*5.1. Boundary Value Problem Reduction to Parabolic-Type Equation*

This section is divided into three parts. The first two sections assume that the deviation *h* is asymptotically small, i.e., function *F*(*s*) is close to symmetric. In Part 1, the parameters *γ*−1, *μ* and *h*<sup>2</sup> are assumed to be of the same order. It is shown that the nonlocal dynamics of specially constructed boundary value problems of parabolic type determine, in general, the dynamic properties of problem (79), (80). The corresponding structures are called slowly oscillating since they are formed mainly on low modes. The diffusive properties of the initial equation are assumed to be small in Part 2. In terms of problem coefficients, this means the qualified smallness of parameters *μ* and *h*<sup>2</sup> compared to *γ*−1. The appropriate range of their change is indicated below. In this case, we demonstrate that the rapidly oscillating (i.e., formed on asymptotically large modes) regimes are distinctive for the boundary value problem (79), (80). In order to find them, special families of nonlinear parabolic boundary value problems are constructed. In Part 3, we study the dynamics of problem (79), (80) under the condition of *F*(*s*) essential dissymmetry when parameter *h* is not small.

Thus, we demonstrate that the boundary value problem (79), (80) dynamics essentially depend on the relations between parameters *γ*−1, *μ*, and *h*.

5.1.1. Slowly Oscillating Structures

Let

$$
\varepsilon = \gamma^{-1}, \quad 0 < \varepsilon \ll 1.
$$

Here, we assume that the parameters *ε*, *μ*, and *h*<sup>2</sup> are of the same order: for some fixed, positive *k* and *h*1, we obtain

$$
\mu = k \varepsilon, \quad h = \varepsilon^{1/2} h\_1. \tag{84}
$$

After dividing (79) by *γ*, we consider the resulting 'main' part, which is the linear boundary value problem

$$\varepsilon \frac{\partial u}{\partial t} = \int\_{-\infty}^{\infty} F(s)u(t, \mathbf{x} + \mathbf{s})ds - u, \quad u(t, \mathbf{x} + 2\pi) \equiv u(t, \mathbf{x}). \tag{85}$$

The characteristic equation of (85) is of the form

$$
\varepsilon \lambda\_m = \exp[i\varepsilon^{1/2} h\_1 m - \varepsilon km^2] - 1, \quad m = 0, \pm 1, \pm 2, \dots \tag{86}
$$

Hence, we obtain the asymptotic formulas

$$
\lambda\_m = i\varepsilon^{-1/2}h\_1m - \left(k + \frac{1}{2}h\_1^2\right)m^2 + O(\varepsilon^{1/2})\tag{87}
$$

for the roots *λ<sup>m</sup>* = *λm*(*ε*). Thus, infinitely many roots of Equation (86) tend to the imaginary axis as *ε* → 0. This gives us reason to regard the considered critical case in the stability problem to be infinite-dimensional. The methodology of studying such systems is developed in [26,28]. We use the appropriate results here. For this purpose, we introduce the formal series

$$\mu = \sum\_{m = -\infty}^{\infty} \xi\_m(t) \exp(imy) + \varepsilon u\_1(t, y) + \dots \,\_{\prime} \tag{88}$$

where *y* = *x* + *ε*1/2*h*1*t*. We insert (88) into (79) and equate the coefficients at the same power of *ε*. We obtain the infinite system of ordinary differential equations to determine the function *ξm*(*t*). As it turns out, this system can be written as one complex parabolic equation of the Ginzburg–Landau type for the function

$$
\tilde{\xi}(t,y) = \sum\_{m = -\infty}^{\infty} \tilde{\xi}\_m(t) \exp(imy)
$$

$$
\frac{\partial \tilde{\xi}}{\partial t} = \left(k + \frac{1}{2}h\_1^2\right) \frac{\partial^2 \tilde{\xi}}{\partial y^2} + r[1 - \tilde{\xi}(t - T, y - \Delta)] \tilde{\xi}\_{\nu} \tag{89}
$$

$$
\xi(t, y + 2\pi) \equiv \xi(t, y),
\tag{90}
$$

where Δ = Δ(*ε*)=(*ε*−1/2*h*1*T*) mod 2*π*.

