Classification and Recursion Operators of Dark Burgers’ Equation

Abstract

With the help of symbolic computation, two types of complete scalar classification for dark Burgers’ equations are derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark Burgers’ systems; so some special equations including symmetry equation and dual symmetry equation are obtained by selecting the free parameter. Furthermore, two kinds of recursion operators for these dark Burgers’ equations are constructed by two direct assumption methods.

Quasi-linearization and stability analysis of some self-dual, dark equations and a new dynamical system †

“…Some twenty years ago, a new class of nonlinear dynamical systems, called 'dark equations' was introduced by Boris Kupershmidt [1,2], and shown to possess unusual properties that were not particularly well-understood at that time. Later, in related developments, some Burgers-type [3][4][5] and also Korteweg-de Vries type [6,7] dynamical systems were studied in detail, and it was proved that they have a finite number of conservation laws, a linearization and degenerate Lax representations, among other properties. In what follows, we provide a description of a class of self-dual dark-type (or just, dark, for short) nonlinear dynamical systems, which a priori allows their quasi-linearization, whose integrability can be effectively studied by means of a geometrically motivated [8,[9][10][11] gradient-holonomic approach [12][13][14].…”

“…Remark 4.4. It is worth mentioning here that the self-dual dynamical system (3.17) as well as those (4.14) constructed above, are not contained within the usual Burgers type hierarchy, which can be easily checked by making use of the analysis done before in[4,7].…”

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

“…Not only because they can help us to understand physical phenomena they describe in nature, but also because they can serve as benchmarks for checking and improving numerical codes developed for studying more complex problems. Therefore, a lot of powerful methods has been developed, such as inverse scattering method, Hirota direct method, Darboux transformation and Bäcklund transformation et al [1][2][3][4][5][6][7][8][9][10]. However, none of these methods is universal due to the diversity and complexity of PDEs.…”