*3.2. Chain Dynamics for σ Values of ε Order*

Here, we assume that the condition (47) holds. Then, the characteristic Equation (20) has the set of roots *λm*(*ε*) and *λm*(*ε*) (*m* = 0, ±1, ±2, ...), the real parts of which tend to zero as *ε* → ∞. The representation

$$\begin{split} d\lambda\_{\mathfrak{m}}(\varepsilon) + \left(\frac{\pi}{2} + \varepsilon^2 r\_1\right) \exp(-\lambda\_{\mathfrak{m}}(\varepsilon)) &= d\left(\int\_{-\infty}^{\infty} F(s, \varepsilon) \exp(ims) ds - 1\right) = \\ d(\cos(\varepsilon m) \exp(-\varepsilon^2 m^2 \sigma\_0^2) - 1). \end{split} \tag{58}$$

holds for these roots.

From here, we obtain that the asymptotic equality

$$
\lambda\_m(\varepsilon) = i\frac{\pi}{2} + \varepsilon^2 \lambda\_1 + \dots, \quad \lambda\_1 = \lambda\_{10} - \left(1 + i\frac{\pi}{2}\right)^{-1} d\left(\frac{1}{2} + \sigma\_0^2\right) m^2
$$

holds for each integer *m*.

Each of the above roots corresponds to the solution *vm*(*t*, *x*) of the boundary value problem (44), (45) for which

$$\upsilon\_m(t, \mathbf{x}) = \exp(i\frac{\pi}{2}t + im\mathfrak{x})\upsilon\_m(\mathbf{r}),$$

where *<sup>ν</sup>m*(*τ*) = *<sup>ν</sup><sup>m</sup>* exp((−*ε*2*λ*<sup>1</sup> + *<sup>O</sup>*(*ε*4))*t*). Let us introduce the formal series

$$\begin{split} u(t, \mathbf{x}, \varepsilon) = & \varepsilon \Big[ \exp \left( i \frac{\pi}{2} t \right) \sum\_{m = -\infty}^{\infty} \xi\_m(\tau) \exp(im\mathbf{x}) + \\ & \exp \left( -i \frac{\pi}{2} t \right) \sum\_{m = -\infty}^{\infty} \xi\_m(\tau) \exp(-im\mathbf{x}) \Big] + \\ & \varepsilon^2 u\_2(t, \tau, \mathbf{x}) + \varepsilon^3 u\_3(t, \tau, \mathbf{x}) + \dots \ . \end{split} \tag{59}$$

Here, *τ* = *ε*2*t* is a slow time, *ξm*(*τ*) are the unknown slowly varying amplitudes, and the functions *uj*(*t*, *τ*, *x*) are periodic with respect to *t* and *x*. We note that Formula (59) defines a solution set of the linear boundary value problem (44), (45) in the linear approximation, i.e., for *uj* ≡ 0.

Expression (59) can be significantly simplified. For this purpose, we assume

$$\xi(\pi,\pi) = \sum\_{m=-\infty}^{\infty} \xi\_m(\pi) \exp(im\pi).$$

Then, it follows from (59) that

$$u(t, \mathbf{x}, \varepsilon) = \varepsilon \left[ \xi(\tau, \mathbf{x}) \exp\left(i\frac{\pi}{2}t\right) + \xi(\tau, \mathbf{x}) \exp\left(-i\frac{\pi}{2}t\right) \right] + \\ \begin{aligned} & \varepsilon^2 u\_2(t, \tau, \mathbf{x}) + \varepsilon^3 u\_3(t, \tau, \mathbf{x}) + \dots \\ & \dots \end{aligned} \tag{60}$$

We insert (60) into (15) and equate the coefficients of the same powers of *ε* in the resulting formal identity. At the first step, the identity holds for *ε*1. At the second step, we collect the coefficients at *ε*<sup>2</sup> and obtain the equality (54), where *ξ* = *ξ*(*τ*, *x*). At the next step, we obtain the boundary value problem for determining *ξ*(*τ*, *x*) from the solvability condition of the resulting equation with respect to *u*3:

$$\begin{split} \frac{\partial \mathfrak{f}}{\partial \mathfrak{r}} &= d\_0 \frac{\partial^2 \mathfrak{f}}{\partial \mathfrak{x}^2} + \lambda\_{10} \mathfrak{f} + \mathfrak{g} \mathfrak{f} |\mathfrak{f}|^2, \\ \mathfrak{f}(t, \mathfrak{x} + 2\pi) &\equiv \mathfrak{f}(\mathfrak{r}, \mathfrak{x}). \end{split} \tag{61}$$

Here, *d*<sup>0</sup> = (1 + *iπ*/2)−1(1/2 + *σ*<sup>2</sup> <sup>0</sup> ) and the coefficients *λ*<sup>10</sup> and *g* are the same as in (51) and (55), respectively.

We formulate the basic results.

**Theorem 5.** *Let the boundary value problem* (61) *have the bounded solution ξ*0(*τ*, *x*) *as τ* → ∞*. Then, the function*

$$\begin{aligned} u\_0(t, \mathbf{x}, \varepsilon) &= \varepsilon \left[ \mathfrak{J}\_0(\mathbf{r}, \mathbf{x}) \exp\left(i\frac{\pi}{2}t\right) + \mathfrak{J}\_0(\mathbf{r}, \mathbf{x}) \exp\left(-i\frac{\pi}{2}t\right) \right] + \\ &\quad \varepsilon^2 \frac{2 - i}{5} \mathfrak{J}^2 \exp(i\pi t) + \frac{2 + i}{5} \mathfrak{J}^2 \exp(-i\pi t) \end{aligned}$$

*satisfies the boundary value problem* (15) *up to O*(*ε*4)*.*

The problem of the existence and stability of an exact solution to (15), which is close to the corresponding solution of the boundary value problem (61) as *ε* → 0, arises. It can be solved, for example, if *ξ*0(*τ*, *x*) is a periodic solution with the property of coarseness. By coarseness, we mean the following: If *ξ*0(*τ*, *x*) ≡ *const* · exp(*iωτ* + *imx*), then only one multiplier of the linearized on *ξ*0(*τ*, *x*) boundary value problem is equal to Modulo 1. In other cases, the coarseness condition is that only two multipliers of the linearized on *ξ*0(*τ*, *x*) boundary value problem are equal to Modulo 1.

**Theorem 6.** *Let ξ*0(*τ*, *x*) *be the coarse periodic solution of the boundary value problem* (61) *with period ω*0*. Then, for all sufficiently small ε, the boundary value problem* (15) *has the periodic with respect to t solution u*0(*t*, *x*,*ε*) *with period ω*<sup>0</sup> + *O*(*ε*) *with the same stability as ξ*0(*τ*, *x*)*. The asymptotic equality*

$$\begin{split} u\_0(t, \mathbf{x}, \varepsilon) &= \varepsilon (\xi\_0((1 + O(\varepsilon))\varepsilon^2 t, \mathbf{x}) \exp\left(i\frac{\pi}{2}t\right) + \\ &\quad \xi\_0((1 + O(\varepsilon))\varepsilon^2 t, \mathbf{x}) \exp\left(-i\frac{\pi}{2}t\right) + \dots + O(\varepsilon^4). \end{split} \tag{62}$$

*holds for this solution.*

The proof of Theorem 5 follows directly from the above construction of the boundary value problem (15) solution asymptotics. The justification of Theorem 6 is standard but cumbersome, so we omit it.
