67 research outputs found
On certain new exact solutions of a diffusive predator-prey system
We construct exact solutions for a system of two nonlinear partial
differential equations describing the spatio-temporal dynamics of a
predator-prey system where the prey per capita growth rate is subject to the
Allee effect. Using the expansion method, we derive
exact solutions to this model for two different wave speeds. For each wave
velocity we report three different forms of solutions. We also discuss the
biological relevance of the solutions obtained.Comment: Accepted for Publication in Commun. Nonlin. Sci. Num. Sim. (2012
On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schr\"{o}dinger equations with distributed coefficients
We construct vector rogue wave solutions of the two-dimensional two coupled
nonlinear Schr\"{o}dinger equations with distributed coefficients, namely
diffraction, nonlinearity and gain parameters through similarity transformation
technique. We transform the two-dimensional two coupled variable coefficients
nonlinear Schr\"{o}dinger equations into Manakov equation with a constraint
that connects diffraction and gain parameters with nonlinearity parameter. We
investigate the characteristics of the constructed vector rogue wave solutions
with four different forms of diffraction parameters. We report some interesting
patterns that occur in the rogue wave structures. Further, we construct vector
dark rogue wave solutions of the two-dimensional two coupled nonlinear
Schr\"{o}dinger equations with distributed coefficients and report some novel
characteristics that we observe in the vector dark rogue wave solutions.Comment: Accepted for publication in The European Physical Journal
Asymptotic dynamics of short-waves in nonlinear dispersive models
The multiple-scale perturbation theory, well known for long-waves, is
extended to the study of the far-field behaviour of short-waves, commonly
called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation
can propagates short-waves. This result contradict the Benjamin hypothesis that
short-waves tends not to propagate in this model and close a part of the old
controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine
equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation
substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation
of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon
equations are understood as an all orders asymptotic behaviour of short-waves.
It is proved that the antikink solution of phi-4 model which was never obtained
perturbatively can be obtained by perturbation expansion in the wave-number k
in the short-wave limit.Comment: to appears in Physical Review E. 4 pages, revtex file
Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation
By considering the long-wave limit of the regularized long wave (RLW)
equation, we study its multiple-time higher-order evolution equations. As a
first result, the equations of the Korteweg-de Vries hierarchy are shown to
play a crucial role in providing a secularity-free perturbation theory in the
specific case of a solitary-wave solution. Then, as a consequence, we show that
the related perturbative series can be summed and gives exactly the
solitary-wave solution of the RLW equation. Finally, some comments and
considerations are made on the N-soliton solution, as well as on the
limitations of applicability of the multiple scale method in obtaining uniform
perturbative series.Comment: 15 pages, RevTex, no figures (to appear in Phys. Rev. E
Dissipative Boussinesq System of Equations in the B\'enard-Marangoni Phenomenon
By using the long-wave approximation, a system of coupled evolution equations
for the bulk velocity and the surface perturbations of a B\'enard-Marangoni
system is obtained. It includes nonlinearity, dispersion and dissipation, and
it can be interpreted as a dissipative generalization of the usual Boussinesq
system of equations. As a particular case, a strictly dissipative version of
the Boussinesq system is obtained. Finnaly, some speculations are made on the
nature of the physical phenomena described by this system of equations.Comment: 15 Pages, REVTEX (Version 3.0), no figure
Perturbative analysis of wave interactions in nonlinear systems
This work proposes a new way for handling obstacles to asymptotic
integrability in perturbed nonlinear PDEs within the method of Normal Forms -
NF - for the case of multi-wave solutions. Instead of including the whole
obstacle in the NF, only its resonant part is included, and the remainder is
assigned to the homological equation. This leaves the NF intergable and its
solutons retain the character of the solutions of the unperturbed equation. We
exploit the freedom in the expansion to construct canonical obstacles which are
confined to te interaction region of the waves. Fo soliton solutions, e.g., in
the KdV equation, the interaction region is a finite domain around the origin;
the canonical obstacles then do not generate secular terms in the homological
equation. When the interaction region is infifnite, or semi-infinite, e.g., in
wave-front solutions of the Burgers equation, the obstacles may contain
resonant terms. The obstacles generate waves of a new type, which cannot be
written as functionals of the solutions of the NF. When an obstacle contributes
a resonant term to the NF, this leads to a non-standard update of th wave
velocity.Comment: 13 pages, including 6 figure
Boussinesq Solitary-Wave as a Multiple-Time Solution of the Korteweg-de Vries Hierarchy
We study the Boussinesq equation from the point of view of a multiple-time
reductive perturbation method. As a consequence of the elimination of the
secular producing terms through the use of the Korteweg--de Vries hierarchy, we
show that the solitary--wave of the Boussinesq equation is a solitary--wave
satisfying simultaneously all equations of the Korteweg--de Vries hierarchy,
each one in an appropriate slow time variable.Comment: 12 pages, RevTex (to appear in J. Math Phys.
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