Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

The Derivative Yajima–Oikawa System: Bright,Dark Soliton and Breather Solutions

In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N ‐bright and N ‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.     

Spinor soliton with electromagnetic field

In this paper, the spinor soliton coupling with its own electromagnetic field is computed by the first order approximation of the energy functional. The numerical calculation disclosed that (1) the soliton do exists, and only a few of meaningful solutions exist, (2) this nonlinear model for an electron implies the abnormal magneton, (3) the structural parameters of the soliton such as the rest massm, the mean radius, the weakly coupling constantw, are determined by empirical data. Besides, the method used in the paper is verified to be an efficient tool for solving the nonlinear spinor equation. The results are compared with those of the dark soliton.     

Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric for some and some derivationD of the Lie algebra ofN, where Ric denotes the Ricci operator of . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. Received August 24, 1999 / Revised October 2, 2000 / Published online February 5, 2001     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

Method of Immersion and Its Parametric Realization for the Korteweg–De Vries Equation

By using the method of immersion (imbedding) proposed in the author's previous works, we describe the space S of initial conditions of the Cauchy problem for the general differential Korteweg–de Vries equation. The space S is called a stationary soliton Korteweg–de Vries manifold because "stationary projections" of solitons fall into the space S. In addition, we introduce the notion of a space of Sturm–Liouville operators over a soliton Korteweg–de Vries manifold. For real functions and parameters, we formulate the spectral theorem for a commutative Lax pair over a real stationary soliton Korteweg–de Vries manifold.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

Soliton model of elementary electric charge

Conclusions Thus, we have shown that in the electrodynamics of the Klein-Gordon field there exist two spectra of three-dimensional electrostatic soliton solutions. One of them is dynamically stable, the other topologically stable. For each of the spectra, the value of the electrostatic potential at the center of the soliton is quantized, while the value of the electric field vanishes. At the periphery of the solitons, the electrostatic potential corresponds to the Coulomb law.The topological charge of the soliton is related to quantization of the value of the electrostatic potential at infinity: ,p=0, 1, 2, .... The topological soliton can obviously be regarded as a model of an elementary electric charge, an attractive feature of which is the absence of divergences of the integrals of the motion.For the dynamical solitons, we have _p =0,p=0, 1, 2, ....A rotating electrostatic soliton can be regarded as a soliton model of an elementary electric charge possessing an intrinsic magnetic moment. The magnetic field at the center of the rotating soliton is quantized, while at its periphery the field has a magneticdipole nature.Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 3, pp. 349–359, March, 1990.     

Numerical experience with a polyhedral-norm CDT trust-region algorithm

In this paper, we study a modification of the Celis-Dennis-Tapia trust-region subproblem, which is obtained by replacing thel ₂-norm with a polyhedral norm. The polyhedral norm Celis-Dennis-Tapia (CDT) subproblem can be solved using a standard quadratic programming code.We include computational results which compare the performance of the polyhedral-norm CDT trust-region algorithm with the performance of existing codes. The numerical results validate the effectiveness of the approach. These results show that there is not much loss of robustness or speed and suggest that the polyhedral-norm CDT algorithm may be a viable alternative. The topic merits further investigation.The first author was supported in part by the REDI foundation and State of Texas Award, Contract 1059 as Visiting Member of the Center for Research on Parallel Computation, Rice University, Houston, Texas, He thanks Rice University for the congenial scientific atmosphere provided. The second author was supported in part by the National Science Foundation, Cooperative Agreement CCR-88-09615, Air Force Office of Scientific Research Grant 89-0363, and Department of Energy Contract DEFG05-86-ER25017.     

Rotating Loop Soliton of the Coupled Dispersionless Equations

We investigate soliton solutions of the coupled dispersionless equations that describe a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space with bilinear equations. We obtain a new type of loop soliton solutions that rotate around the Z axis. We also investigate the two-soliton interaction.     

Darboux transformation and solitons of Yang-Mills-Higgs equations in R2,1

The Darboux transformations for soliton equations are applied to the Yang-Mills-Higgs equations. New solutions can be obtained from a known one via universal and purely algebraic formulas. SU(N) soliton solutions are constructed with explicit formulas. The interaction of solitons is described by the splitting theorem: each p-soliton is splitting into p single solitons asymptotically as t → ±∞. The main contents of this work were reported at the Moshé Flato memorial conference (Dijon, September 2000) and W. L. Chow and K. T. Chen memorial conference (Tianjin, October 2000).     

Some properties of the spinor soliton

In this paper, the solitons of nonlinear Dirac equation are discussed in detail, and several functions which reflect their characteristics are computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 1/2-spin and positive mass; the spinor soliton has two kinds of parity states, and each parity state has two kinds of energy states; the larger the self-coupling coefficientw, the more the excitation states, and ifw is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.     

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

Subspace choices for the Celis-Dennis-Tapia problem

Grapiglia et al. (2013) proved subspace properties for the Celis-Dennis-Tapia (CDT) problem. If a subspace with lower dimension is appropriately chosen to satisfy subspace properties, then one can solve the CDT problem in that subspace so that the computational cost can be reduced. We show how to find subspaces that satisfy subspace properties for the CDT problem, by using the eigendecomposition of the Hessian matrix of the objection function. The dimensions of the subspaces are investigated. We also apply the subspace technologies to the trust region subproblem and the quadratic optimization with two quadratic constraints.     

Dynamics of Strongly Nonlinear Kinks and Solitons in a Two‐Layer Fluid

Perturbation theory is developed for interaction of strongly nonlinear solitary waves close to the limiting, tabletop solitons (Π‐solitons). The method is based on representing each soliton as a compound of two kinks so that the interaction of N solitons is treated as the interaction of 2N kinks. As an example the Miyata–Choi–Camassa equations for a two‐layer fluid is considered. Equations for kink coordinates are obtained and analyzed. Some nontrivial features of two‐soliton interaction characteristic of the strongly nonlinear case are established.     

General soliton and (semi-)rational solutions to the nonlocal Mel'nikov equation on the periodic background

In this paper, the Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi-)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram-type determinants. When N is even, soliton, line breather and (semi-)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi-rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.