# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Integrability Analysis: Diffusion Class Dynamical Dark Type Systems
#### 3.2.2 A Second Diffusion Class Dynamical System  
As an interesting enough another example, let us analyze the next of diffusion class dynamical system  
\[v_{t}=\ u^{-2}(v_{xx}-\ v^{-1}v_{x}^{2}):=K[v,u] \tag{68}\]  
on the manifold \(M_{\flat}\), parameterized by a functional variable \(u\in M_{u}\). The determining Lax-Noether Equation (59) on the extended cotangent space \(T^{*}(M_{\flat})\otimes\mathbb{C}\) looks as  
\[\varphi_{t}+\Big{[}u^{-2}(v_{x}/v)_{x}+v^{-1}\partial^{2}u^{-2}v\Big{]}\varphi=0 \tag{69}\]  
and allows the following asymptotic as \(\lambda\to\infty\) solution:  
\[\varphi=\exp[-\lambda^{2}t+\partial^{-1}\sigma(x;\lambda)],\ \ \sigma(x;\lambda)\sim\sum_{j\in\mathbb{Z}_{+}\cup\{-1\}}\lambda^{-j}\,\sigma_{j} [u,v]. \tag{70}\]From this expressions above one ensues the infinite hierarchy of recurrent differential-algebraic relationships:  
\[ \begin{split}-\delta_{j,-2}+\partial^{-1}\sigma_{j,t}+u^{-2}\sigma_ {j,x}+u^{-2}&\sum\limits_{k\in\mathbb{Z}_{+}\cup\{-1\}}\sigma_{k} \sigma_{j-k}+\\ +2u^{-2}v^{-1}v_{x}\sigma_{j}+2u^{-2}v^{-1}v_{xx}\delta_{j,0}-u^ {-2}v^{-2}v_{x}^{2}\delta_{j,0}=0\end{split}  \tag{71}\]  
for \(j\in\mathbb{Z}_{+}\cup\{-2,-1\}\), whose solutions  
\[ \begin{split}\sigma_{-1}=u,&\sigma_{0}=-\tfrac{1}{ 2}u\partial^{-1}u_{t}-v_{x}^{-1}v_{x}-\tfrac{1}{2}u^{-1}u_{x}=\\ =-1/2p_{x}-v^{-1}v_{x}-1/2u^{-1}u_{x}=-1/2(p+\ln(uv^{2}))_{x}, \end{split}  \tag{72}\]  
give rise to the expressions \(\sigma_{j}=0\) for all \(j\in\mathbb{Z}_{+}\), iff the following nonlinear and smooth evolution flow  
\[u_{t}=-u^{-2}u_{x}p_{x}+u^{-1}\ p_{xx}\ :=F[u,p], \tag{73}\]  
depending parametrically on a functional variable \(p\in M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\), holds jointly with its evolution relationship  
\[p_{t} =-(\ln(v/u))_{t}+2\Big{[}u^{-2}(v_{x}/v)_{x}+v^{-1}(v/u^{2})_{x} \Big{]}- \tag{74}\] \[-u^{-2}[p+\ln(v/u)]_{xx}+1/2([p+\ln(v/u)]_{x})^{2}.\]  
The latter can be equivalently rewritten as the next nonlinear smoothdynamical system  
\[p_{t}=-2u^{-2}p_{xx}+1/2u^{-2}p_{x}^{2}+2u^{-3}u_{x}p_{x}-u^{-3}u_{xx}+3/2u^{- 4}u_{x}^{2}:=P[u,p] \tag{75}\]  
on the extended functional manifold \(M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\). It is easy now to observe that the nonlinear evolution flows (73) and (75) are closed on the extended functional manifold \(M_{u}\times M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\), posing an additional question about their joint integrability. To demonstrate that this is the case, we applied once more yet now to the joint nonlinear dynamical system  
\[ \begin{split} u_{t}=-u^{-2}u_{x}p_{x}+u^{-1}\ p_{xx}\\ p_{t}=-2u^{-2}p_{xx}+1/2u^{-2}p_{x}^{2}+2u^{-3}u_{x}p_{x}-u^{-3}u_{ xx}+3/2u^{-4}u_{x}^{2}\end{split}  \tag{76}\]  
on the extended functional manifold \(M_{u}\times M_{p}\) the mentioned above gradient-holonomic approach and proved the following lemma. Lemma 3: _The nonlinear dynamical system (76) determines a completely integrable Lax linearized diffusion type flow on the extended functional manifold \(M_{u}\times M_{p}\), possessing only a finite number of suitably ordered conservation laws._  
The latter guarantees the integrability of the combined nonlinear diffusion flow (58) with that of (76):  
\[\left. \begin{split} v_{t}&=\ u^{-2}v_{xx}-\ u^{-2}v^{- 1}v_{x}^{2}\\ u_{t}&=-u^{-2}u_{x}p_{x}+u^{-1}\ p_{xx}\\ p_{t}&=-2u^{-2}p_{xx}+1/2u^{-2}p_{x}^{2}+2u^{-3}u_{ x}p_{x}-u^{-3}u_{xx}+3/2u^{-4}u_{x}^{2}\end{split} \right\}=Q[v,u,p] \tag{77}\]  
on the extended functional manifold \(M_{u}\times M_{v}\times M_{p}\). The obtained result can be now formulated as the following theorem.