# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Properties of the Periodic Spectral Problem: Floquet Theory Aspects  
Let us study in detail the spectral properties of the problem (3) for the uniform and periodic dynamical system (1) within the classical [25; 26] Floquet theory. To do this, we will consider the differential operator \(L[u;\lambda]:L_{\infty}(\mathbb{R};\mathbb{C})\to L_{\infty}(\mathbb{R}; \mathbb{C})\), preliminarily equivalently represented in the following canonical differential-matrix form of the first order:  
\[\mathcal{L}[u;\lambda]=\partial/\partial x-l[u;\lambda], \tag{5}\]  
where \(l:=l[u;\lambda]\in\mathrm{End}(L_{\infty}(\mathbb{R};\mathbb{C}^{k}))\) is the corresponding \(2\pi\)-periodic matrix functional on the manifold \(M\). The Equation (3) can be then rewritten in the form of the linear matrix differential equation \(\mathcal{L}[u;\lambda]f=0\) in the space \(L_{\infty}(\mathbb{R};\mathbb{C}^{k})\), or equivalently as  
\[\partial f/\partial x=l[u;\lambda]f \tag{6}\]  
for \(f\in L_{\infty}(\mathbb{R};\mathbb{C}^{k})\). Let \(F(x,x_{0};\lambda)\) be the fundamental solution of the Equation (6) normalized by the unit matrix at the point \(x=x_{0}\in\mathbb{R}\), that is \(F(x_{0},x_{0};\lambda)=I\in\mathrm{End}\ \mathbb{C}^{k}\) for all \(x_{0}\in\mathbb{R}\), \(\lambda\in\mathbb{C}\). Evidently, any solution of Equation (6) can be represented in the form  
\[f=F(x,x_{0};\lambda)f_{0}, \tag{7}\]  
where \(f_{0}\in\mathbb{C}^{k}\) is an initial value at the point \(x_{0}\in\mathbb{R}\). Consider the value \(f(x,x_{0};\lambda)\) at \(x=x_{0}\pm 2\pi N\), \(x_{0}\in\mathbb{R}\), where \(2\pi\) is the period of the element \(u\in M\) and \(N\ \in\ \mathbb{Z}_{+}\). By virtue of periodicity in the variable \(x\in\mathbb{R}\) of the matrix \(l[u;\lambda]\in\mathrm{End}\ L_{\infty}(\mathbb{R};\mathbb{C}^{k})\), we have  
\[f(x_{0}\pm 2\pi N,x_{0};\lambda)=S^{\pm N}(x_{0};\lambda)f_{0}, \tag{8}\]  
where \(S:=S(x_{0};\lambda)=F(x_{0}+2\pi,x_{0};\lambda)\in\mathrm{End}\ \mathbb{C}^{k},x_{0}\in\mathbb{R}\), is the so-called _monodromy matrix_ of the differential operator (5). Let \(\xi(\lambda)\in\sigma(S)\subset\mathbb{C}\) be an eigenvalue of the matrix \(S(x_{0};\lambda)\in\mathrm{End}\ \mathbb{C}^{k},x_{0}\in\mathbb{R}\). Then from (8) it directly follows [14; 15; 27] that the solution \(f(x,x_{0};\lambda)\in C(\mathbb{R};\mathbb{C}^{k})\) is bounded on the whole axis \(\mathbb{R}\) if and only if any eigenvalue \(\xi(\lambda)\in\sigma(S)\) of the matrix \(S(x_{0};\lambda)\) has the absolute value equal to one, that is \(|\xi(\lambda)|=1\). Lemma 1: _The spectrum \(\sigma(S)\subset\mathbb{C}\) of the periodic problem (6) does not depend on a point \(x_{0}\in\mathbb{R}\)._  
Proof: To show that the eigenvalues \(\xi(\lambda)\in\sigma(S)\) don't depend on a point \(x_{0}\in\mathbb{R}\), we consider the corresponding differential equation for matrix \(S(x_{0};\lambda)\in\mathrm{End}\ \mathbb{C}^{k}\) :  
\[\partial S/\partial x_{0}=[I,S], \tag{9}\]  
where \([\cdot,\cdot]\) denotes here the usual matrix commutator in the space \(\mathrm{End}\ \mathbb{C}^{k}\). From the Equation (9) we conclude that the traces \(\mathrm{tr}\ S^{m}(x_{0};\lambda)\), \(m\in\mathbb{Z}\), don't depend on a point \(x_{0}\in\mathbb{R}\). It means, equivalently, that the eigenvalue \(\xi(\lambda)\in\sigma(S)\) of the matrix \(S(x_{0};\lambda)\) does not depend on a point \(x_{0}\in\mathbb{R}\), that is  
\[\partial\xi(\lambda)/\partial x_{0}=0,\]  
proving the statement. Let \(\tilde{f}\in C(\mathbb{R};\mathbb{C}^{k})\) be an eigenfunction of the problem (6), satisfying the classical Bloch condition [14; 27]  
\[\tilde{f}(x+2\pi,x_{0};\lambda)=\xi(\lambda)\tilde{f}(x,x_{0};\lambda), \tag{10}\]  
that is the vector-function \(\tilde{f}(x,x_{0};\lambda)\in\mathbb{C}^{k}\) is an eigenfunction of the monodromy matrix \(S(x;\lambda)\):  
\[S(x;\lambda)\tilde{f}(x,x_{0};\lambda)=\xi(\lambda)\tilde{f}(x,x_{0};\lambda)\]  
for all \(x\in\mathbb{R}\).