*3.3. Chain Dynamics for σ* = *O*(*ε*2)

Here, we assume that the condition (48) holds. We distinguish the roots of the characteristic Equation (20) with real parts tending to zero as *ε* → 0. This equation's roots *λ* = *λ*(*k*) are calculated from the formula

$$\begin{split} \lambda + (r\_0 + \varepsilon^2 r\_1) \exp(-\lambda) &= d \left( \int\_{-\infty}^{\infty} F(s, \varepsilon) \exp(iks) ds - 1 \right) = \\ d(\cos(z) \exp(-\varepsilon^4 \sigma\_0^2 z^2) - 1), \end{split} \tag{63}$$

where *z* = *εk*. The vanishing of the right-hand side in (63) up to *O*(1) (as *ε* → 0) is responsible for the real parts of the roots tending to zero. Those numbers *k* = *k*(*ε*) satisfy this condition for which cos(*z*) ∼ 1. We introduce the notation to describe such numbers. We fix an arbitrary integer *n* and let *θ<sup>n</sup>* = *θn*(*ε*) ∈ [0, 1) be the expression that complements the value 2*πnε*−<sup>1</sup> to an integer. It appears that the function *θn*(*ε*) can be considered identically zero. The point is that the parameter *ε* introduced above is determined as *ε* = 2*π*(1 + 2*N*)−1. Therefore, 2*πnε*−<sup>1</sup> = *n*(1 + 2*N*), which is an integer.

Then, the set of numbers *k*(*ε*) of the roots *λ*(*k*(*ε*)) consists of the values

$$k(\varepsilon) = 2\pi n \varepsilon^{-1} + m, \quad m, n = 0, \pm 1, \pm 2, \dots \tag{64}$$

in the considered case.

It is convenient to denote these roots by *λm*,*n*(*ε*). We obtain the asymptotic expression

$$\lambda\_{m,n}(\varepsilon) = i\frac{\pi}{2} - \varepsilon^2 \left(1 + i\frac{\pi}{2}\right)^{-1} \left(m^2 + 4\pi^2 \sigma\_0^2 n^2\right) + O(\varepsilon^4),$$

for them.

We follow the algorithm investigated above and introduce the formal series

$$\begin{split} u &= \varepsilon \left[ \exp \left( i \frac{\pi}{2} t \right) \sum\_{m = -\infty}^{\infty} \sum\_{n = -\infty}^{\infty} \xi\_{m, n}(\tau) \exp \left( i (2 \pi n \varepsilon^{-1} + m) x \right) + \\ &\varepsilon \left( -i \frac{\pi}{2} t \right) \sum\_{m = -\infty}^{\infty} \sum\_{n = -\infty}^{\infty} \bar{\xi}\_{m, n}(\tau) \exp \left( -i (2 \pi n \varepsilon^{-1} + m) x \right) \right] + \\ &\varepsilon^2 u\_2(t, \tau, x) + \varepsilon^3 u\_3(t, \tau, x) + \dots \,, \end{split} \tag{65}$$

where *τ* = *ε*2*t*, and the functions *uj*(*t*, *τ*, *x*) are periodic with respect to *t* and *x*.

Let *y* = 2*πε*−1*x* and

$$\xi(\tau, x, y) = \sum\_{m = -\infty}^{\infty} \sum\_{n = -\infty}^{\infty} \xi\_{m, n}(\tau) \exp(iny + imx).$$

Then, we can simplify expression (65)

$$u = \varepsilon \left[ \exp\left(i\frac{\pi}{2}t\right)\overline{\xi}(\tau, x, y) + \exp\left(-i\frac{\pi}{2}t\right)\overline{\xi}(\tau, x, y)\right] + \varepsilon^2 u\_2 + \varepsilon^3 u\_3 + \dots \tag{66}$$

We insert (66) into (15) and equate the coefficients at the same powers of *ε*. First, we determine *u*2(*τ*, *t*, *x*). Then, from the solvability condition of the equation with respect to *u*3, we obtain the expression for *ξ*(*τ*, *x*, *y*), determining:

$$\frac{\partial \tilde{\xi}}{\partial \tau} = \left(1 + i\frac{\pi}{2}\right)^{-1} \cdot \left(\frac{\partial^2 \tilde{\xi}}{\partial \mathbf{x}^2} + 4\pi^2 \sigma\_0^2 \frac{\partial^2 \tilde{\xi}}{\partial y^2}\right) + \lambda\_{10} \tilde{\xi} + g\tilde{\xi} |\tilde{\xi}|^2,\tag{67}$$

$$\xi(\mathbf{r}, \mathbf{x} + 2\pi, y) \equiv \xi(\mathbf{r}, \mathbf{x}, y + 2\pi) \equiv \xi(\mathbf{r}, \mathbf{x}, y),\tag{68}$$

where the coefficients *λ*<sup>10</sup> and *g* are the same as in (52).

Ideologically, the basic results of this subsection repeat Theorems 5 and 6. We cite an analogue of Theorem 5 as an example.

**Theorem 7.** *Let ξ*0(*τ*, *x*, *y*) *be the bounded solution of the boundary value problem* (67)*,* (68) *as τ* → ∞*. Then, the function*

$$\begin{split} u\_0(t, \mathbf{x}, \boldsymbol{\varepsilon}) &= 1 + \varepsilon \Big[ \exp \left( i \frac{\boldsymbol{\pi}}{2} t \right) \mathbb{f}\_0(\boldsymbol{\varepsilon}^2 t, \mathbf{x}, 2\boldsymbol{\pi}^{-1} \mathbf{x}) + \\ & \quad \exp \left( -i \frac{\boldsymbol{\pi}}{2} t \right) \mathbb{f}\_0(\boldsymbol{\varepsilon}^2 t, \mathbf{x}, 2\boldsymbol{\pi} \boldsymbol{\varepsilon}^{-1} \mathbf{x}) \Big] + \varepsilon^2 \mu\_2 \end{split} \tag{69}$$

*satisfies the boundary value problem* (15) *up to O*(*ε*3)*.*

The boundary value problems (61) and (67), (68) have been numerically investigated by many authors (see, for example, [30]). It has been shown that complicated and irregular oscillations are typical for such boundary value problems, especially for (67), (68). The formulas (60) and (66), which couple these boundary value problem solutions with the boundary value problem (15) solutions, allow us to formulate the same conclusion about the solutions of (15).
