Loop Soliton Research Articles

Stretching and Shrinking of the Loop Soliton Interacting with a External Field

The integrable equation of motion of the loop soliton interacting with an external field is considered from the standpoint of stretching and/or shrinking of the loop. To study the role of the elastic force and the nonlinear forces, the basic equation is divided into three equations. We obtain stationary solutions for these equations and numerically solve their initial value problems to seek stability of the loop soliton.

One loop renormalization of soliton quantum mass corrections in (1+1)-dimensional scalar field theory models

The bare one loop soliton quantum mass corrections can be expressed in two ways: as a sum over the zero-point energies of small oscillations around the classical configuration, or equivalently as the (Euclidean) effective action per unit time. In order to regularize the bare one loop quantum corrections (expressed as the sum over the zero-point energies) we subtract and add from it the tadpole graph that appear in the expansion of the effective action per unit time. The subtraction renders the one loop quantum corrections finite. Next, we use the renormalization prescription that the added tadpole graph cancels with adequate counterterms, obtaining in this way a finite unambiguous expression for the one loop soliton quantum mass corrections. When we apply the method to the solitons of the sine-Gordon and φ 4 kink models we obtain results that agree with known results. Finally we apply the method to compute the soliton quantum mass corrections in the recently introduced φ 2cos 2ln( φ 2) model.

Hyperelliptic loop solitons with genus g: investigations of a quantized elastica

In the previous work [J. Geom. Phys. 39 (2001) 50], the closed loop solitons in a plane, i.e., loops whose curvatures obey the modified Korteweg–de Vries equations, were investigated for the case related to algebraic curves with genera 1 and 2. This paper is a generalization of the previous paper to those of hyperelliptic curves with general genera. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al-function for a given hyperelliptic curve y 2= f( x) with genus g.

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.

Closed loop solitons and sigma functions: classical and quantized elasticas with genera one and two

Closed loop solitons in a plane, whose curvatures obey the modified Korteweg–de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker’s sigma functions for the genus two case. This study is closely related to classical and quantized elastica problems.

The N loop soliton solution of the Vakhnenko equation

An exact N loop soliton solution to the Vakhnenko equation (VE) is found, where N2 is an arbitrary positive integer. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit N-soliton solution by using Hirota's method. The exact N loop soliton solution to the VE is then found in implicit form by means of a transformation back to the original independent variables. The shifts that occur when the solitons interact are found. The general results and details of the interaction between solitons are illustrated for the case N = 3.

Loop Soliton Solutions of String Interacting with External Field

We give a physical model of the O (3)-invariant coupled integrable dispersionless equations where a string with internal structure is interacting to the external field. We derive loop soliton solutions through the bilinear equations in which solitons can move to positive and negative directions. We study two soliton interactions in detail.

The two loop soliton solution of the Vakhnenko equation

An exact two loop soliton solution to the Vakhnenko equation is found. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit 2-soliton solution by use of Hirota's method. The exact two loop soliton solution to the Vakhnenko equation is then found in implicit form by means of a transformation back to the original independent variables. The nature of the interaction between the two loop solitons depends on the ratio of their amplitudes.

Discrete Modeling of a String and Analysis of a Loop Soliton

A discrete model for an extensible string is proposed and analyzed by a discrete soliton theory and computer simulations. The relation between tension of the string and the size of a loop propagating on the string is obtained analytically by using the soliton theory. We use this relation to investigate dynamics and stability of loops, and it is found that one loop is stable against various kinds of perturbation. It is confirmed numerically that the loop can be formed by moving a boundary along a semicircle. If the moment of the string is introduced, behaviors of the formation are drastically changed and there is a critical value of stiffness of the string beyond which the loop cannot be formed. As for collision of two loops, we, found that two loops do not break after collision if the two are similar. This result of collision can be well explained by our former analysis of a continuous string theory (Nishinari, 1997).

Anomaly on a submanifold system-new index theorem related to a submanifold system

Recently the submanifold quantum system has been studied. In this article, after we confine a Dirac field in a thin curved rod, we find an anomalous term in the fermionic field theory related to the extrinsic curvature. In other words, we find a new Atiyah-Singer-type index theorem related to the geometry or the submanifold. As the anomaly is associated with the nonlinear Schrodinger equation as a loop soliton, we also discuss it.

N-loop solitons and their link with the complex Harry Dym equation

N-loop solitons are constructed by means of N-cusp soliton solutions of the Harry Dym equation. The proposed approach is based on a novel link between the 'loop soliton equation' and the complex Harry Dym equation considered on a special curve.

Stabilisation of optical solitons by an acousto-optic modulator and filter

Continuously shifting the optical frequency by acousto-optic modulators may substantially improve the stability of a periodically amplified soliton transmission system with filtering. This method, which is analogous to the sliding filter technique, may also be applied to fibre loop soliton generation and storage.

Computer two dimensional maps of loop soliton lattice systems using the new approach to the no integrability Aesthetic field equations

We show that there are varieties of somewhat different loop soliton lattices when we specify an integration path in No Integrability Aesthetic Field Theory. These are illustrated using two dimensional computer maps. We have previously studied several such systems using the new approach to non‐integrable systems developed in previous papers. [1‐3]. The results of these earlier papers indicated that the solitons were rearranged by the new integration scheme in an erratic looking manner. However, we were restricted to regions close to the origin in these studies. With additional computer time made available and the use of tapes to store large amounts of information we have studied the above loop soliton systems and have been able to map considerable larger regions of the x, y plane. Symmetries for locations of the planar maxima and minima have been uncovered within a particular quadrant, although the symmetry found is not as great as the lattice. The type symmetry found is not maintained when more than one quadrant is involved. We also study a system that can be looked at as a perturbation of a loop lattice system. A brief discussion of the background material appears in the appendix.

The connection between fluid and elastostatical turbulence

This work represents a preliminary attempt to establish some connection between loop solitons and the onset of spatial irregularity of motion due to the movement of a cylinder in a fluid. It is shown here that the path taken by certain fluid particles corresponds to a spatial homoclinic orbit and may be rendered chaotic by spatial deterministic fluctuations. Subsequently, a hypothetical model is constructed for an elastic string whose deformation resembles some of the vortex wake produced by the movement of the cylinder.

Soliton chaos models for mechanical and biological elastic chains

The possibility of purely spatial chaos of loop and envelope soliton localization in long elastic strings is considered. Possible connections to some problems in molecular biology are discussed.

Some Remarkable Properties of Two Loop Soliton Solutions

It is shown that, depending on the ratio of the two eigenvalues in the inverse scattering method, the two loop soliton solution has some remarkable properties in collision processes.

On the Modified Korteweg-de Vries Soliton and the Loop Soliton

It is shown that the soliton solution of the modified Korteweg-de Vries (mKdV) equation is essentially the same as the loop soliton solution found by Konno, Ichikawa and Wadati. Another scheme of the inverse scattering transformation is also presented for the mKdV equation.

A Loop Soliton Propagating along a Stretched Rope

A loop solitary wave propagating along a stretched rope has been obtained as one soliton solution of a modified type of the new integrable nonlinear evolution equation discovered by the present authors.