1,980 research outputs found
Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications
In this paper we discuss some exact results related to the fractional
Klein--Gordon equation involving fractional powers of the D'Alembert operator.
By means of a space-time transformation, we reduce the fractional Klein--Gordon
equation to a fractional hyper-Bessel-type equation. We find an exact analytic
solution by using the McBride theory of fractional powers of hyper-Bessel
operators. A discussion of these results within the framework of linear
dispersive wave equations is provided. We also present exact solutions of the
fractional Klein-Gordon equation in the higher dimensional cases. Finally, we
suggest a method of finding travelling wave solutions of the nonlinear
fractional Klein-Gordon equation with power law nonlinearities
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
Coupled KdV equations derived from atmospherical dynamics
Some types of coupled Korteweg de-Vries (KdV) equations are derived from an
atmospheric dynamical system. In the derivation procedure, an unreasonable
-average trick (which is usually adopted in literature) is removed. The
derived models are classified via Painlev\'e test. Three types of
-function solutions and multiple soliton solutions of the models are
explicitly given by means of the exact solutions of the usual KdV equation. It
is also interesting that for a non-Painlev\'e integrable coupled KdV system
there may be multiple soliton solutions.Comment: 19 pages, 2 figure
Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices
The longstanding problem of moving discrete solitary waves in nonlinear
Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal
lattices with saturable nonlinearity whose grand-canonical energy barrier
vanishes for isolated coupling strength values. {\em Genuinely localised
travelling waves} are computed as a function of the system parameters {\it for
the first time}. The relevant solutions exist only for finite velocities.Comment: 5 pages, 4 figure
Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein--Gordon lattices
We construct small amplitude breathers in 1D and 2D Klein--Gordon infinite
lattices. We also show that the breathers are well approximated by the ground
state of the nonlinear Schroedinger equation. The result is obtained by
exploiting the relation between the Klein Gordon lattice and the discrete Non
Linear Schroedinger lattice. The proof is based on a Lyapunov-Schmidt
decomposition and continuum approximation techniques introduced in [7],
actually using its main result as an important lemma
Towards soliton solutions of a perturbed sine-Gordon equation
We give arguments for the existence of {\it exact} travelling-wave (in
particular solitonic) solutions of a perturbed sine-Gordon equation on the real
line or on the circle, and classify them. The perturbation of the equation
consists of a constant forcing term and a linear dissipative term. Such
solutions are allowed exactly by the energy balance of these terms, and can be
observed experimentally e.g. in the Josephson effect in the theory of
superconductors, which is one of the physical phenomena described by the
equation.Comment: 16 pages, 4 figures include
Numerical study of the generalised Klein-Gordon equations
24 pages, 10 figures, 56 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets
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