**Theorem 9.** *Let, for some fixed* Δ = Δ<sup>0</sup> ∈ [0, 2*π*]*, the boundary value problem* (89)*,* (90) *be bounded together with the time derivative solution ξ*0(*t*, *y*) *as t* → ∞*. Then, as determined from the equality* Δ(*ε*) = Δ0*, for the sufficiently small εn, the function*

$$N(t, \mathbf{x}, \varepsilon\_n) = \xi\_0(t, \mathbf{x} + \varepsilon\_n^{-1/2} h\_1 t)$$

*satisfies the boundary value problem* (79)*,* (80) *up to O*(*ε* 1/2 *<sup>n</sup>* )*.*

**Theorem 10.** *On conditions of Theorem 9, let ξ*0(*t*, *y*) *be a periodic solution of the problem* (89)*,* (90)*, and let only two of its multipliers be equal to Modulo 1. Then, for all sufficiently small εn, the boundary value problem* (79)*,* (80) *has a periodic solution N*0(*t*, *x*,*ε*) *of the same stability as ξ*0(*t*, *y*)*, and*

$$N\_0(t, \mathfrak{x}, \mathfrak{e}) = \mathfrak{z}\_0((1 + o(\mathfrak{e}\_n^{1/2}))t, \mathfrak{x} + \mathfrak{e}\_n^{-1/2}h\_1t) + O(\mathfrak{e}^{1/2}).$$

5.1.2. Rapidly Oscillating Structures

Here, we demonstrate that the decrease of diffusion coefficients (*k* and *h*1) can lead to the appearance of rapid oscillation with respect to spatial and time variable families of structures. Conditionally, we can divide them into two types. Proportional decrease of coefficients *k* and *h*1, which play the role of diffusion, leads to the appearance of the first type. The structures are formed in the neighborhood of asymptotic large modes. However, the product of diffusion coefficients and the (asymptotically large) value of the corresponding modes is of an asymptotically small quantity. In other words, the squares of the corresponding modes coincide in order with the reciprocal of deviation of spatial variables. The second type of structures arises only when one diffusion coefficient, *k*, decreases. Here also, rapidly oscillating structures are formed due to the interaction of a large number of modes far apart from each other. However, the principal difference from structures of the first type is that the values of modes themselves coincide in order with the reciprocal of the deviation of the spatial variable but not the values of the squares of the modes. Let us consider these cases separately.

**Case 1.** *Structures of the 'first' type.*

*Let*

$$
\mu = \varepsilon^2 k\_1, \quad h = \varepsilon h\_2 \tag{91}
$$

*for some fixed positive k*<sup>1</sup> *and h*2*.*

*We arbitrarily fix z as real, and let θ* = *θ*(*ε*, *z*) *be the value from the semi-open interval* [0, 1) *that complements the expression zε*−1/2 *to an integer. We consider the integer set*

$$(z\varepsilon^{-1/2} + \theta)m + n\_\prime \quad m, n = 0, \pm 1, \pm 2, \dots, \pm n$$

*For these numbers, the characteristic Equation* (86) *has the set of roots similar to* (87)

$$
\lambda\_{m,n} = i\varepsilon^{-1/2}h\_2zm + i(\theta m + n)h\_2 - \left(k\_1 + \frac{1}{2}h\_2^2\right)z^2m^2 + O(\varepsilon^{1/2}).
$$

