We have derived a family of equations related to the untwisted affine Lie algebras $A_r^{\left(1\right)}$ using a Coxeter ℤ_r+1 reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. Applying additional ℤ₂-involutions, we also obtain the real Hamiltonian forms of these equations. We also give some particular examples and impose additional reductions.

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 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 that 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 requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties.

Acquiring adequate sensory information and using it to provide motor control are important issues in the process of creating walking robots. The objective of this article is to present control algorithms for the optimization of the walking cycle of an innovative walking robot named “Big Foot”. The construction of the robot is based on minimalist design principles—only two motors are used, with which Big Foot can walk and even overcome obstacles. It is equipped with different types of sensors, with some of them providing information necessary for the realization of an optimized walk cycle. We examine two laws of motion—sinusoidal and polynomial—where we compare the results with constant angular velocity motion. Both proposed laws try to find balance between minimizing shock loads and maximizing walking speed for a given motor power. Experimental results are derived with the help of a 3D-printed working prototype of the robot, with the correct realization of the laws of motion being ensured by the use of a PD controller receiving data from motor encoders and tactile sensors. The experimental results validate the proposed laws of motion and the results can be applied to other walking robots with similar construction.

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 derive a new exactly solvable multi-component system of non-linear evolution equations (NLEEs). The system consists of three $1+1$-dimensional evolution equations—one first-order and two second-order in the spatial variable. We review their Lax representation, formulate the scattering problem, and derive the soliton-like solutions of the system.

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra $A_2^{\left(1\right)}$. The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair $L,M$.