Two new classes of exact solutions for the KdV equation via Bäcklund transformations

On the well-posedness of the exp-Rabelo equation

“…In [15,16], the authors used the notion of differential equation for a function u(t, x) that describes a pseudo-spherical surface, and they derived some Bäcklund transformations for nonlinear evolution equations which are the integrability condition sl(2, R)-valued linear problems [13,14,18,19,26].…”

Bäcklund transformation, auto-Bäcklund transformation and exact solutions for KdV-type equations

“…In the past decades, there has been significant progression in the development of methods such as the inverse scattering method [1][2], Hirotas bilinear method [3], Painlevé expansions [4][5][6][7], truncated Painlevé [8], homogeneous balance method [9][10], the linearized transformation method [11], tanh function method [12][13][14][15]and several ansatz method [16][17]. The Bäcklund transformation (BT) technique is one of the direct methods for generating a new solution of NLEEs from a known solution of that equation (see, for example, [18][19][20]). Previously, Konno and Wadati [21], for example, had derived some BTs for the NEEs of the AKNS class [2,21].…”

Conservation law and exact solutions for ion acoustic waves in a magnetized quantum plasma

“…Methods used to construct the analytic and exact solutions of the NLPDEs have been proposed, such as the Hirota bilinear method [33], Bäcklund transformation (BT) [34][35], inverse scattering transform [36], and Painlevé analysis [37][38][39]. BT is the connection of several analytic solutions and auto-BT can connect the different solutions of the same NLPDE [40][41][42][43].…”