5.1.3. Case of Essentially Asymmetric Function

Here, we assume parameter *h* to not be small but to be close to some number rationally commensurate with *π*. This means, that for some relatively prime integers *m*<sup>1</sup> and *m*2,

$$h = \frac{\pi m\_1}{m\_2} - \varepsilon^{1/2} h\_1. \tag{96}$$

We assume that *m* = *Mn*, where *n* = 0, ±1, ±2, ..., *M* = 2*m*<sup>2</sup> if *m*<sup>1</sup> is even, or *M* = 2*m*<sup>2</sup> if *m*<sup>1</sup> is odd. Here, we repeat the constructions from Section 5.1.1 and obtain, similar to (89), (90), the boundary value problem

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

$$\tilde{\xi}(t, y + 2\pi) \equiv \tilde{\xi}(t, y).$$

The formula

$$N(t, x, \varepsilon) = \xi \left( t, M(x - \varepsilon^{-1/2} h\_1 t) \right) + O(\varepsilon^{1/2})$$

establishes a relation between its solutions and the solutions of problem (79), (80). Accordingly, for (91), we arrive at the boundary value problem

$$\begin{array}{ll}\frac{\partial\widetilde{\xi}}{\partial t} = z^2 M^2 \Big(k\_1 + \frac{1}{2}h\_2^2\Big) \frac{\partial^2 \widetilde{\xi}}{\partial y^2} + h\_2 M \frac{\partial \widetilde{\xi}}{\partial v} + r[1 - \widetilde{\xi}(t - T, y - \delta, v)]\widetilde{\xi},\\\widetilde{\xi}(t, y + 2\pi, v) \equiv \widetilde{\xi}(t, y, v) \equiv \widetilde{\xi}(t, y, v + 2\pi). \end{array} \tag{97}$$

The solutions of this boundary value problem and of the boundary value problem (79), (80) are coupled by the equality

$$N(t, \mathbf{x}, \varepsilon) = \tilde{\boldsymbol{\xi}}\left(t, M(z\varepsilon^{-1/2} + \theta)\mathbf{x} + M(\varepsilon^{-1/2}h\_2\mathbf{z} + h\_2\theta), M\mathbf{x}\right) + \mathcal{O}(\varepsilon^{1/2}).$$

Satisfaction of only one condition *μ* = *ε*2*k*<sup>1</sup> (*h* = *ε*1/2*h*1) is a more interesting situation. Let *<sup>θ</sup>* = *<sup>θ</sup>*(*ε*) ∈ [0, *<sup>M</sup>*) be a value which complements the expression *<sup>π</sup>m*1(*m*2*h*1*ε*1/2)−<sup>1</sup> to an integer multiple of *<sup>M</sup>*. We examine the set of integers *<sup>K</sup>* = {(*πm*1(*m*2*h*2*ε*1/2)−<sup>1</sup> + *<sup>θ</sup>*)*p*}, *p* = 0, ±1, ±2, ... . Then, for the roots of Equation (86) with numbers *m* from *K* and *n* = 0, ±1, ±2, . . ., we obtain

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

Here, the boundary value problem

$$\begin{cases} \frac{\partial \xi}{\partial t} = \frac{\pi^2 m\_1^2}{m\_1^2} k\_1 \frac{\partial^2 \xi}{\partial y^2} + \frac{1}{2} h\_1^2 \left( \theta \frac{\partial}{\partial y} + \frac{\partial}{\partial v} \right) \xi + r [1 - \xi(t - T, y - \delta, v - \kappa)] \xi, \\\ \xi(t, y + 2\pi, v) \equiv \xi(t, y, v) \equiv \xi(t, y, v + 2\pi) \end{cases}$$

is the analogue of problem (97). Definitely in the situations under consideration, there is also a connection between the solutions of the constructed boundary value problems and the asymptotic (with respect to residual) solutions of the boundary value problem (79), (80). As in Section 5.1.1, boundary value problems similar to those given but with a large number of spatial variables appear when taking into account the larger number of modes. We do not dwell on this in more detail.
