-parametric family of analogous solutions (3.10) Thus, it was proved the statement.

Bifurcations of the automodel solutions
Consider the local bifurcations problem for the solutions u n (x) = η n exp(inx), η n = (1 − dn 2 ) 1/2 in the case when the stability changes. Here, n = ±1, ±2, . . . , and the case n = 0 we do not consider since the corresponding solution u 0 (x) = 1 does not change the stability. We remind that the critical value is d = d n = 2/(6n 2 − 1).
As the phase space of the solutions of the boundary value problem (4.4), (4.5) we assume H 2 2,2 , i.e., the space of the two-dimensional periodic vector-functions (f 1 (x), f 2 (x)) with a period 2π , where f j (x) ∈ W 2 2 [−π; π] , i.e., belongs to the Sobolev space. From the results of section 1, it follows that a linear differential operator A n has a triple zero eigenvalue with the corresponding eigenfunctions Consider the subspace H * ⊂ H 2 2,2 consisting the vector-functions f (x) ∈ H 2 2,2 which have as a first component an even function, and as the second one an odd function. This subspace is invariantly for the solutions of the boundary value problem (4.4), (4.5). In H * the linear operator A n has a simple zero eigenvalue with the corresponding eigenfunction E 1 (x) We consider in this space the question about local bifurcations in a neighborhood of a zero equilibrium state. From the theorem of existence and properties of the central invariant manifold M 1 (ε) (in the given case dimM 1 (ε) = 1) it follows that the question about local bifurcations may be reduced to the analysis of an ordinary differential equation (normal form) z = εΨ n (z) + o(ε), (4.6) where a form of the function Ψ n (z) will be defined below, z = z(t). Briefly remind the algorithm allowing to form the right part of the equation (4.6). We will find the solutions belonging to M 1 (ε) in the following form (see [8][9]) w(x, z, ε) = ε 1/2 Q 1 (x, z) + εQ 2 (x, z) + ε 3/2 Q 3 (x, z) + o(ε 3/2 ), (4.7) where z = z(t) are solutions of (4.6), Q 1 (x, z) = zE 1 (x) vector-functions Q 2 (x, z), Q 3 (x, z) as the functions of x for all z belong to H * . These equalities satisfy −π π (Q 1k Q 11 + Q 2k Q 21 )dx = 0, k = 2, 3, Q k = Q k (x, z) = colon(Q 1k (x, z), Q 2k (x, z)).