Solitons

We consider some nonlinear phenomena in metamaterials with negative refractive index properties. Our consideration includes a survey of previously known results as well as identification of the phenomena that are important for applications of this new field. We focus on optical behavior of thin films as well as multi-wave interactions.

In this paper, the topological (dark) soliton solutions to the Camassa-Holm type equations are obtained by the solitary wave ansatz method. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Camassa-Holm type models.

We have investigated the nonlinear optical interaction of uniform and kink states of a nematic and a ferrofluid-doped nematic (ferronematic) liquid crystal with an incident laser field. We find that the transition between the permitted uniform oreintational states of these systems is of first order in the case of nematics, and of second order in the case of ferronematics. In the latter case we also find the phenomenon of reentrance. We find new kink states in a magnetic field with topological winding different from π in the case of nematics, and 2π in the case of ferronematics. In ferronematics, due to grain segregation the phase diagrams for uniform and kink states are entirely different. In these systems we find a first or second order structural transformation from a single kink into a pair of kinks. Further, we obtain a rich variety of kink states as the intensity of the laser field is varied.

Explicit expressions for the generators of the quantum superalgebra U q [gl(n/m)} acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a GePfand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of ^-number identities.

In this thesis study, it is aimed to cover the surface of Ti-based materials with Ag by using new
experimental processes with cold substrate technique and to examine the TiO2-Ag binary
structure. Titanium and its alloys (Ti-6Al-4V) are generally preferred due to their stable
structure, high strength/weight ratios, high corrosion resistance, processability and
biocompatibility in a wide range of applications. Thin film formation process is realized by
soliton growth mechanism in the cold substrate technique used in this thesis. The most
important feature of the soliton mechanism is the coating of the surface with equal size nano
particles. Silver ions were preferred which are used especially in the photoelectronic technology
also in the health field due to its antibacterial properties as coating material. The morphological
and structural properties of the samples were examined after preparation of Ag coated samples
at the substrate temperature of 100-300 K with specially prepared apparatus. It was determined
from the X-ray diffraction patterns where Ag layers were grown in cubic structure and the
intensity of Ag structure peaks increased as the substrate temperature decreased. Parallel to this,
energy dispersive X-ray spectroscopy measurements was determined where the amount of Ag
coated increased as the substrate temperature decreased. As the substrate temperature decreases,
scanning electron microscope measurements showed that the particle sizes decreased in the Ag
layers and the Ag films prepared in the 100-200 K temperature range were coated in equal sized
nano particles. Higher corrosion resistance was observed in this temperature range, the coatings
which has grown. On the other hand, nano-sized Ag coatings obtained in the 100-200 K
temperature zone showed a decrease in the hardness value as expected.

The nonlinear Klein-Gordon equation is used as a vehicle to employ the tanh method and the sine-cosine method to formally derive a number of travelling wave solutions. The study features a variety of solutions with distinct physical structures. The work shows that one method complements the other, and each method gives solutions of formal properties. The obtained solutions include compactons, solitons, solitary patterns, and periodic solutions.

A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg — de Vries, nonlinear Schrodinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamil-tonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative eft'ects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered. CONTENTS

An overview of optical solitons was presented in this project. Starting from the nonlinear effects on the refractive index and the wave equation, the Nonlinear Schrodinger Equation (NLSE) was developed. The NLSE is capable of describing both temporal and spatial solitons of both the bright and dark types. The solutions to these NLSE were also given, and their physical properties were explored. The crux of soliton theory is that the 𝛽2 linear dispersive effects (GVD or diffraction) can be balanced by intensity dependent nonlinear effects, creating a field that does not change in shape as it propagates. This places a defining condition between the pulse width and amplitude. Simulation results of soliton propagation through a dispersive fiber were presented, and it was found that the exact initial conditions of a soliton do not have to be matched, since solitons are self-adjusting. The implementation of optical solitons for fiber optic communications were studied, and the relations between bit-rate, transmission distance, and pulse widths in a typical setup were determined. There are several challenges in the implementation, the most important is power loss in the fiber, which causes an exponential broadening of pulse width during propagation. Finally, a brief description of previous experimental results on soliton transmission in fiber optic communications was presented.

It is shown that Lorentz Invariance is a wave phenomenon. The relativistic mass, length contraction and time dilation all follow from the assumption that energy-momentum is constrained to propagate at the speed of light, c, in all contexts, matter as well as radiation. Lorentz Transformations, and both of the usual postulates, then follow upon adopting Einstein clock synchronisation. The wave interpretation proposed here is paradox free and it is compatible with quantum nonlocality.

