# 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
#### 3.2.3 A Third Diffusion Class Dynamical System: Not Dark Type Example  
Herewith we will consider then next nonlinear dynamical system [6] of diffusion class  
\[v_{t}=-vv_{x}-(u^{-2}v_{x})_{x}:=K[v,u], \tag{78}\]  
which is defined on a functional manifold \(M_{v}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\), depends parametrically on a functional parameter \(u\in M_{u}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\) proves to be of not-dark type. The corresponding Lax-Noether equation  
\[\varphi_{t}+v\varphi_{x}+(u^{-2}\varphi_{x})_{x}=0, \tag{79}\]  
on the extended cotangent space \(T^{*}(M_{v})\otimes\mathbb{C}\) possesses the asymptotic as \(\lambda\to\infty\) solution  
\[\varphi=\exp(-\lambda^{2}t+\partial^{-1}\sigma), \tag{80}\]  
where \(\sigma\simeq\sum_{j\in\mathbb{Z}_{+}\cup\{-1\}}\sigma_{j}\lambda^{-j}\), whose coefficients satisfy an infinite hierarchy of the recurrent relationships  
\[-\delta_{j,-2}+\partial^{-1}\sigma_{j,t}+v\sigma_{j}+(u^{-2}\sigma_{j})_{x}+u ^{-2}\sigma_{j-k}\sigma_{k}=0 \tag{81}\]  
for \(j\in\mathbb{Z}_{+}\cup\{-2,-1\}\). In particular at \(j=-2\) we obtain \(\sigma_{-1}=u\in M_{u}\) and at \(j=-1\)  
\[\partial^{-1}u_{t}+uv-u^{-2}u_{x}+2u^{-1}\sigma_{0}=0. \tag{82}\]  
To determine the conserved quantity \(\sigma_{0}:J(\mathbb{R};\mathbb{R}^{2})\to\mathbb{R}\), we put, by definition,  
\[u_{t}=-(uv)_{x}+\left(u^{-1}p\right)_{x}-\left(u^{-1}\right)_{xx}, \tag{83}\]  
where have introduced an additional functional variable \(p\in M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\), allowing to determine from the recurrent hierarchy (81) the next density \(\sigma_{0}:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R}\):  
\[\sigma_{0}=-p/2. \tag{84}\]  
Herewith, to close the system of two evolution flows (78) and (83) we assume that the additional functional variable \(p\in M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\) satisfies the following closing evolution relationship:  
\[p_{t}:=P[v,u,p]=4\Big{(}u^{-1}h_{x}\Big{)}_{x}-(vp)_{x}-(pu^{-2})_{x}+(u^{-1}p ^{2}/2)_{x}, \tag{85}\]  
where a functional variable \(h\in M_{h}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\) makes it possible to determine the trivial conserved density \(\sigma_{1}=h_{x}\) on the extended functional manifold \(M_{v}\times M_{u}\times M_{p}\times M_{h}\subset C(\mathbb{R}/\{2\pi\mathbb{Z }\};\mathbb{R}^{4})\). The latter gives rise to the next alleged conserved density  
\[\sigma_{2}=\Big{(}u^{-1}h_{x}p-u(u^{-2}h_{x})_{x}-uvh_{x}+uh_{t}\Big{)}/2, \tag{86}\]  
which can be compatibly retrieved if and only if \(\sigma_{2}=\partial\eta_{2}/\partial x\) for some density mapping \(\eta_{2}:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{4})\to\mathbb{R}\) for the deemed evolution flow  
\[h_{t}=H[v,u,p,h], \tag{87}\]generated by some smooth mapping \(H:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{4})\to\mathbb{R}\). As a result of simple calculations one easily ensued that the density \(\sigma_{2}=-f[u]_{x}/2\) and  
\[\varphi_{t}+v\varphi_{x}+(u^{-2}\varphi_{x})_{x}=0, \tag{79}\]0  
for some smooth mapping \(f:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\to\mathbb{R}.\) Moreover, based on the recurrent hierarchy (81), one obtains successively that the next density  
\[\varphi_{t}+v\varphi_{x}+(u^{-2}\varphi_{x})_{x}=0, \tag{79}\]1  
is not conserved for any smooth mapping \(f:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\to\mathbb{R}\), that is \(\sigma_{3,t}\
eq\partial\eta_{3}/\partial x\) on the functional manifold \(M_{v}\times M_{u}\times M_{p}\times M_{h}.\) Thus we have stated that the closed dynamical system  
\[\left. \begin{array}{c}v_{t}=-vv_{x}+(u^{-2}v_{x})_{x}\\ u_{t}=-(uv)_{x}+\left(u^{-1}p\right)_{x}-\left(u^{-1}\right)_{xx}\\ p_{t}=4\left(u^{-1}h_{x}\right)_{x}-(\bar{v}p)_{x}-(pu^{-2})_{x}+(u^{-1}p^{2}/2 )_{x}\\ h_{t}=u^{-2}h_{x}p-(u^{-2}h_{x})_{x}-vh_{x}+u^{-1}f[u]_{x}\end{array}