24 research outputs found
Conservation laws for some compacton equations using the multiplier approach
AbstractThis paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers
Symmetry and double reduction for exact solutions of selected nonlinear partial differential equations
Amongst the several analytic methods available to obtain exact solutions of non-linear differential equations, Lie symmetry reduction and double reduction technique are proven to be most effective and have attracted researcher from different areas to utilize these methods in their research. In this research, Lie symmetry analysis and double reduction are used to find the exact solutions of nonlinear differential equations. For Lie symmetry reduction method, symmetries of differential equation will be obtained and hence invariants will be obtained, thus differential equation will be reduced and exact solutions are calculated. For the method of double reduction, we first find Lie symmetry, followed by conservation laws using ‘Multiplier’ approach. Finally, possibilities of associations between symmetry with conservation law will be used to reduce the differential equation, and thereby solve the differential equation. These methods will be used on some physically very important nonlinear differential equations; such as Kadomtsev- Petviashvili equation, Boyer-Finley equation, Short Pulse Equation, and Kortewegde Vries-Burgers equations. Furthermore, verification of the solution obtained also will be done by function of PDETest integrated in Maple or comparison to exist literature
Exotic traveling waves for a quasilinear Schr\"odinger equation with nonzero background
We study a defocusing quasilinear Schr\"odinger equation with nonzero
conditions at infinity in dimension one. This quasilinear model corresponds to
a weakly nonlocal approximation of the nonlocal Gross--Pitaevskii equation, and
can also be derived by considering the effects of surface tension in
superfluids. When the quasilinear term is neglected, the resulting equation is
the classical Gross-Pitaevskii equation, which possesses a well-known stable
branch of subsonic traveling waves solution, given by dark solitons.
Our goal is to investigate how the quasilinear term affects the
traveling-waves solutions. We provide a complete classification of finite
energy traveling waves of the equation, in terms of the two parameters: the
speed and the strength of the quasilinear term. This classification leads to
the existence of dark and antidark solitons, as well as more exotic localized
solutions like dark cuspons, compactons, and composite waves, even for
supersonic speeds. Depending on the parameters, these types of solutions can
coexist, showing that finite energy solutions are not unique. Furthermore, we
prove that some of these dark solitons can be obtained as minimizers of the
energy, at fixed momentum, and that they are orbitally stable
Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations
Different approaches to construct first integrals for ordinary differential equations
and systems of ordinary differential equations are studied here. These
approaches can be grouped into three categories: direct methods, Lagrangian
or partial Lagrangian formulations, and characteristic (multipliers) approaches.
The direct method and symmetry conditions on the first integrals correspond to
first category. The Lagrangian and partial Lagrangian include three approaches:
Noether’s theorem, the partial Noether approach, and the Noether approach for
the equation and its adjoint as a system. The characteristic method, the multiplier
approaches, and the direct construction formula approach require the
integrating factors or characteristics or multipliers. The Hamiltonian version of
Noether’s theorem is presented to derive first integrals. We apply these different
approaches to derive the first integrals of the harmonic oscillator equation. We also
study first integrals for some physical models. The first integrals for nonlinear
jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic
system with quadratic nonlinearities are derived. Moreover, solutions via first
integrals are also constructed
Generalized curvature-matter couplings in modified gravity
In this work, we review a plethora of modified theories of gravity with
generalized curvature-matter couplings. The explicit nonminimal couplings, for
instance, between an arbitrary function of the scalar curvature and the
Lagrangian density of matter, induces a non-vanishing covariant derivative of
the energy-momentum tensor, implying non-geodesic motion and consequently leads
to the appearance of an extra force. Applied to the cosmological context, these
curvature-matter couplings lead to interesting phenomenology, where one can
obtain a unified description of the cosmological epochs. We also consider the
possibility that the behavior of the galactic flat rotation curves can be
explained in the framework of the curvature-matter coupling models, where the
extra-terms in the gravitational field equations modify the equations of motion
of test particles, and induce a supplementary gravitational interaction. In
addition to this, these models are extremely useful for describing dark
energy-dark matter interactions, and for explaining the late-time cosmic
acceleration.Comment: 55 pages, to appear as a review paper in a Special Issue of Galaxies:
"Beyond Standard Gravity and Cosmology". V2: minor corrections and references
added. Matches published versio