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N -soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions. periodic initial data, based on both the inverse spectral theory and algebrogeometric methods, was developed by Novikov, Dubrovin, Lax, Its, Matveev et al. . For more recent reviews on the KdV equation one may refer, for instance, to Refs. .

In the course of the past several years, a new level of understanding has been achieved about conditions for the existence, stability, and generation of spatiotemporal optical solitons, which are nondiffracting and nondispersing wavepackets propagating in nonlinear optical media. Experimentally, effectively two-dimensional (2D) spatiotemporal solitons that overcome diffraction in one transverse spatial dimension have been created in quadratic nonlinear media. With regard to the theory, fundamentally new features of light pulses that self-trap in one or two transverse spatial dimensions and do not spread out in time, when propagating in various optical media, were thoroughly investigated in models with various nonlinearities. Stable vorticity-carrying spatiotemporal solitons have been predicted too, in media with competing nonlinearities (quadratic–cubic or cubic–quintic). This article offers an up-to-date survey of experimental and theoretical results in this field. Both achievements and outstanding difficulties are reviewed, and open problems are highlighted. Also briefly described are recent predictions for stable 2D and 3D solitons in Bose–Einstein condensates supported by full or low-dimensional optical lattices.

We discuss the early history of an important field of ''sturm and drang'' in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.

A generally covariant refractive medium interpretation of gravity, isomorphic to the time independent Einstein-Yilmaz variation of General Relativity, is developed from the assumption that mass energy propagates at the characteristic velocity in a dielectric medium.

Using a modified version of the split-step Fourier method, we analyze the effect of noise on soliton propagation inside erbium-doped fiber amplifiers. In fact, noise from forward-propagating amplified spontaneous emission, associated with a Markov immigration process, is included in the analysis of soliton amplification. Moreover, this algorithm accounts for the real spectral gain profile of the fiber amplifier. The frequency jitter, induced during soliton amplification, is compared with the Gordon-Haus effect where optical amplifiers are considered as noisy point-like devices

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N -soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions. periodic initial data, based on both the inverse spectral theory and algebrogeometric methods, was developed by Novikov, Dubrovin, Lax, Its, Matveev et al. . For more recent reviews on the KdV equation one may refer, for instance, to Refs. .

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

We consider the multicomponent Yajima-Oikawa (YO) system and show that the two-component YO system can be derived in a physical setting of a three-coupled nonlinear Schrödinger (3-CNLS) type system by the asymptotic reduction method. The derivation is further generalized to the multicomponent case. This set of equations describes the dynamics of nonlinear resonant interaction between a one-dimensional long wave and multiple short waves. The Painlevé analysis of the general multicomponent YO system shows that the underlying set of evolution equations is integrable for arbitrary nonlinearity coefficients which will result in three different sets of equations corresponding to positive, negative, and mixed nonlinearity coefficients. We obtain the general bright N-soliton solution of the multicomponent YO system in the Gram determinant form by using Hirota's bilin-earization method and explicitly analyze the one-and two-soliton solutions of the multicomponent YO system for the above mentioned three choices of nonlinearity coefficients. We also point out that the 3-CNLS system admits special asymptotic solitons of bright, dark, anti-dark, and gray types, when the long-wave–short-wave resonance takes place. The shortwave component solitons undergo two types of energy-sharing collisions. Specifically, in the two-component YO system, we demonstrate that two types of energy-sharing collisions—(i) energy switching with opposite nature for a particular soliton in two components and (ii) similar kind of energy switching for a given soliton in both components—result for two different choices of nonlinearity coefficients. The solitons appearing in the long-wave component always exhibit elastic collision whereas those of shortwave components exhibit standard elastic collisions only for a specific choice of parameters. We have also investigated the collision dynamics of asymptotic solitons in the original 3-CNLS system. For completeness, we explore the three-soliton interaction and demonstrate the pairwise nature of collisions and unravel the fascinating state restoration property.

