Singularity Structure Analysis Research Articles

Painlevé analysis, infinite dimensional symmetry group and symmetry reductions for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finite-dimensional subalgebra of physical transformations is studied. The commutation relations of the subalgebra, which have been established by selecting the Laurent polynomials in t, are calculated. This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics. Meanwhile, the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.

Brane-world singularities and asymptotics of five-dimensional bulk fluids.

We review studies on the singularity structure and asymptotic analysis of a 3-brane (flat or curved) embedded in a five-dimensional bulk filled with a 'perfect fluid' with an equation of state [Formula: see text], where [Formula: see text] is the 'pressure' and [Formula: see text] is the 'density' of the fluid, depending on the fifth space coordinate. Regular solutions satisfying positive energy conditions in the bulk exist only in the cases of a flat brane for [Formula: see text] or of AdS branes for [Formula: see text]. More cases can be found by gluing two regular brunches of solutions at the position of the brane. However, only a flat brane for [Formula: see text] leads to finite Planck mass on the brane and thus localizes gravity. In a more recent work, we showed that a way to rectify the previous findings and obtain a solution for a flat brane and a range of [Formula: see text], which is both free from finite-distance singularities and compatible with the physical conditions of energy and finiteness of four-dimensional Planck mass, is by introducing a bulk fluid component that satisfies a nonlinear equation of state of the form [Formula: see text] with [Formula: see text] and [Formula: see text]. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.

Polarized waveguide excitations in microwave ferrites: The singularity structure analysis

In this work, we delve into the structure of some typical microwave ferrites while investigating the propagation of nonlinear circularly polarized waveguide excitations. In the standpoint, we consider a ferromagnetic slab insulator of 0.5 mm thickness submitted to an external transverse magnetic field. Combining the Landau–Lifshitz-Gilbert formalism of evolution of the magnetization to Maxwell equations, we survey the interaction process of an electromagnetic wave perturbation with the slab. As a result, using the perturbative scaling approach suitable to high-frequency excitations, we derive some new evolution system describing the propagation of circularly polarized waves in the medium. Pursuing further with the analysis, we unwrap the integrability properties of the new system using the singularity structure method where sufficient arbitrary functions are generated. Taking advantage of such properties, we construct a rich variety of nonlinear excitations, solutions to the ferrite dynamics. Additionally, we address some physical implications of the results obtained previously.

Singularity structure analysis of lower-dimensional ferrites within inhomogeneous exchange

In the wake of the derivation of some new evolution equation of inhomogeneous exchange within lower-dimensional ferrites Kuetche et al. (2015) [30], we investigate the complete integrability of such a system by surveying its singularity structure with the aid of some powerful method known as the PainlevȨ-analysis. Looking forward to mitigating the huge mathematical development associated to this analysis where the singularity manifold is expressed in its general form, we address the Kruskal’simplification which presents one interesting advantage to generating in a straightforward manner some typical solutions to the system above. As a result, we find the system passes the PainlevȨ-test. Besides, while constructing some additional integrability properties of the system such as the Bäcklund transformation and the Hirota’s bilinearization, we show that the system is completely integrable. We further discuss the physical implications of some solutions derived from the previous analysis.

On the integrability aspects of nonparaxial nonlinear Schrödinger equation and the dynamics of solitary waves

The integrability nature of a nonparaxial nonlinear Schrödinger (NNLS) equation, describing the propagation of ultra-broad nonparaxial beams in a planar optical waveguide, is studied by employing the Painlevé singularity structure analysis. Our study shows that the NNLS equation fails to satisfy the Painlevé test. Nevertheless, we construct one bright solitary wave solution for the NNLS equation by using the Hirota's direct method. Also, we numerically demonstrate the stable propagation of the obtained bright solitary waves even in the presence of an external perturbation in a form of white noise. We then numerically investigate the coherent interaction dynamics of two and three bright solitary waves. Our study reveals interesting energy switching among the colliding solitary waves due to the nonparaxiality.

Painlevé integrability of the supersymmetric Ito equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11975156 and 11775146) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LY18A050001).

A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation. To investigate the integrability of the N = 1 supersymmetric Ito (sIto) equation, a singularity structure analysis for this system is carried out. Through a detailed analysis in two cases by using Kruskal’s simplified method, the sIto system is found to pass the Painlevé test, and thus is Painlevé integrable.

Inhomogeneous exchange within higher-dimensional ferrites: The singularity structure analysis and pattern formations

Abstract In this work the propagation of nonlinear electromagnetic short waves in a ferromagnetic medium with inhomogeneous exchange is discussed. It is shown that such waves propagate perpendicular to the magnetization density. The evolution of waves under the influence of perturbations in one transverse dimension is considered. As result, we derive a new (2+1)-dimensional evolution system in which painleve analysis reveals its integrability properties. In the wake of such analysis, using the arbitrary functions to enter into the Laurent series of solutions to the above system, we present some typical class of excitations.

