Approximate symmetry analysis and approximate conservation laws of perturbed KdV equation (Q1629221)

In this paper, with the application of approximate symmetry method and partial Lagrangian method, the authors consider the perturbed Korteweg-de Vries (KdV) equation : \[ u_t-6uu_x+u_{xxx}+\varepsilon\alpha(u+2xu_x)-\varepsilon\beta(2u+xu_x)=0, \] where \(\varepsilon\) is a small parameter, while \(\alpha\) and \(\beta\) are arbitrary constants. The authors show its all first-order approximate symmetries. Furthermore, they will also construct several approximate solutions and approximate conservation laws of the equation. This paper is organized as follows : the first section is an introduction to the subject. Section 2 gives some basic concepts and notations. Section 3 analyzes approximate symmetries of the perturbed KdV equation by applying the approximate symmetry method, developed by \textit{V. A. Baĭkov} et al. [Math. USSR, Sb. 64, No. 2, 427--441 (1989; Zbl 0683.35004); translation from Mat. Sb., Nov. Ser. 136(178), No. 4(8), 435--450 (1988)]. The authors compute the optimal system of the presented approximate symmetries. In Section 4 they generate the approximate invariants of presented approximate symmetries and construct corresponding approximately invariant solutions. Section 5 presents approximate conservation laws by using partial Lagrangian method.