This article presents a three-layer index guided lead silicate (SF57) photonic crystal fiber which simultaneously promises to
yield large effective optical nonlinear coefficient and low anomalous dispersion that makes it suitable for supercontinuum
(SC) generation. At an operating wavelength 1550 nm, the typical optimized value of anomalous dispersion and effective
nonlinear coefficient turns out to be ~4 ps/km/nm and ~1078 W−1km−1, respectively. Through numerical simulation, it is
realized that the designed fiber promises to exhibit three octave spanning SC from 900 to 7200 nm using 50 fs ‘sech’
optical pulses of 5 kW peak power. Due to the cross-phase modulation and four-wave mixing processes, a long range of
red-shifted dispersive wave generated, which assists to achieve such large broadening. In addition, we have investigated
the compatibility of SC generation with input pulse peak power increment and briefly discussed the impact of nonlinear
processes on SC generation.

The creation of stable 1D and 2D localized modes in lossy nonlinear media is a fundamental problem in optics and plasmonics. This article gives a mini review of theoretical methods elaborated on for this purpose, using localized gain applied at one or several hot spots (HS). The introduction surveys a broad class of models for which this approach was developed. Other sections focus in some detail on basic 1D continuous and discrete systems, where the results can be obtained, partly or fully, in an analytical form (and verified by comparison with numerical results), which provides deeper insight into the nonlinear dynamics of optical beams in dissipative nonlinear media. Considered, in particular, are the single and double HS in the usual waveguide with the self-focusing (SF) or self-defocusing (SDF) Kerr nonlinearity, which gives rise to rather sophisticated results in spite of apparent simplicity of the model, solitons attached to a PT -symmetric dipole embedded into the SF or SDF medium, gap solitons pinned to an HS in a Bragg grating, and discrete solitons in a 1D lattice with a hot site.

are discussed in terms of how they are influenced, but not suppressed, by a metamaterial background. It is strongly discussed that metamaterials and optical rogue waves have both been making headlines in recent years and that they are, separately, large areas of research to study. A brief background of the inevitable linkage of them is considered and important new possibilities are discussed. After this background is revealed some new rogue wave configurations combining the two areas are presented alongside a discussion of the way forward for the future. 5 Nanotechnology 28 (2017) 444001 A D Boardman et al w ( ) z z t t z k V t t z z t z z

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schrödinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schrödinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schrödinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions. †

The present study aims to develop a hybrid scheme using trigonometric cubic B-spline basis functions with differential quadrature method for solving nonlinear Schrödinger equation in both one and two dimensions. This method reduces the nonlinear equation into a set of ordinary differential equations which can be further solved by the modified form of Ruge-Kutta method. This proposed method has been applied to this equation using two different approaches and also has been tested for proficiency on seven numerical examples. The obtained numerical results found to be synonymous when related with the exact solution. The obtained numerical results are also in good agreement with the results available in the literature. Comparison of numerical and the exact solution is depicted in the form of figures and tables.

Through investigating history, evolution of the concept, and development in the theories of electrons, I am convinced that what was missing in our understanding of the electron is a structure, into which all attributes of the electron could be incorporated in a self-consistent way. It is hereby postulated that the topological structure of the electron is a closed two-turn Helix (a so-called Hubius Helix) that is generated by circulatory motion of a mass-less particle at the speed of light. A formulation is presented to describe an isolated electron at rest and at high speed. It is shown that the formulation is capable of incorporating most (if not all) attributes of the electron, including spin, magnetic moment, fine structure constant, anomalous magnetic moment, and charge quantization into one concrete description of the Hubius Helix. The equations for the description emerge accordingly. Implications elicited by the postulate are elaborated. Inadequacy of the formulation is discussed.

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

We study theoretically and experimentally actively modelocked fiber lasers that are used in high repetition rate optical communication systems. Using an innovative numerical technique and a reduced model, we have found that the laser can operate in four different operating regimes when the laser intensity was changed; three of the regimes were experimentally observed in a laser with a sigma configuration. An excellent quantitative agreement between the theoretical and the experimental results was obtained. The use of dispersion management in the sigma laser was found to significantly improve the laser performance.

We propose a design of tellurite and As 2 S 3 -based chalcogenide tapered photonic crystal fibers (TPCF) for broadband coherent mid-infrared supercontinuum (SC) generation in the few-optical-cycle regime. We optimize the soliton self-compression by injecting pre-chirped femtosecond pulses. More than one octave-spanning SC spectra were generated in 8 mm-long in both tellurite and chalcogenide TPCF with low input pulse energies of 1 nJ and 100 pJ at pump wavelengths of 2.9 and 4.7 μm, respectively.

