99 research outputs found

### Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov-Pinney equations of the
form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the
nonlinear dissipation g(v) is based on the corresponding Chiellini integrable
Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, general
solutions are obtained following the Abel equation route. Based on particular
solutions, we also provide general solutions containing a factor with the phase
of the Milne type. In addition, the same kinds of general solutions are
constructed for the cases of higher-order Reid nonlinearities. The Chiellini
dissipative function is actually a dissipation-gain function because it can be
negative on some intervals. We also examine the nonlinear case
h_0(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic
caseComment: 15 pages, 5 figures, 1 appendix, 21 references, published versio

### Dissipative periodic and chaotic patterns to the KdV--Burgers and Gardner equations

We investigate the KdV-Burgers and Gardner equations with dissipation and
external perturbation terms by the approach of dynamical systems and
Shil'nikov's analysis. The stability of the equilibrium point is considered,
and Hopf bifurcations are investigated after a certain scaling that reduces the
parameter space of a three-mode dynamical system which now depends only on two
parameters. The Hopf curve divides the two-dimensional space into two regions.
On the left region the equilibrium point is stable leading to dissapative
periodic orbits. While changing the bifurcation parameter given by the velocity
of the traveling waves, the equilibrium point becomes unstable and a unique
stable limit cycle bifurcates from the origin. This limit cycle is the result
of a supercritical Hopf bifurcation which is proved using the Lyapunov
coefficient together with the Routh-Hurwitz criterion. On the right side of the
Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using
Shil'nikov's theorem which requires the construction of a homoclinic orbit,
while for the Gardner equation the supercritical Hopf bifurcation leads only to
a stable periodic orbit.Comment: 13 pages, 12 figure

### Ermakov-Lewis Invariants and Reid Systems

Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant
alpha are equivalent to an integrable Emden-Fowler equation. The standard
Ermakov-Lewis invariant is discussed from this perspective, and a closed
formula for the invariant is obtained for the higher-order Reid systems (m\geq
3). We also discuss the parametric solutions of these systems of equations
through the integration of the Emden-Fowler equation and present an example of
a dynamical system for which the invariant is equivalent to the total energyComment: 8 pages, published versio

### Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between
the dissipative nonlinear second order differential equations and the Abel
equations which in its first kind form have only cubic and quadratic terms.
Then, employing an old integrability criterion due to Chiellini, we introduce
the corresponding integrable dissipative equations. For illustration, we
present the cases of some integrable dissipative Fisher, nonlinear pendulum,
and Burgers-Huxley type equations which are obtained in this way and can be of
interest in applications. We also show how to obtain Abel solutions directly
from the factorization of second-order nonlinear equationsComment: 6 pages, 7 figures, published versio

### Integrable Abel equations and Vein's Abel equation

We first reformulate and expand with several novel findings some of the basic
results in the integrability of Abel equations. Next, these results are applied
to Vein's Abel equation whose solutions are expressed in terms of the third
order hyperbolic functions and a phase space analysis of the corresponding
nonlinear oscillator is also providedComment: 12 pages, 4 figures, 17 references, online at Math. Meth. Appl. Sci.
since 7/28/2015, published 4/201

### Traveling Wave Solutions to Kawahara and Related Equations

Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using an elliptic function method which is more general than the tanh-method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to the KdV equation with only a cubic dispersion term, while for the KE which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. By solving the elliptic equation with coefficients that depend on the boundary conditions, velocity of the traveling waves, nonlinear strength, and dispersion coefficients, in the case of KdV equation we find the well-known solitary waves (solitons) for zero boundary conditions, as well as wave-trains of cnoidal waves for nonzero boundary conditions. Both solutions are either compressive (bright) or rarefactive (dark), and either propagate to the left or right with arbitrary velocity. In the case of KE with nonzero boundary conditions and zero cubic dispersion, we obtain cnoidal wave-trains which represent solutions to the TL equation. For KE with zero boundary conditions and all the dispersion terms present, we obtain again solitary waves, while for KE with all coefficients present and nonzero boundary condition, the solutions are written in terms of Weierstrass elliptic functions. For all cases of the KE we only find bright waves that are propagating to the right with velocity that is a function of both dispersion coefficients

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