391 research outputs found
Symmetry Properties of a Generalized Korteweg-de Vires Equation and some Explicit Solutions
The symmetry group method is applied to a generalized Korteweg-de Vries
equation and several classes of group invarint solution for it are obtained by
means of this technique. Polynomial, trigonometric and elliptic function
solutions can be calculated. It is shown that this generalized equation can be
reduced to a first-order equation under a particular second-order differential
constraint which resembles a Schrodinger equation. For a particular instance in
which the constraint is satisfied, the generalized equation is reduced to a
quadrature. A condition which ensures that the reciprocal of a solution is also
a solution is given, and a first integral to this constraint is found
Connections of Zero Curvature and Applications to Nonlinear Partial Differential Equations
A general formulation of zero curvature connections in a principle bundle is
presented and some applications are discussed. It is proved that a related
connection based on a prolongation in an associated bundle remains zero
curvature as well. It is also shown that the connection coefficients can be
defined so that the partial differential equation to be studied appears as the
curvature term in the structure equations. It is discussed how Lax pairs and
Backlund transformations can be formulated for such equations. It is discussed
how Lax pairs and Backlund transformations can be formulated for such equations
that occur as zero curvature terms.Comment: 2
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