In this work, we present a comparative study of meshless method, modified Bernstein polynomials (BP) and B-Spline finite element method (BS-FEM) for the numerical solution of two different models of Korteweg–de Vries (KdV) equation. The multiquadric (MQ) and Gaussian (GA) Radial basis functions (RBFs) are used due to exponential convergence rate. Excellent agreement is found between exact and RBFs solutions and same accuracy is obtained as Bernstein polynomials and B-Spline finite element method.

During July and August of 1996, the summer component of the New England Shelfbreak Front PRIMER Experiment was fielded in the Mid-Atlantic Bight, at a site due south of Martha's Vineyard, MA. This study produced acoustic transmission data from a network of moored sources and receivers in conjunction with very-high-resolution oceanography measurements. This paper analyzes receptions at the northeast array receiver from two 400-Hz acoustic tomography sources, with the transmission paths going from the continental slope onto the continental shelf. These data, along with forward acoustic-propagation modeling based on moored oceanographic data, SeaSoar hydrography measurements, and bottom measurements, reveal many new and interesting aspects of acoustic propagation in a complicated slope-shelf environment. For example, one sees that both the shelfbreak front and tidally generated soliton internal wave packets produce stronger mode coupling than previously expected, leading to an interesting time-and-range-variable population of the acoustic normal modes. Additionally, the arrival time wander and the signal spread of acoustic pulses show variability that can be attributed to the presence of a frontal meander and variability in the soliton field. These and other effects are discussed in this paper, with an emphasis on creating a strong connection between the environmental measurements and the acoustic field characteristics.

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a “fitted operator finite difference method (FOFDM)” which is first order ε-uniformly convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the “fitted mesh finite difference method (FMFDM)”. This FMFDM is a direct method and ε  -uniformly convergent with the nodal error as O(n-2ln2n)O(n-2ln2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n-1lnn)O(n-1lnn) where n denotes the total number of sub-intervals of [0, 1]. Comparative numerical results are presented in support of the theory.

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schro ̈dinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schro ̈dinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schro ̈dinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

The exact bright one-and two-soliton solutions of a particular type of coherently coupled nonlinear Schrödinger equations, with alternate signs of nonlinearities among the two components, are obtained using the non-standard Hirota's bilinearization method. We find that in contrary to the coherently coupled nonlinear Schrödinger equations with same signs of nonlinearities the present system supports only coherently coupled solitons arising due to an interplay between dispersion and the nonlinear effects, namely, self-phase modulation, cross-phase modulation, and four-wave mixing process, thereby depend on the phases of the two co-propagating fields. The other type of soliton, namely, incoherently coupled solitons which are insensitive to the phases of the co-propagating fields and arise in a similar kind of coherently coupled nonlinear Schrödinger equations but with same signs of nonlinearities are not at all possible in the present system. The present system can support regular solution for the choice of soliton parameters for which mixed coupled nonlinear Schrödinger equations admit only singular solution. Our analysis on the collision dynamics of the bright solitons reveals the important fact that in contrary to the other types of coupled nonlinear Schrödinger systems the bright solitons of the present system can undergo only elastic collision in spite of their multicomponent nature. We also show that regular two-soliton bound states can exist even for the choice for which the same system admits singular one-soliton solution. Another important effect identified regarding the bound solitons is that the breathing effects of these bound solitons can be controlled by tuning the additional soliton parameters resulting due to the multicom-ponent nature of the system which do not have any significant effects on bright one soliton propagation and also in soliton collision dynamics. C 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4772611]

This paper obtains solutions to the Zakharov-Kuznetsov modified equal width equation with power law nonlinearity. The Lie symmetry approach and the simplest equation method are used to obtain these solutions. Moreover, conservation laws are derived for the underlying equation by employing two approaches: the new conservation theorem and the multiplier method.

This paper implemented the tanh method to solve a few coupled nonlinear wave equations in (2 ? 1)dimensions. They are the Konopelchenko-Dubrovsky equation, dispersive long wave equation and the Riemann wave equation. Additionally, the traveling wave hypothesis is used to extract a few more solutons to some of these equations. Finally, the numerical simulations supplement these analytical results.

Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things, such as x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating its first relations to mereotopology. Mereotopological axioms and definitions are then discussed in detail from a phasefield perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected, and wasConnected. Such relations introduce dynamics and thus physics into mereotopology, as transitions from isDisconnected to isPartOf can be described.