Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces

On the well-posedness of the exp-Rabelo equation

“…1 We begin by proving that u, defined in (3.2), is an entropy solution of (1. 16) or (1.17) in the sense of Definition 1.3. Let φ ∈ C ∞ (R 2 ) be a positive text function with a support, and let us consider a compactly supported entropy-entropy flux pair (η, q).…”

“…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].…”

“…In [17], the authors used Bäcklund transformations derived in [15,16] in the construction of exact soliton solutions for some nonlinear evolution equations describing pseudo-spherical surfaces which are beyond the AKNS class. In particular, they analyzed the following equation [3]:…”

Travelling Wave Solutions of Kaup-Kupershmidt Equation Which Describes Pseudo Spherical Surfaces

“…We can cite the inverse scattering transform [6], the Bäcklund and Darboux transform [7]- [10], Hirota's bilinear method [11], the homogeneous balance method [12], Jacobi elliptic function method [13], the tanh method and extended tanh-function method [14]- [20], F-expansion method [21]- [23] and so on. The notion of conservation laws is important in the study of nonlinear evolution equations (NLEEs) appearing in mathematical physics [24].…”

Conservation Laws and Exact Solutions for some Nonlinear Partial Differential Equations

“…The symmetries of a NLPDE satisfy a linear determining equation, for which there is a standard algorithmic method (Mikhailov, 1981;Ablowitz and Clarkson, 1991;Bluman, 2001, 2002;Wolf, 2002;Khater et al, 2004a) to seek all solutions. There is also an invariance condition, involving just the NLPDE and its symmetries, which is necessary and sufficient for a symmetry of a NLPDE with a variational principle to correspond to a symmetry of the action.…”