*Then, the family of the boundary value problems depending on the parameter z*

$$\frac{\partial \tilde{\xi}}{\partial t} = z^2 \left( k\_1 + \frac{1}{2} h\_2^2 \right) \frac{\partial^2 \tilde{\xi}}{\partial y^2} + h\_2 \left( \theta \frac{\partial \tilde{\xi}}{\partial y} + \frac{\partial \tilde{\xi}}{\partial v} \right) + r \tilde{\xi} \left[ 1 - \tilde{\xi} \left( t - T, y - \delta, v \right) \right], \tag{92}$$

$$\xi(t, y + 2\pi, v) \equiv \xi(t, y, v) \equiv \xi(t, y, v + 2\pi) \tag{93}$$

*plays a role in the boundary value problem* (89)*,* (90) *in the considered case. Here, δ* = (*ε*−1/2*h*2*z* + *h*2*θ*) mod 2*π.*

**Theorem 11.** *Let, for some fixed δ* = *δ*<sup>0</sup> *and z* = *z*0*, the boundary value problem* (92)*,* (93) *be bounded together with a derivative with respect to t solution ξ*0(*t*, *y*, *v*) *as t* → ∞*. Then, as determined from the equality δ*(*ε*) = *δ*0*, for the sufficiently small εn, the function*

$$N(t, \mathbf{x}, \boldsymbol{\varepsilon}) = \xi\_0(t, (z\_0 \varepsilon^{-1/2} + \theta)\mathbf{x} + (\varepsilon^{-1/2} h\_2 z\_0 + h\_2 \theta)\mathbf{t}, \mathbf{x})$$

*satisfies the boundary value problem* (79)*,* (80) *up to O*(*ε*1/2)*.*

The 'first' type of structure can also be formed by the interaction of a larger number of modes. We use constructions from [32] to demonstrate this.

We fix an arbitrarily natural number *m*<sup>0</sup> and real numbers *z*1, ... , *zm*<sup>0</sup> . We consider the set of integer numbers

$$\sum\_{j=1}^{m\_0} (z\_j \varepsilon^{-1/2} + \theta\_j) m\_j + n\_{j\prime} \quad m\_{j\prime}, n\_j = 0, \pm 1, \pm 2, \dots, \dots$$

For modes with these numbers, the equation

$$\begin{split} \frac{\partial \mathfrak{F}}{\partial t} &= \left[ z\_1 \frac{\partial}{\partial y\_1} + \dots + z\_{m\_0} \frac{\partial}{\partial y\_{m\_0}} \right]^2 \left( k\_1 + \frac{1}{2} h\_2^2 \right) \mathfrak{F} + h\_2 \left( \theta\_1 \frac{\partial \mathfrak{F}}{\partial y\_1} + \dots + \theta\_{m\_0} \frac{\partial \mathfrak{F}}{\partial y\_{m\_0}} + \dots \right) \\ &\frac{\partial \mathfrak{F}}{\partial v\_1} + \dots + \frac{\partial \mathfrak{F}}{\partial v\_{m\_0}} \Big) + r \left[ 1 - \mathfrak{F}(t - T, y\_1 - \delta\_1, \dots, y\_{m\_0} - \delta\_{m\_0}, v\_{1}, \dots, v\_{m\_0}) \right] \mathfrak{F} \end{split}$$

plays the role of the boundary value problem (92), (93) with 2*π*-periodic boundary conditions with respect to each spatial variable, and

$$
\delta\_{\dot{j}} = (\epsilon^{-1/2} h\_2 z\_{\dot{j}} + h\_2 \theta\_{\dot{j}}) \mod 2\pi \cdot \zeta
$$

Similarly to Theorem 11, the corresponding statement about coupling with the solutions of the boundary value problem (79), (80) is formulated for the solutions of this boundary value problem.

