5.2.3. Reduction to Spatially Two-Dimensional Parabolic Equations

As noted above, the local dynamics of the problem (79), (80) essentially depend on the relations between the parameters *ε*, *h*, and *μ*. Two of the most important cases are analyzed in Sections 5.2.1 and 5.2.2 with some model 'relations' between small parameters. Here, we dwell on another important scenario that is fundamentally different from the previous ones. We construct nonlinear parabolic equations with two spatial variables called the quasinormal forms to study local dynamics of the problem (79), (80) in *N*<sup>0</sup> neighborhood.

We consider a model situation where

$$
\hbar = \varepsilon \hbar\_{2'} \quad \mu = k\_0 \varepsilon^{7/2}, \quad T = \varepsilon^{1/2} T\_1. \tag{107}
$$

Let *m* = *m*(*ε*) be such asymptotically large modes for which

$$m(\varepsilon) = \frac{2\pi\hbar}{h\_2}\varepsilon^{-1} + c\varepsilon^{-1/2} + \theta\_1 + b\varepsilon^{-1/4} + \theta\_2,\tag{108}$$

where *n* = 0, ±1, ±2, ... , the parameter *c* is determined below, the values of *b* are arbitrary, the value *θ*<sup>1</sup> = *θ*1(*ε*) ∈ [0, 1) complements the sum of two previous terms to an integer, and *<sup>θ</sup>*<sup>2</sup> = *<sup>θ</sup>*2(*ε*) ∈ [0, 1) complements the expression *<sup>b</sup>ε*−1/4 to an integer. For the roots of the characteristic Equation (98) with numbers (108), the asymptotic formulas

$$\lambda = -r \exp\left(-\lambda \varepsilon^{-1/2} T\_1\right) + i h\_2 c \varepsilon^{-1/2} + i h\_2 b \varepsilon^{-1/4} + i(\theta\_1 + \theta\_2) h\_2 - \frac{1}{2} h\_2^2 c^2 + \dots$$

hold. In order to determine the stability boundary of *N*<sup>0</sup> for the problem (79), (80) in the range of parameter *T*1, we assume

$$\lambda = i\hbar \,\_2\varepsilon \varepsilon^{-1/2} + i\hbar \,\_2\mathfrak{b} \varepsilon^{-1/4} + i\omega + \varepsilon^{1/4} \lambda\_1 + \varepsilon^{1/2} \lambda\_2 + \dots \tag{109}$$

At the first step of the asymptotic analysis, from (109), we arrive at the equality

$$
\dot{\omega}\omega = -r\exp(-i\hbar\omega T\_1) - \frac{1}{2}\hbar^2\_2 c^2.\tag{110}
$$

The least value of *T*<sup>1</sup> = *T*<sup>0</sup> <sup>1</sup> for which Equation (110) is solvable for some *c* = *c*<sup>0</sup> is determined from the relations (103) as *k*<sup>1</sup> = 0, i.e.,

$$\begin{array}{rcl} \mathfrak{c}\_0 &=& (-r \cos \mathfrak{x}\_0)^{1/2} 2^{-1/2} h\_{2\prime} \\ T\_1^0 &=& 2^{-3/2} \mathfrak{x}\_0 h\_2 (-r \cos \mathfrak{x}\_0)^{1/2} \end{array}$$

Here, the statements of Lemmas 9 and 10 also hold. Thus, we assume below that a case close to critical in the stability problem of *N*<sup>0</sup> is realized. Let, for some constant *T*11, the equality

$$T\_1 = T\_1^0 + \varepsilon^{1/2} T\_{11}$$

hold. Similar to the previous part, infinitely many roots of Equation (98) tend to the imaginary axis as *ε* → 0, and there are no roots with a positive zero-separated as *ε* → 0 real part. However, there are 'essentially' more such roots in this case. Let us explain the above. For this purpose, we obtain expressions for *λ*<sup>1</sup> and *λ*2:

$$\begin{array}{rcl}\lambda\_1 &=& i\Delta\_0 b\_\prime \quad \Delta\_0 = r h\_2 T\_1^0 \exp(-\mathbf{x}\_0) + i\varepsilon\_0 h\_{2\prime}^2 \quad \Diamond\Delta = 0, \\\lambda\_2 &=& -r b^2 - k\_0 h\_2^{-2} 4\pi^2 n^2 + \beta\_{0\prime} \\\beta\_0 &=& -c\_0 \theta\_1(\mathbf{c}\_0) h\_2^2 + r \exp(-\mathbf{x}\_0) \left(i\omega\_0 T\_1^0 + i\hbar\_2 \mathbf{c}\_0 T\_{11}\right), \\\dot{\imath}\omega\_0 &=& -r \exp(-\mathbf{x}\_0) - \frac{1}{2} h\_2^2 \mathbf{c}\_0^2 \\\ \sigma\_0 &=& \frac{1}{2} h\_2^2 \left(1 + \frac{1}{2} \mathbf{c}\_0^2 h\_2^2 (T\_1^0)^2\right) + i\omega\_0 \frac{1}{2} h\_2^2 (T\_1^0)^2. \end{array} \tag{111}$$

The expression (111) differs from the similar formula for *λ*<sup>2</sup> in the previous part by the presence of parameter *n*, which takes all integer values *n* = 0, ±1, ±2, ... . Hence, we obtain that the quantity *ξ<sup>b</sup>* is also an *h*2-periodic function with respect to *x* in the basic formula of the form (105) in the considered case: *ξ<sup>b</sup>* = *ξb*(*τ*, *x*). Thus, we repeat the Equation (106) construction technique and obtain the more complicated equation

$$\frac{\partial \tilde{\xi}}{\partial \tau} = \sigma\_0 \frac{\partial^2 \tilde{\xi}}{\partial y^2} + k\_0 \frac{\partial^2 \tilde{\xi}}{\partial x^2} + \beta\_0 \tilde{\xi} + d\_0 |\tilde{\xi}|^2 \tilde{\xi} \tag{112}$$

with *h*2-periodic boundary conditions with respect to *x*, and *d*<sup>0</sup> differs from the value *d* appearing in (106) only by the presence of *k*<sup>1</sup> = 0 in the appropriate formula.

For the dynamics of (79), (80), Equation (112) plays the same role as Equation (106) with the conditions from Section 5.1. We do not dwell on this in more detail.
