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By the nonlocal symmetry approach, Hernández-Heredero and Reyes [J. Phys. A: Math. Theor. 42, 182002 (2009)] and Bies et al. [J. Math. Phys. 53, 073710 (2012)] obtained Darboux Transformations (DTs) of the Camassa–Holm equation and the modified Camassa–Holm equation. However, the wave function does not appear in their DTs explicitly. We introduce wave functions to the DTs and show how they are related to the binary DT of the first negative flow in the Korteweg–de Vries (KdV) hierarchy. Furthermore, we connect nonlocal symmetries of the Camassa–Holm equation and the modified Camassa–Holm equation with those of the negative KdV equation and the negative modified KdV equation, respectively. [ABSTRACT FROM AUTHOR]

In an external cavity diode laser, the rate-dependent nonlinearity of the piezoelectric actuator based optical resonator cavity (ORC) restricts its application in high-precision measurements. This paper proposes a rate-dependent nonlinear active suppression method for an ORC system based on the Hammerstein model. Using a back propagation neural network and system identification algorithm, a high-precision rate-dependent Hammerstein model of the ORC system is established. The B-spline curve is used to simplify the input voltage curve, and the optimized voltage curve that can obtain the linear displacement output is solved by the particle swarm optimization algorithm. The experimental results show that in the frequency range of 2–50 Hz, the regression coefficients (R2) of the ascent and descent stages at each experimental frequency of the ORC system increased to over 0.9998 and optimized the curve at the inflection point to suppress the vibration at the inflection point. [ABSTRACT FROM AUTHOR]

Complex fluids always possess obvious non-Newtonian properties that facilitate the occurrence and development of vortex instability in porous media, which is of critical significance in many natural and industrial processes. It is widely known that this flow instability is regulated by both fluid flow and solid structure. However, the quantitative understanding of how structural characteristics of porous space affect the evolution of vortex instability is still nascent, especially in the case of fluids with varying rheological properties. Herein, the flow of polymer solutions with distinct non-Newtonian properties through microcavity arrays is experimentally studied, by which we systematically explore the effect of structural parameters of the cavity array on vortex instability. We find that, for both Newtonian and shear-thinning fluids with negligible elasticity, the vortex evolution behavior in each cavity of the cavity array is identical to those in an isolated cavity. In contrast, for viscoelastic fluids, the vortex instability is visibly affected by cavity number and cavity–cavity interval, and this effect exhibits different forms when the fluid shear-thinning participates or not. Multiplex vortex instabilities are observed under these tested conditions. By multiplex, we mean the vortex formation dynamics and evolution patterns are diversified. These unusual evolution phenomena are then interpreted in terms of the interplay between the elongation and relaxation of polymers as they navigate among neighboring cavities. These results can help us to further understand the flow instability of complex fluids in porous media and evoke new strategies for microfluidic applications of efficient mixing. [ABSTRACT FROM AUTHOR]

Reciprocal transformations are introduced for two super Camassa–Holm (CH) equations. Under these transformations and appropriate changes of dependent variables, the super CH equation, proposed by Geng et al. [Stud. Appl. Math. 130, 1 (2013)], is converted to a negative member of the super Korteweg–de Vries (KdV) hierarchy studied by Geng and Wu in 2010 [Appl. Math. Lett. 23, 716 (2010)], while the other super CH equation, due to Zhang and Zuo [J. Math. Phys. 52, 073503 (2011)], is related to a new super KdV hierarchy. In the latter case, algebraic properties of this new super KdV hierarchy are established, including Hamiltonian operators, a recursion operator, and conserved quantities. [ABSTRACT FROM AUTHOR]

The N = 2 supercomformal transformations are employed to study supersymmetric integrable systems. It is proved that two known N = 2 supersymmetric Harry Dym equations are transformed into two N = 2 supersymmetric modified Korteweg-de Vries equations, thus are connected with two N = 2 supersymmetric Korteweg-de Vries equations. [ABSTRACT FROM AUTHOR]

A linear eigenvalue problem is constructed for the gradients of the eigenvalue of the Wadati-Konno-Ichikawa (WKI) spectral problem. This in turn leads to a compatible pair of Hamiltonian operators, which endows the WKI hierarchy with a bi-Hamiltonian structure. Furthermore, a proper recombination of these two Hamiltonian operators, or the duality of this bi-Hamiltonian structure, generates a coupled system of Camassa-Holm type, which is nothing but a system proposed by Song, Qu, and Qiao [J. Math. Phys. 52, 013503 (2011)] recently. [ABSTRACT FROM AUTHOR]

The supersymmetric analog of the reciprocal transformation is introduced. This is used to establish a transformation between one of the supersymmetric Harry Dym equations and the supersymmetric modified Korteweg-de Vries equation. The reciprocal transformation, as a Bäcklund-type transformation between these two equations, is adopted to construct a recursion operator for the supersymmetric Harry Dym equation. By proper factorization of the recursion operator, a bi-Hamiltonian structure is found for the supersymmetric Harry Dym equation. Furthermore, a supersymmetric Kawamoto equation is proposed and is associated with the supersymmetric Sawada-Kotera equation. The recursion operator and odd bi-Hamiltonian structure of the supersymmetric Kawamoto equation are also constructed. [ABSTRACT FROM AUTHOR]