35 research outputs found
Meromorphic exact solutions of the generalized Bretherton equation
The generalized Bretherton equation is studied. The classification of the
meromorphic traveling wave solutions for this equation is presented. All
possible exact solutions of the generalized Brethenton equation are given
Exact solutions of equations for the Burgers hierarchy
Some classes of the rational, periodic and solitary wave solutions for the
Burgers hierarchy are presented. The solutions for this hierarchy are obtained
by using the generalized Cole - Hopf transformation
Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations
We consider a family of non-autonomous second-order differential equations,
which generalizes the Li\'enard equation. We explicitly find the necessary and
sufficient conditions for members of this family of equations to admit
quadratic, with the respect to the first derivative, first integrals. We show
that these conditions are equivalent to the conditions for equations in the
family under consideration to possess Lax representations. This provides a
connection between the existence of a quadratic first integral and a Lax
representation for the studied dissipative differential equations, which may be
considered as an analogue to the theorem that connects Lax integrability and
Arnold--Liouville integrability of Hamiltonian systems. We illustrate our
results by several examples of dissipative equations, including generalizations
of the Van der Pol and Duffing equations, each of which have both a quadratic
first integral and a Lax representation.Comment: To appear in Journal of Mathematical Analysis and Application