We construct integrable hierarchies of flows for curves in centroaffine R through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in RP induces...

Defect states in different dielectric materials with crystalline and amorphous structures were investigated with two models, the Broer model and the Bloch model. For crystalline structures as in hafnium oxide, defects were found to fit well with the Broer model due to strong local magnetic-field-orientation dependency. In amorphous materials, for example, organosilicate glass, defects were foun...

We develop a Cauchy matrix reduction technique that enables us to obtain solutions for the reduced local and nonlocal complex equations from of original nonreduced systems. Specifically, by imposing reductions on some Ablowitz–Kaup–Newell–Segur-type equations, we study involving modified Korteweg–de Vries equation, sine-Gordon potential nonlinear Schrödinger equation. matrix-type soliton Jordan...

By using gauge transformations, we manage to obtain new solutions of (2+1)-dimensional Kadomtsev-Petviashvili(KP), Kaup-Kuperschmidt(KK) and Sawada-Kotera(SK) equations from non-zero seeds. For each the preceding equations, a Galilean type transformation between these $u_2$ previously known $u_2^{\prime}$ generated zero seed is given. We present several explicit formulas single-soliton for $u_2...

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton this equation are derived by solving determining set. Furthermore, applying an appropriate continuum limit we obtain semidiscrete AKNS and after second derive the nonlinear The reductions discrete, semi-discrete continuous s...

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutio...