26 research outputs found
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
Dispersive shock waves and modulation theory
There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs
Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation
Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo
Does solitary wave solution persist for the long wave equation with small perturbations?
In this paper, persistence of solitary wave solutions of the regularized long
wave equation with small perturbations are investigated by the geometric
singular perturbation theory. Two different kinds of the perturbations are
considered in this paper: one is the weak backward diffusion and dissipation,
the other is the Marangoni effects. Indeed, the solitary wave persists under
small perturbations. Furthermore, the different perturbations do affect the
proper wave speed ensuring the persistence of the solitary waves. Finally,
numerical simulations are utilized to confirm the theoretical results
Solitons, Breathers and Rogue Waves in Nonlinear Media
In this thesis, the solutions of the Nonlinear Schrödinger
equation (NLSE) and its
hierarchy are studied extensively. In nonlinear optics, as the
duration of optical pulses
get shorter, in highly nonlinear media, their dynamics become
more complex, and,
as a modelling equation, the basic NLSE fails to explain their
behaviour. Using the
NLSE and its hierarchy, this thesis explains the ultra-short
pulse dynamics in highly
nonlinear media. To pursue this purpose, the next higher-order
equations beyond
the basic NLSE are considered; namely, they are the third order
Hirota equation and
the fifth order quintic NLSE. Solitons, breathers and rogue wave
solutions of these
two equations have been derived explicitly. It is revealed that
higher order terms
offer additional features in the solutions, namely, ‘Soliton
Superposition’, ‘Breather
Superposition’ and ‘Breather-to-Soliton’ conversion.
How robust are the rogue wave solutions against perturbations? To
answer this
question, two types of perturbative cases have been considered;
one is odd-asymmetric
and the other type is even-symmetric. For the odd-asymmetric
perturbative case,
combined Hirota and Sasa-Satsuma equations are considered, and
for the latter case,
fourth order dispersion and a quintic nonlinear term combined
with the NLSE are
considered. Indeed, this thesis shows that rogue waves survive
these perturbations
for specific ranges of parameter values.
The integrable Ablowitz-Ladik (AL) equation is the discrete
counterpart of the
NLSE. If the lattice spacing parameter goes to zero, the discrete
AL becomes the
continuous NLSE. Similar rules apply to their solutions. A list
of corresponding
solutions of the discrete Ablowitz-Ladik and the NLSE has been
derived. Using
associate Legendre polynomial functions, sets of solutions have
been derived for the
coupled Manakov equations, for both focusing and defocusing
cases. They mainly
explain partially coherent soliton (PCS) dynamics in Kerr-like
media. Additionally,
corresponding approximate solutions for two coupled NLSE and AL
equations have
been derived. For the shallow water case, closed form breathers,
rational and
degenerate solutions of the modified Kortweg-de Vries equation
are also presented
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Nonlinear classical and quantum integrable systems with PT -symmetries
A key feature of integrable systems is that they can be solved to obtain exact analytical solutions. In this thesis we show how new models can be found through generalisations of some well known nonlinear partial differential equations including the Korteweg-de Vries, modified Korteweg-de Vries, sine-Gordon, Hirota, Heisenberg and Landau-Lifschitz types with joint parity and time symmetries whilst preserving integrability properties.
The first joint parity and time symmetric generalizations we take are extensions to the complex and multicomplex fields, such as bicomplex, quaternionic, coquaternionic and octonionic types. Subsequently, we develop new methods from well-known ones, such as Hirota’s direct method, Bäcklund transformations and Darboux-Crum transformations to solve for these newsystems to obtain exact analytical solutions of soliton and multi-soliton types. Moreover, in agreement with the reality property present in joint parity and time symmetric non-Hermitian quantum systems, we find joint parity and time symmetries also play a key role for reality of conserved charges for the new systems, even though the soliton solutions are complex or multicomplex.
Our complex extensions have proved to be successful in helping one to obtain regularized degenerate multi-soliton solutions for the Korteweg-de Vries equation, which has not been realised before. We extend our investigations to explore degenerate multi-soliton solutions for the sine-Gordon equation and Hirota equation. In particular, we find the usual time-delays from degenerate soliton solution scattering are time-dependent, unlike the non-degenerate multi-soliton solutions, and provide a universal formula to compute the exact time-delay values for scattering of N-soliton solutions.
Other joint parity and time symmetric extensions of integrable systems we take are of nonlocal nature, with nonlocalities in space and/or in time, of time crystal type. Whilst developing new methods for the construction of soliton solutions for these systems, we xiv find new types of solutions with different parameter dependence and qualitative behaviour even in the one-soliton solution cases. We exploit gauge equivalence between the Hirota system with continuous Heisenberg and Landau-Lifschitz systems to see how nonlocality is inherited from one system to another and vice versa. In the final part of the thesis, we extend some of our investigations to the quantum regime. In particularwe generalize the scheme of Darboux transformations for fully timedependent non-Hermitian quantum systems, which allows us to create an infinite tower of solvable models