Lumps and rogue waves of generalized Nizhnik–Novikov–Veselov equation

We investigate the generalized $$(2+1)$$ Nizhnik–Novikov–Veselov equation and construct its linear eigenvalue problem in the coordinate space from the results of singularity structure analysis thereby dispelling the notion of weak Lax pair. We then exploit the Lax pair employing Darboux transformation and generate lumps and rogue waves. The dynamics of lumps and rogue waves is then investigated.

Painlevé analysis, nonlocal symmetry and explicit interaction solutions for supersymmetric mKdVB equation

The N=1 supersymmetric mKdVB system is transformed to a coupled bosonic system by using the bosonization approach. By a singularity structure analysis, the bosonized supersymmetric mKdVB (BSmKdVB) equation admits the Painlevé property. Starting from the standard truncated Painlevé method, the nonlocal symmetry for the BSmKdVB equation is obtained. To solve the first Lie’s principle related with the nonlocal symmetry, the nonlocal symmetry is localized to the Lie point symmetry by introducing multiple new fields. Thanks to localization processes, similarity reductions for the prolonged systems are studied by the Lie point symmetry method. The interaction solutions among solitons and other complicated waves including Painlevé II waves and periodic cnoidal waves are given through the reduction theorems. The soliton-cnoidal wave interaction solutions are explicitly given by using the mapping and deformation method. The concrete soliton-cnoidal interaction solutions are displayed both in analytical and graphical ways.

The Integrability of an Extended Fifth-Order KdV Equation in 2+1 Dimensions: Painlevé Property, Lax Pair, Conservation Laws, and Soliton Interactions

Abstract In this article, we apply the singularity structure analysis to test an extended 2+1-dimensional fifth-order KdV equation for integrability. It is proven that the generalized equation passes the Painlevé test for integrability only in three distinct cases. Two of those cases are in agreement with the known results, and a new integrable equation is first given. Then, for the new integrable equation, we employ the Bell polynomial method to construct its bilinear forms, bilinear Bäcklund transformation, Lax pair, and infinite conversation laws systematically. The N-soliton solutions of this new integrable equation are derived, and the propagations and collisions of multiple solitons are shown by graphs.

The integrability for a generalized seventh-order KdV equation: Painlevé property, soliton solutions, Lax pairs and conservation laws

This paper investigates the integrability of a generalized seventh-order Korteweg–de Vries equation arising in fluids and plasmas. By means of singularity structure analysis, it is proven that this equation passes the Painlevé test for integrability in only three distinct cases. Under three sets of Painlevé integrable conditions, the soliton solutions are obtained by using Hirotaʼs bilinear method; the pseudopotentials and Lax pairs are derived by virtue of the method developed by Nucci. Finally, the infinite conservation laws are found by using its Lax pair, and all conserved densities and fluxes are presented with explicit recursion formulas.

Painlevé integrability of a generalized fifth-order KdV equation with variable coefficients: Exact solutions and their interactions

By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painlevé test for integrability only for three distinct cases. Moreover, the multi-soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.

Fractal structure of ferromagnets: The singularity structure analysis

Following the Weiss-Tabor-Carnevale approach [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983)10.1063/1.525721; J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 25, 13 (1984).]10.1063/1.526009 designed for studying the integrability properties of nonlinear partial differential equations, we investigate the singularity structure of a (2+1)-dimensional wave-equation describing the propagation of polariton solitary waves in a ferromagnetic slab. As a result, we show that, out of any damping instability, the system above is integrable. Looking forward to unveiling its complete integrability, we derive its Bäcklund transformation and Hirota's bilinearization useful in generating a set of soliton solutions. In the wake of such results, using the arbitrary functions to enter into the Laurent series of solutions to the above system, we discuss some typical class of excitations generated from the previous solutions in account of a classification based upon the different expressions of a generic lower dimensional function. Accordingly, we unearth the nonlocal excitations of lowest amplitudes, the dromion and lump patterns of higher amplitudes, and finally the stochastic pattern formations of highest amplitudes, which arguably endow the aforementioned system with the fractal properties.

INTEGRABLE ASPECTS AND SOLITON-LIKE SOLUTIONS OF AN INHOMOGENEOUS COUPLED HIROTA–MAXWELL–BLOCH SYSTEM IN OPTICAL FIBERS WITH SYMBOLIC COMPUTATION

For describing wave propagation in an inhomogeneous erbium-doped nonlinear fiber with higher-order dispersion and self-steepening, an inhomogeneous coupled Hirota–Maxwell–Bloch system is considered with the aid of symbolic computation. Through Painlevé singularity structure analysis, the integrable condition of such a system is analyzed. Via the Painlevé-integrable condition, the Lax pair is explicitly constructed based on the Ablowitz–Kaup–Newell–Segur scheme. Furthermore, the analytic soliton-like solutions are obtained via the Darboux transformation which makes it exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some obtained analytic one- and two-soliton-like solutions, our concerns are mainly on the envelope soliton excitation, the propagation features of optical solitons and their interaction behaviors in actual fiber communication. Finally, the conservation laws for the system are also presented.

