Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Abstract

Abstract. Two-component second and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New symmetry integrable systems with their master symmetries are obtained. Some third order systems are observed to possess conservation laws. Bi-Poisson structures of systems possessing conservation laws are given.

Master symmetries of non-linear systems

“…Notice that the required constant of integrations are 0 so that the following conditions are satisfied So far all the two-component systems that we have considered have appeared in [17]. The following two examples are from current work (in progress) of Wang et al [25].…”

“…Mikhailov, Shabat, and Yamilov in [21] initiated the systematic classification of multi-component integrable equations. Talati and Tuhan provided a classification for (1, 1)-homogeneous systems in [17] that were previously unclassified with respect to the matrix of leading order terms in [22][23][24]. We will compute master symmetries for three systems in Examples 2, 3 and 4.…”

“…In their paper [17], Talati and Turhan classified systems of the Burger type that feature a non-diagonal constant matrix for the leading-order term. The first three examples in section 2 we examine in this work were introduced as new systems in their study.…”

Some new integrable systems of two-component fifth-order equations

“…Solutions of the compatibility condition are given in the following theorems. Theorem 2.1 A coupled fifth-order system of two-component evolution equations of the forms (13) and (14) that possesses a seventh-order generalized symmetry of form (13) with γ 1 = γ 2 = 1 have a lower order symmetry or can be transformed by a linear change of variables to one of the following two systems (11) and (12). Theorem 2.2 Every coupled fifth-order system of two-component evolution equations of form (13) and (14) that possesses a seventh-order generalized symmetry of form (13) with γ 1 = 1, γ 2 = 0 have a lower order symmetry.…”

“…Here we mention papers pertaining to multi-component generalizations of fifth order systems only. For the other integrable systems and their properties, we refer the readers to the useful papers [5,1,4,11] and the some of the references therein. So far the only known integrable systems of fifth order two-component equations are as follows…”