# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### Integrability Analysis: Dispersion Class Dynamical Dark Type Systems
#### 3.3.2. A Second Diffusion-Dispersion Class Dynamical System
Having assumed, for simplicity, that \(H:J^{1}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\to\mathbb{R}\), we obtain right away that  
\[H(v,v_{x})=\ \ k^{\prime}(v)v_{x} \tag{124}\]for arbitrary smooth mapping \(k:J^{0}(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\in\mathbb{R}.\) Thus, the resulting dynamical system looks as follows  
\[u_{t} = -u_{xx}-\frac{1}{2}c_{1}\ (u^{2}v)_{x}+\frac{1}{2}c_{2}u^{2}v_{x}+2 \alpha u_{x}-v^{-1}A_{x}+v^{-1}k^{\prime}(v)v_{x}, \tag{125}\] \[v_{t} = v_{xx}-(v^{2}u)_{x}+2\alpha(u+v)_{x},\]  
depending on the up to now not determined smooth mapping \(A:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R}\) and which can be formally retrieved from the necessary condition that the next functional \(\gamma_{1}:=\int\sigma_{1}[u,v]dx\) generates a conserved quantity, that is \(d\gamma_{1}/dt=0\) along the evolution flow (125) on the functional manifold \(M_{v}\times M_{u}.\) As this conserved quantity depends a priori on the unknown smooth mapping \(A:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R},\) this problem can be strongly simplified, if to extend the present phase space \(M_{v}\times M_{u}\) of the evolution flow (125) to the phase space \(M_{v}\times M_{u}\times M_{p},\) having introduced an additional functional variable \(p:=A+\int_{0}^{v}k(v)dv\in M_{p}\subset\mathbb{C}(\mathbb{R}/\{2\pi\mathbb{Z} \};\mathbb{R}).\) Then, it is enough to find such an evolution flow  
\[p_{t}:\overset{?}{=}P[v,u,p] \tag{126}\]  
on the functional manifold \(M_{p},\) for which the functional \(\gamma_{1}:=\int\sigma_{1}[u,v]dx\) persists to be conserved on the extended phase space \(M_{v}\times M_{u}\times M_{p}\) already with respect to the combined evolution flow  
\[F^{\prime,*}_{v}\cdot v =2\alpha u_{x}-u_{xx}-2 uvu_{x}-A[v,u],\]0 \[F^{\prime,*}_{v}\cdot v =2\alpha u_{x}-u_{xx}-2 uvu_{x}-A[v,u],\]1  
on the functional manifold \(M_{v}\times M_{u}\times M_{p}\) under the condition that \(c_{j}\in\mathbb{R},j=\overline{1,2}\) and \(c_{1}+c_{2}=2.\) Taking into account that the density  
\[F^{\prime,*}_{v}\cdot v =2\alpha u_{x}-u_{xx}-2 uvu_{x}-A[v,u],\]2  
is conserved if and only if its temporal derivative satisfies the relationship  
\[F^{\prime,*}_{v}\cdot v =2\alpha u_{x}-u_{xx}-2 uvu_{x}-A[v,u],\]3  
for some jet-density mapping \(\rho_{1}:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R},\) it is enough to apply to the left-hand side of (129) the gradient operator and to check that it vanishes on the extended manifold \(M_{v}\times M_{u}\times M_{p}\):  
\[\sigma_{1,t}^{\prime*}(1)=\left( \begin{array}{c}-\frac{P_{x}^{\prime*}(1)}{ \varepsilon^{2}}+2\alpha\frac{P_{x}u}{v^{2}}+\frac{P_{xx}}{v}+up_{x}\\ -\frac{P_{x}^{\prime*}(1)}{\varepsilon^{2}}+vp_{x}-2\alpha\frac{P_{x}}{v}\\ -\frac{P_{x}^{\prime*}(1)}{2}+2\alpha\big{(}\frac{p}{v}\big{)}_{x}+\big{(} \frac{p_{x}}{v}\big{)}_{x}-(uv)_{x}\end{array} \right)=0. \tag{130}\]  
System (130) proves to be compatible if  
\[P=-4\alpha uv^{-1}p_{x}+2p_{xx}\ln v+2 uvp_{x}, \tag{131}\]  
giving rise to the following closed nonlinear dark type integrable dynamical system  
\[u_{t} = -u_{xx}-\frac{1}{2}c_{1}\ (u^{2}v)_{x}+\frac{1}{2}c_{2}u^{2}v_{x}+2 \alpha u_{x}-v^{-1}p_{x},\] \[v_{t} = v_{xx}-(v^{2}u)_{x}+2\alpha(u+v)_{x},\] \[p_{t} = -4\alpha uv^{-1}p_{x}+2p_{xx}\ln v+2 uvp_{x} \tag{132}\]  
on the extended manifold \(M_{v}\times M_{u}\times M_{p},\) if \(c_{j}\in\mathbb{R},j=\overline{1,2}\) and \(c_{1}+c_{2}=2.\)As a simplest inference from the flow (132) one easily observes that the functional reduction on the submanifold \(\{(v,u,p)\in M_{v}\times M_{u}\times M_{p}:p=0\in M_{p}\}\) is compatible, thus determining a nonlinear dynamical system on the reduced phase \(M_{v}\times M_{u}\). The obtained above new integrability results can be now formulated as the following theorem.