251 research outputs found
Multiple lump solutions and their interactions for an integrable nonlinear dispersionless PDE in vector fields
In this article, lump solutions, lump with I-kink, lump with II- kink, periodic, multiwaves, rogue waves and several other interactions such as lump interaction with II-kink, interaction between lump, lump with I-kink and periodic, interaction between lump, lump with II-kink and periodic are derived for Pavlov equation by using appropriate transformations. Additionally, we also present 3-dimensional, 2-dimensional and contour graphs for our solutions
Extractions of some new travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation
In this paper, complex and combined dark-bright characteristic properties of nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable are extracted by using two powerful analytical approaches. Many graphical representations such as 2D, 3D and contour are also reported. Finally, general conclusions of about the novel findings are introduced at the end of this manuscript
d-dimensional Oscillating Scalar Field Lumps and the Dimensionality of Space
Extremely long-lived, time-dependent, spatially-bound scalar field
configurations are shown to exist in spatial dimensions for a wide class of
polynomial interactions parameterized as . Assuming spherical symmetry and if
for a range of values of , such configurations exist if: i) spatial
dimensionality is below an upper-critical dimension ; ii) their radii are
above a certain value . Both and are uniquely
determined by . For example, symmetric double-well potentials only
sustain such configurations if and . Asymmetries may modify the value of . All
main analytical results are confirmed numerically. Such objects may offer novel
ways to probe the dimensionality of space.Comment: In press, Physics Letters B. 6 pages, 2 Postscript figures, uses
revtex4.st
The -dressing Method for Two (2+1)-dimensional Equations and Combinatorics
The soliton solutions of two different (2+1)-dimensional equations with the
third order and second order spatial spectral problems are studied by the
-dressing method. The 3-reduced and 5-reduced equations of
CKP equation are obtained for the first time by the binary Bell polynomial
approach. The -dressing method is a very important method to
explore the solution of a nonlinear soliton equation without analytical
regions. Two different types of equations, the (2+1)-dimensional
Kaup-Kuperschmidt equation named CKP equation and a generalized
(2+1)-dimensional nonlinear wave equation, are studied by analyzing the
eigenfunctions and Green's functions of their Lax representations as well as
the inverse spectral transformations, then the new problems
are deduced to construct soliton solutions by choosing proper spectral
transformations. Furthermore, once the time evolutions of the spectral data are
determined, we will be able to completely obtain the solutions formally of the
CKP equation and the generalized (2+1)-dimensional nonlinear wave equation. The
reduced problem of the (2+1)-dimensional Kaup-Kuperschmidt is discussed. The
main method is binary Bell polynomials related about the
-constraints and -conditions which turns out to be an effective
tool to represent the bilinear form. On the basis of this method, the bilinear
form is determined under the scaling transformation. Starting from the bilinear
representation, the CKP equation is reduced to Kaup-Kuperschmidt~equation and
bidirectional Kaup-Kuperschmidt equation under the reduction of and
, respectively
Quasideterminant solutions of noncommutative integrable systems
Quasideterminants are a relatively new addition to
the field of integrable systems. Their simple structure disguises a wealth of interesting and useful properties, enabling solutions of noncommutative integrable equations to be expressed in a straightforward and aesthetically pleasing manner. This thesis investigates the derivation and quasideterminant solutions of two noncommutative integrable equations - the Davey-Stewartson (DS) and
Sasa-Satsuma nonlinear Schrodinger (SSNLS) equations.
Chapter 1 provides a brief overview of the various concepts
to which we will refer during the course of the thesis. We begin by explaining the notion of an integrable system, although no concrete definition has ever been explicitly stated. We then move on to discuss Lax pairs, and also introduce the Hirota bilinear form of an integrable equation, looking at the Kadomtsev-Petviashvili (KP)
equation as an example. Wronskian and Grammian determinants will play an important role in later chapters, albeit in a noncommutative setting, and, as such, we give an account of their widespread use in integrable systems.