Integrability aspects with optical solitons of a generalized variable-coefficient N-coupled higher order nonlinear Schrödinger system from inhomogeneous optical fibers

For describing the long-distance communication and manufacturing problems of N fields propagation in inhomogeneous optical fibers, we consider a generalized variable-coefficient N-coupled nonlinear Schrödinger system with higher order effects such as the third-order dispersion, self-steepening and self-frequency shift. Using the Painlevé singularity structure analysis, we obtain two cases for this system to admit the Painlevé property. Then for case (1) we derive the optical dark solitons via solving the Hirota bilinear equations; and based on the obtained (2N+1)×(2N+1) Lax pair, we construct the Darboux transformation to obtain the optical bright solitons (including the multisoliton profiles) for case (2). Finally, the features of optical solitons (both dark and bright ones) in inhomogeneous optical fibers are analyzed and graphically discussed.

Painlevé singularity structure analysis of three component Gross–Pitaevskii type equations

In this paper, we have studied the integrability nature of a system of three-coupled Gross–Pitaevskii type nonlinear evolution equations arising in the context of spinor Bose–Einstein condensates by applying the Painlevé singularity structure analysis. We show that only for two sets of parametric choices, corresponding to the known integrable cases, the system passes the Painlevé test.

Painlevé-integrability of a(2+1)-dimensional reaction-diffusion equation: Exact solutions and their interactions

We investigate the singularity structure analysis of a (2+1)-dimensional coupled nonlinear extension of the reaction-diffusion (NLERD) equation by means of the Painlevé (P) test. Following the Weiss et al.'s formalism [J. Math. Phys. 24, 522 (1983)], we prove the arbitrariness of the expansion coefficients of the observables. Thus, without the use of the Kruskal's simplification, we obtain a Bäcklund transformation of the coupled NLERD equation via a consistent truncation procedure stemming from the Weiss 's methodology [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 25, 13 (1984)]. In the wake of such results, we unveil a typical spectrum of localized and periodic coherent patterns. We also investigate the scattering properties of such structures and we unearth two peculiar soliton phenomena, namely, the fusion and the fission.

Integrability of anN-coupled nonlinear Schrödinger system for polarized optical waves in an isotropic medium via symbolic computation

Considering the simultaneous propagation of multicomponent fields in an isotropic medium, an N-coupled nonlinear Schrödinger system with the self-phase modulation, cross-phase modulation, and energy exchange terms is investigated in this paper. First, via symbolic computation, the Painlevé singularity structure analysis shows that such a system admits the Painlevé property. Then, with the Ablowitz-Kaup-Newell-Segur scheme, the linear eigenvalue problem (Lax pair) associated with this model is constructed in the frame of the block matrices. With the Hirota bilinear method, the bright one- and two-soliton solutions of this system are presented. In addition, the bright multisoliton solutions of the system for N=2 are straightforwardly derived by the linear superposition of soliton solutions of two independent scalar nonlinear Schrödinger equations. Furthermore, through the analysis for the soliton solutions, the corresponding propagation behavior and applications for soliton pulses in nonlinear optical fibers are considered. Finally, three significant conserved quantities, i.e., energy, momentum, and Hamiltonian, are also given.

Comment on “Loop Soliton Solutions of String Interacting with External Field”

where q, r, and s are physical observables, x and t representing spaceand time-like independent variables. Taking q 1⁄4 Z, r 1⁄4 X þ iY , and s 1⁄4 X iY with i 1⁄4 1, eq. (4) is transformed to eq. (1) provided 1⁄4 xþ t and 1⁄4 x t. Singularity structure analysis of eq. (4) has recently been investigated. The associated Backlund transformation has been constructed and Hirota’s bilinearization also given through dependent variable transformations. As a result, q, r, and s are expressed as follows r 1⁄4 G=F; s 1⁄4 H=F; q 1⁄4 2@tðlnFÞ: ð5Þ We assume r 1⁄4 s, ð?Þ denoting complex conjugation, and we transform eq. (5) to a new one by looking for some soliton solutions with the following asymptotic behavior jrj ! 0; q ! x; as jxj ! 1: ð6Þ We may then write

Solitary wave solution for inhomogeneous nonlinear Schrödinger system with loss/gain

This paper deals with the propagation of solitons in real fibres, governed by the system of inhomogeneous nonlinear Schrödinger (INLS) equations. The Painlevé singularity structure analysis is utilized to check for the integrability of the system and from the analysis, the system is found to admit soliton-type lossless wave propagation. The system is transformed to its homogeneous counterpart using a suitable variable transformation and the soliton solutions are obtained through Bäcklund transformation after constructing the explicit Lax pair for the system. The one-soliton solutions are plotted for different choices of inhomogeneity parameters and the evolutionary characteristics of the solutions are analyzed.