# 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  
This diffusion-dispersion class dynamical system  
\[v_{t}=v_{xx}-(v^{2}u)_{x}+2\alpha(u+v)_{x}:=K[v,u] \tag{111}\]  
is defined on a functional manifold \(M_{v}\subset C(\mathbb{R}/\left\{2\pi\mathbb{Z}\right\};\mathbb{R})\), where \(u\in M_{u}\subset C(\mathbb{R}/\left\{2\pi\mathbb{Z}\right\};\mathbb{R})\) is a functional parameter and \(\alpha\in\mathbb{R}\) is arbitrary. To determine a kind of evolution  
\[u_{t}\overset{?}{=}F[v,u,p], \tag{112}\]  
on the functional manifold \(M_{u}\), ensuring the existence for (111) conservation laws within the gradient-holonomic integrability scheme, one needs to find an asymptotic as \(\lambda\rightarrow\infty\) solution \(\varphi\in T^{*}(M_{u})\otimes\mathbb{C}\) to the corresponding Lax-Noether equation  
\[\varphi_{t}+\varphi_{xx}+2(uv-\alpha)\varphi_{x}=0, \tag{113}\]where  
\[\varphi=\exp(-\lambda^{2}t+\ \partial^{-1}\sigma),\ \ \sigma\sim\sum_{j\in\mathbb{Z}_{+} \cup\{-1\}}\sigma_{j}\lambda^{-j}. \tag{114}\]  
Substitution of (114) into (113) gives rise to the following infinite hierarchy of recurrent equations:  
\[\partial^{-1}\sigma_{j,t}+\sum_{k\in\mathbb{Z}_{+}\cup\{-1\}}\sigma_{j-k} \sigma_{k}+\sigma_{j,x}+2(uv-\alpha)\delta_{j,-1}+2 uv\sigma_{j}=0 \tag{115}\]  
for \(j\in\mathbb{Z}_{+}\cup\{-1\}\), whose first two coefficients are  
\[\sigma_{-1}=1,\sigma_{0}=\alpha-uv,\sigma_{1}=[\partial^{-1}(uv)_{t}+(uv)^{2}- \alpha^{2}+(uv)_{x}]/2. \tag{116}\]  
Now we will take into account that the quantity \(\gamma_{1}:=\int\sigma_{1}dx\) should be a conservation law for the combined dynamical system  
\[ \begin{array}{c}v_{t}=v_{xx}-(v^{2}u)_{x}\\ u_{t}=F[v,u]\end{array} \bigg{\}}:=Q[v,u]. \tag{117}\]  
The latter condition is equivalent to the following functional constraint:  
\[ \begin{array}{c}(uv)_{t}=2\alpha uv_{x}+2\alpha uu_{x}+uv_{xx}-2u^{2}vv_{x}- \\ -uu_{x}v^{2}+vF[v,u]=\partial\rho_{0}[v,u]/\partial x\end{array}  \tag{118}\]  
for some smooth density \(\rho_{0}:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R}\). Having applied to the left-hand side of (118) the gradient operators with respect to the functional variables \((u,v)\in M_{u}\times M_{v}\), one easily obtain the following system of differential-functional relationships:  
\[F^{\prime,*}_{u}\cdot v =-2\alpha v_{x}-v_{xx}+2 uvv_{x}, \tag{119}\] \[F^{\prime,*}_{v}\cdot v =2\alpha u_{x}-u_{xx}-2 uvu_{x}-A[v,u],\]  
where \(A:J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\to\mathbb{R}\) is some smooth jet-mapping, \(F^{\prime}_{u}:T(M_{u})\to T(M_{u})\) and \(F^{\prime}_{v}:T(M_{v})\to T(M_{u})\) are the corresponding Frechet derivatives of the smooth mapping \(F:M_{v}\times M_{u}\to T(M_{u})\) and the star \"\({}^{*}\)\" denotes their corresponding adjoint mappings. From the first relationship of (119) one easily derives that the linear operator  
\[F^{\prime,*}_{u}=-\partial^{2}++c_{1}uv\partial+c_{2}uv_{x}+A^{\prime,*}_{u} \partial v^{-1}, \tag{120}\]  
where \(c_{1},c_{2}\in\mathbb{R}\) are constant parameters, satisfying the constraint \(c_{1}+c_{2}=2\), or, equivalently,  
\[F^{\prime}_{u}=2\alpha\partial-\partial^{2}\ -c_{1}\partial\circ uv+c_{2}uv_{x} -v^{-1}\partial\circ A^{\prime}_{u}, \tag{121}\]  
From the expressions (121) one easily ensues that  
\[F[v,u]=-u_{xx}-\ \frac{1}{2}c_{1}\ (u^{2}v)_{x}+\frac{1}{2}c_{2}u^{2}v_{x}+2 \alpha u_{x}-v^{-1}A_{x}+H \tag{122}\]  
satisfies the second relationship of (119) identically, if the following functional-operator constraint  
\[H^{\prime,*}_{v}v+H=0, \tag{123}\]  
holds on the functional manifold \(M_{v}\).