**Remark 1.** *The structures considered here that rapidly oscillate with respect to spatial variables arise when the coefficients k and h*<sup>1</sup> *in* (89) *are also asymptotically small.*

In this part, the case of *k* = *εk*<sup>1</sup> and *h*<sup>1</sup> = *ε*1/2*h*<sup>2</sup> is considered. Of course, one can investigate a more general case, when for some positive *α*, *k*<sup>1</sup> and *h*2, the equalities

$$k = \varepsilon^{1+a} k\_{1\prime} \quad h\_1 = \varepsilon^{\frac{1+a}{2}} h\_2$$

hold. For such *k* and *h*, the changes in the corresponding constructions are not fundamental, so we do not dwell on them.

**Case 2.** *Structures of the 'second' type.*

*Here, the coefficient h is assumed to be the same as in* (84)*. Further,*

$$
\mu = \varepsilon^2 k\_1.
$$

*We consider the integer set* (2*πh*−<sup>1</sup> <sup>1</sup> *<sup>ε</sup>*−1/2 + *<sup>θ</sup>*)*<sup>m</sup>* + *n, where <sup>θ</sup>* ∈ [0, 1) *is such that the expression in brackets is an integer, m*, *n* = 0, ±1, ±2, ... *. The characteristic Equation* (86) *has the set of roots λm*,*<sup>n</sup> for which*

$$
\lambda\_{m,n} = i h\_1 (\theta m + n) \varepsilon^{-1/2} - \left( \frac{4 \pi^2}{h\_1^2} k\_1 m^2 + \frac{1}{2} h\_1^2 (\theta m + n)^2 \right) + O(\varepsilon^{1/2}).
$$

*We apply the above construction and obtain the resulting boundary value problem*

$$\frac{\partial \tilde{\xi}}{\partial t} = 4\pi^2 h\_1^{-2} k\_1 \frac{\partial^2 \tilde{\xi}}{\partial y^2} + \frac{h\_1^2}{2} \left(\theta \frac{\partial}{\partial y} + \frac{\partial}{\partial v}\right)^2 \tilde{\xi} + r[1 - \tilde{\xi}(t - T, y - \delta, v - \kappa)] \tilde{\xi},\tag{94}$$

$$\xi(t, y + 2\pi, v) \equiv \xi(t, y, v) \equiv \xi(t, y, v + 2\pi), \tag{95}$$

*where*

$$\delta = (h\_1 T \theta \varepsilon^{-1/2}) \mod 2\pi\prime \quad \kappa = (h\_1 T \varepsilon^{-1/2}) \mod 2\pi\prime$$

*We introduce some notation in order to formulate an analogue of Theorem 9 in the considered situation. We fix an arbitrary κ* = *κ*<sup>0</sup> ∈ [0, 1) *and the sequence ε <sup>p</sup>* → 0 *of the roots of equation κ*(*ε <sup>p</sup>*) = *κ*0*. Let θ*<sup>0</sup> *be the arbitrary limit point of the sequence θ*(*ε <sup>p</sup>*)*. Let θ*(*ε pq* ) → *θ*<sup>0</sup> *for the sequence ε pq . Finally, σ*<sup>0</sup> ∈ [0, 1) *denotes the arbitrary limit point of the sequence σ*(*ε pq* )*, and let ε<sup>R</sup>* ⊂ *ε pq and σ*(*εR*) → *σ*0*.*

**Theorem 12.** *Let, for some θ*0, *σ*0*, and κ*0*, the boundary value problem* (94)*,* (95) *be bounded together with a derivative with respect to t solution ξ*0(*t*, *y*, *v*)*. Then, for σ* = *σ*0*, κ* = *κ*0*, θ* = *θ*0*, and ε* = *ε<sup>R</sup>* → 0*, the function*

$$N(t, \mathbf{x}, \varepsilon) = \xi\_0 \left( t, (2\pi h\_1^{-1} \varepsilon\_R^{-1/2} + \theta\_0) \mathbf{x} + h\_1 \theta\_0 \varepsilon\_R^{-1/2} t, \mathbf{x} + h\_1 \varepsilon^{-1/2} t \right)$$

*satisfies the boundary value problem* (79)*,* (80) *up to O*(*ε*1/2)*.*
