Integrability of One Bilinear Equation: Singularity Analysis and Dimension

The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra D(1)4 is studied by means of the singularity analysis. This equation is shown to pass the Painlevé test in three distinct cases of its coefficients, exactly when the equation is effectively a three-dimensional one, equivalent to the BKP equation.

Complex singularity analysis for vortex layer flows

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier–Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff–Rott singularity seems to suggest that vortex layers, in the limit $Re\rightarrow \infty$ , behave differently from vortex sheets.

Optimum Seeking of Redundant Actuators for M-RCM 3-UPU Parallel Mechanism

The singularity problem brings troubles to the design and application for the parallel mechanism. Currently, redundant actuation is one of the useful methods to solve this singularity problem. However, faced to the numerous joints in a parallel mechanism, how to make a quantitative criterion of seeking the most efficient joints added actuators for letting the mechanism passes through singularity is a necessarily open issue. This paper focuses on a 2R1T 3-UPU (U for universal joint and P for prismatic joint) parallel mechanism (PM) with two rotational and one translational (2R1T) degrees of freedom (DOFs) and the ability of multiple remote centers of motion (M-RCM). The singularity analysis based on the indexes of motion/force transmissibility and constraint shows that this PM has transmission singularity, constraint singularity, mixed singularity and limb singularity. To solve these singular problems, the quantifiable redundancy transmission index (RTI) and the redundancy constraint index (RCI) are proposed for optimum seeking of redundant actuators for this PM. Then the appropriate redundant actuators are selected and the working scheme for redundant actuators near the corresponding singular configuration are given to help the PM passes through the singularity. This research proposes a quantitative criterion to optimum seeking of redundant actuators for the parallel mechanism to solve its singularity.

One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation

We apply the Lie theory to determine the infinitesimal generators of the one-parameter point transformations which leave invariant the 3 + 1 Kudryashov–Sinelshchikov equation. We solve the classification problem of the one-dimensional optimal system, while we derive all the possible independent Lie invariants; that is, we determine all the independent similarity transformations which lead to different reductions. For an application, the results are applied to prove the existence of travel-wave solutions. Furthermore, the method of singularity analysis is applied where we show that the 3 + 1 Kudryashov–Sinelshchikov equation possess the Painlevé property and its solution can be written by using a Laurent expansion.

On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain Ω belongs to R^N (N >= 2) for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary \partial Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

Kinematics and Singularity Analysis of a 7-DOF Redundant Manipulator

A new method of kinematic analysis and singularity analysis is proposed for a 7-DOF redundant manipulator with three consecutive parallel axes. First, the redundancy angle is described according to the self-motion characteristics of the manipulator, the position and orientation of the end-effector are separated, and the inverse kinematics of this manipulator is analyzed by geometric methods with the redundancy angle as a constraint. Then, the Jacobian matrix is established to derive the conditions for the kinematic singularities of the robotic arm by using the primitive matrix method and the block matrix method. Then, the kinematic singularities conditions in the joint space are mapped to the Cartesian space, and the singular configuration is described using the end poses and redundancy angles of the robotic arm, and a singularity avoidance method based on the redundancy angles and end pose is proposed. Finally, the correctness and feasibility of the inverse kinematics algorithm and the singularity avoidance method are verified by simulation examples.