Similarity reduction, nonlocal and master symmetries of sixth order Korteweg–deVries equation

Abstract

A systematic investigation to derive the Lie point symmetries, nonlocal and master symmetries of sixth order Korteweg–de Vries equation (KdV6) is presented. Using the obtained point symmetries, similarity reductions are derived and constructed their particular solutions wherever possible. It is shown that KdV6 admits infinitely many nonlocal and master symmetries. The existence of infinitely many master symmetries ensures that KdV6 is completely integrable.

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms

“…An adjusted version of this equation with real ψ has also already been presented, with Lax pair, conservation laws, and N-soliton solutions all generated as well [31]. However, along with its sixth-order relative for which symmetries have been investigated [32], it is regarded in this case as an extended Korteweg-de Vries equation, applicable to shallow-depth fluid studies. We shall instead provide the Lax pair for the full complex-valued quintic NLSE in Sec.…”

New bilinearization, Bäcklund transformation and infinite conservation laws for the KdV6 equation with Bell polynomials

“…Yao and Zeng further found that the deformed KdV6 equation is equivalent to the Rosochatius deformation of the KdV equation with a self‐consistent source . Sahadevan and Nalinidevi proved that the KdV6 equation admits infinitely many nonlocal and master symmetries . The bilinear representation and bilinear BT were given in references .…”

“…Its Lax pair and auto-Bäcklund transformation have been obtained via the truncated singular expansion method [25]. As KdV6 equation does not belong to any recognizable theory, its integrable properties and exact solutions captured the interests of many researchers [26][27][28][29][30][31][32][33][34][35]. Kupershmidt described the deformed KdV6 equation as a nonholonomic perturbations of bi-Hamiltonian systems [26].…”

Integrability of Certain Deformed Nonlinear Partial Differential Equations

“…(1.4) admits a bi-Hamiltonian representation while Ramani et al [15] have shown that it can be written in bilinear form. Recently we have shown that the coupled or deformed equations (1.4) possess other integrability structures such as the existence of infinitely many generalized symmetries, polynomial conserved quantities, nonlocal symmetries, master symmetries and a recursion operator [10,16] . In this article we report that the deformed NLS, Hirota and AKNS equations with (1 + 1) dimensions, respectively, given by …”