Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations converting the defined Lax pairs to some simpler forms and reducing the variety of the Lax pairs this way are found. The direct method and its possible applications to other types of evolution equations are discussed.

In this paper, it is shown that the bivariate beta statistical manifold of the first kind with three parameters has a gradient system Hamiltonian and is completely integrable. It is shown that this system admits a Lax pair representation. The question here is how to construct a gradient system and show that it is completely integrable while determining the integrable curves on the three-parameter bivariate beta manifold of the first kind admitting a potential function?To do this, we prove the existence of the potential on the manifold using ovidiu in Ovidiu, then using Amari's theorems in Amari, we show that this coordinate system admits a dual coordinate system. This will allow us to determine the Riemannian metric on the variety and to construct the gradient system. We linearise this system using Nakamura's method. We prove that it is Hamiltonian and completely integrable using the theorem in \cite{mama-proceeding}. It is shown that the Bivariate Beta of the first kind with three parameters, is an exponential function. The gradient system is linearizable. It is proven that the associate gradient system is Hamiltonian with $ \mathcal{H} $ which is in involution and satisfying $d\mathcal{H}=0.$ Therefore, the gradient system obtained is a sub-dynamical system of a $4$-dimensional system and is a completely integrable system. We show that the gradient system on the statistical model has the following Lax pair representation: $ \dot{L}= \left[L, N\right] $ where $L$ is a symmetric matrix and $N$ is the diagonal matrix. The gradient system defined by the Bivariate Beta family of the first kind with three parameters odd-dimensional manifold is a completely integrable Hamiltonian system.

The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: 1) Lax representation $[L,M]=0$; 2)~construction of fundamental analytic solutions (FAS); 3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) $\xi^+(x,t,\lambda) =\xi^-(x,t,\lambda) G(x,t\lambda)$ on a contour $\Gamma$ with sewing function $G(x,t,\lambda)$; 4) soliton solutions and possible applications.Step 1) involves several assumptions: the choice of the Lie algebra $\mathfrak{g}$ underlying $L$, as well as its dependence on the spectral parameter, typically linear or quadratic in $\lambda$. In the present paper we propose another approach which substantially extends the classes of integrable NLEE.Its first advantage is that one can effectively use any polynomial dependence in both $L$ and $M$. We use the following steps: A) Start with canonically normalized RHP with predefined contour $\Gamma$; B) Specify the $x$ and $t$ dependence of the sewing function defined on $\Gamma$; C) Introduce convenient parametrization for the solutions $ \xi^\pm(x,t,\lambda)$ of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); D) use Zakharov-Shabat dressing method to derive their soliton solutions. This needs correctly taking into account the symmetries of the RHP. E) Define the resolvent of the Lax operator and use it analyze its spectral properties.

We present three reduced integrable hierarchies of nonlocal integrable NLS type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. Three similarity transformations are taken to keep the invariance of the transformed zero curvature equations. The key step is to formulate a solution to a reduced stationary zero curvature equation so that the zero curvature formulation works for a reduced case.

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. Usually, such two-component integrable nonlinear evolutionary equations are considered, the form of which does not depend on the replacement of one component with another. In this paper, we propose for consideration and usage a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. Among the hierarchy equations there are analogues of the two-component nonlinear Schrödinger equation (second equation from hierarchy) and the two-component modified Korteweg-de Vries equation (fourth equation from hierarchy). The third equation of the hierarchy is a combination of the nonlinear Schrödinger equation for one component and the modified Korteweg-de Vries equation for the second component. The equations for the individual components are very different from each other, even if they have the same order. Let us note that even hierarchy equations can be reduced to well-known variants of the scalar derivative nonlinear Schrödinger equations, and odd equations can’t be reduced. To construct the hierarchy equations, we use the monodromy matrix method, first proposed by B.A. Dubrovin. Knowledge of the monodromy matrix makes it possible to construct spectral curves of multiphase solutions, as well as to find stationary equations that these solutions satisfy. The last section presents the simplest solutions in the form of solitons and periodic one-phase waves, as well as spectral curves corresponding to these solutions.

The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. For example, the duality issue for multi-criteria and vector optimization problems could be solved using the complete-lattice approach, compare [11]. So far, it has been applied to set-valued dynamic risk measures (in the stochastic case), as discussed in Feinstein, Rudloff etc. (see [11], for example), but it has not been applied to deterministic calculus of variations and optimal control problems. In this paper, the following problem of set-valued optimization is considered: minimize the functional $$ \overline J_t[y]=\int_0^t \overline L(s,y(s),\dot y(s))\ ds + U_0(y(0)) $$ over all admissible arcs $y$, where $\overline L$ is the associated multifunction to a vector-valued Lagrangian $L$, the integral is in the Aumann sense and $U_0$ is the initial cost. A new concept of \emph{value function}, for which a Bellman's optimality principle holds, is introduced. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time $t$ and a directional derivative with respect to $x$ of the value function are defined, based on ideas close to the concepts in [12]. The value function is proved to be solution of a suitable Hamilton-Jacobi equation.

This work deals with a numerical implementation of a fifth order CENTRAL WENO-TVD (\textit{Weighted Essentially Non-Oscillatory-Total Variation Dimimishing}) of Haschem (2006) scheme applied to the convective terms of some hyperbolic conservation laws problems, in a volume finite framework. The WENO-TVD scheme is used to solve the 1D advection and Burgers equations. For this case is implemented two different numerical fluxes: The Lax-Friedrichs and TVD fluxes. In the TVD fluxes the schemes applied are in flux-limiter form. The schemes implemented for this flux are: Van Albada-1 (van Albada et al.,1982), van Albada-2 (Kermani et al., 2003), van Leer (Hassanzadeh, 2009) and MINMOD (Hirsch, 2007). The WENO type schemes are characterized for their high order approximation, and do not produce spurious oscilations near discontinuities, shocks and higher gradients. A third order Runge-Kutta TVD for the temporal variable is used. Qualitative and quantitative comparison are presented. The numerical solutions are computed with an in-house computer code developed in MATLAB software. In future works, it will develope a paralelization of computer code for solving systems of conservation laws, e.g. Euler equations of gas dynamics.