# 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  
#### 3.3.1 A First Dispersion Class Dynamical System  
We consider the dispersion class dynamical system  
\[v_{t}=-uv_{x}+v_{xxx}=K[v,u] \tag{91}\]  
of the Korteweg-de Vries type on a functional manifold \(M_{v}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\), parametrically depending on a functional variable \(u\in M_{u}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}).\) The determining Lax-Noether equation  
\[\varphi_{t}-\varphi_{xxx}+(u\varphi)_{x}=0, \tag{92}\]  
on the complexified cotangent space \(T^{*}(M_{v})\otimes\mathbb{C}\) allows the following asymptotic as \(\lambda\to\infty\) solution  
\[\varphi=\exp(\lambda^{3}t+\partial^{-1}\sigma), \tag{93}\]  
where \(\sigma\sim\sum_{j\in\mathbb{Z}_{+}\cup\{-1\}}\sigma_{j}\lambda^{-j}\), whose coefficients satisfy the following recurrent relationships:  
\[\delta_{j,-3}+\partial^{-1}\sigma_{j,t}-\sigma_{j,xx}-3\sigma_{j-k}\sigma_{k, x}-\sigma_{j-k}\sigma_{k-s}\sigma_{s}+u_{x}\delta_{j,0}+u\sigma_{j}=0 \tag{94}\]  
for \(j\in\mathbb{Z}_{+}\cup\{-3,-2,-1\}.\) From (94) one easily obtains that  
\[\sigma_{-1}=1,\sigma_{0}=0,\sigma_{1}=\frac{u}{3},\sigma_{2}=0 \tag{95}\]  
and  
\[\partial^{-1}u_{t}-u_{xx}-9\sigma_{3}=0. \tag{96}\]Since the quantity \(\gamma_{3}:=\int\sigma_{3}dx\) should be a conservation law, we put the functional coefficient \(\sigma_{3}:=p/9\in M_{p}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R})\). Then we obtained a first supplementing dynamical system  
\[u_{t}=u_{xxx}+\ p_{x} \tag{97}\]  
on the functional manifold \(M_{u}\). Following the above recurrent scheme at \(j=2\), one finds that  
\[\sigma_{4}=-\left(\frac{1}{18}u^{2}+\frac{p}{9}\right)_{x}. \tag{98}\]  
The simplest way to close the hierarchy consists in making the density (98) trivial, putting by definition, that  
\[p=-u^{2}/2. \tag{99}\]  
The latter gives rise to the next conservation law density:  
\[\sigma_{5}=\frac{1}{27}\partial^{-1}p_{t}+\frac{1}{9}(uu_{x})_{x}-\frac{1}{81} u^{3}-\frac{1}{27}pu+\frac{2}{27}p_{xx}=\frac{1}{18}(u_{x}^{2}+u^{3}/3), \tag{100}\]  
generating a true conservation law \(\gamma_{5}:=\int\sigma_{5}dx\) if the following functional constraint  
\[\varphi_{t}-\varphi_{xxx}+(u\varphi)_{x}=0, \tag{92}\]0  
holds on the functional manifold \(M_{u}\). Respectively on easily calculates the next densities:  
\[\varphi_{t}-\varphi_{xxx}+(u\varphi)_{x}=0, \tag{92}\]1 \[\varphi_{t}-\varphi_{xxx}+(u\varphi)_{x}=0, \tag{92}\]2  
and so on, that is the functional constraint (99) proves to be compatible for the whole infinite hierarchy of recurrent relationships (94). Thus, the resulting dynamical system  
\[\left. \begin{array}{c}v_{t}=-uv_{x}+v_{xxx}\\ u_{t}=u_{xxx}-\ uu_{x}\end{array} \right\}:=Q[v,u] \tag{102}\]  
is compatible and presents a closed nonlinear dynamical system on the extended functional manifold \(M_{p}\times M_{u}\subset C(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{2})\). In addition, as it can be verified by means of the gradient-holonomic integrability scheme, the supplementing closed dynamical system (102) proves to be completely integrable on the functional manifold \(M_{p}\times M_{u}\). The latter makes it possible to formulate subject to the combined dynamical system (102) the following theorem. Theorem 4: _The combined nonlinear dynamical system (102) presents a completely integrable dark type flow on the extended functional manifold \(M_{p}\times M_{u}\), it is Lax type linearized and possesses an infinite number of suitably ordered conservation laws._  
The second interesting case arises when the density (98) is not trivial.