Chapter 2 provides further background information, now focusing on noncommutativity. We explain how noncommutativity can be defined and implemented, both
specifically using a star product formalism, and also in a more general manner. It is this general definition to which we will allude in the remainder of the thesis. We then give the definition of a quasideterminant, introduced by Gel'fand and Retakh in 1991, and provide some examples and properties of these noncommutative determinantal analogues. We also explain how to calculate the
derivative of a quasideterminant. The chapter concludes by
outlining the motivation for studying our particular choice of noncommutative integrable equations and their quasideterminant solutions.
We begin with the DS equations in Chapter 3, and derive a noncommutative version of this integrable system using
a Lax pair approach. Quasideterminant solutions arise in a natural way by the implementation of Darboux and binary Darboux transformations, and, after describing these transformations in detail, we obtain two types of quasideterminant solution to our system of noncommutative DS equations - a quasi-Wronskian solution from the application of the ordinary Darboux transformation, and a
quasi-Grammian solution by applying the binary transformation. After verification of these solutions, in Chapter 4 we select the quasi-Grammian solution to allow us to determine a particular class of solution to our noncommutative DS equations. These solutions, termed dromions, are lump-like objects decaying exponentially in all directions, and are found at the intersection of two perpendicular plane waves. We extend earlier
work of Gilson and Nimmo by obtaining plots of these dromion
solutions in a noncommutative setting. The work on the
noncommutative DS equations and their dromion solutions constitutes our paper published in 2009.
Chapter 5 describes how the well-known Darboux and
binary Darboux transformations in (2+1)-dimensions discussed in the previous chapter can be dimensionally-reduced to enable their application to (1+1)-dimensional integrable equations. This reduction was discussed briefly by Gilson, Nimmo and Ohta in reference to the self-dual Yang-Mills (SDYM) equations, however we explain these results in more detail, using a reduction from the DS to the nonlinear Schrodinger (NLS) equation as a specific
example. Results stated here are utilised in Chapter 6, where we consider higher-order NLS equations in
(1+1)-dimension. We choose to focus on one particular equation, the SSNLS equation, and, after deriving a noncommutative version of this equation in a similar manner to the derivation of our noncommutative DS system in Chapter 3, we apply the dimensionally-reduced Darboux transformation to the noncommutative SSNLS equation. We see that this ordinary Darboux transformation does not preserve the properties of the equation and its Lax pair,
and we must therefore look to the dimensionally-reduced binary Darboux transformation to obtain a quasi-Grammian solution. After calculating some essential conditions on various terms appearing in our solution, we are then able to determine and obtain plots of soliton solutions in a noncommutative setting.
Chapter 7 seeks to bring together the various results obtained in earlier chapters, and also discusses some open questions arising from our work
The Method of Hirota Bilinearization
Bilinearization of a given nonlinear partial differential equation is very
important not only to find soliton solutions but also to obtain other solutions
such as the complexitons, positons, negatons, and lump solutions. In this work
we study the bilinearization of nonlinear partial differential equations in
-dimensions. We write the most general sixth order Hirota bilinear form
in -dimensions and give the associated nonlinear partial differential
equations for each monomial of the product of the Hirota operators ,
, and . The nonlinear partial differential equations
corresponding to the sixth order Hirota bilinear equations are in general
nonlocal. Among all these we give the most general sixth order Hirota bilinear
equation whose nonlinear partial differential equation is local which contains
12 arbitrary constants. Some special cases of this equation are the KdV, KP,
KP-fifth order KdV, and Ma-Hua equations. We also obtain a nonlocal nonlinear
partial differential equation whose Hirota form contains all possible triple
products of , , and . We give one- and two-soliton
solutions, lump solutions with one, two, and three functions, and hybrid
solutions of local and nonlocal -dimensional equations. We proposed also
solutions of these equations depending on dynamical variables.Comment: 39 pages, 48 figure
- …