One-loop kink mass shifts: a computational approach

In this paper we develop a procedure to compute the one-loop quantum correction to the kink masses in generic (1+1)-dimensional one-component scalar field theoretical models. The procedure uses the generalized zeta function regularization method helped by the Gilkey-de Witt asymptotic expansion of the heat function via Mellin's transform. We find a formula for the one-loop kink mass shift that depends only on the part of the energy density with no field derivatives, evaluated by means of a symbolic software algorithm that automates the computation. The improved algorithm with respect to earlier work in this subject has been tested in the sine-Gordon and $\lambda(\phi)_2^4$ models. The quantum corrections of the sG-soliton and $\lambda(\phi^4)_2$-kink masses have been estimated with a relative error of 0.00006% and 0.00007% respectively. Thereafter, the algorithm is applied to other models. In particular, an interesting one-parametric family of double sine-Gordon models interpolating between the ordinary sine-Gordon and a re-scaled sine-Gordon model is addressed. Another one-parametric family, in this case of $\phi^6$ models, is analyzed. The main virtue of our procedure is its versatility: it can be applied to practically any type of relativistic scalar field models supporting kinks.Comment: 35 pages, 6 figures, to be published in Nuclear Physics

Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes

We generalize the method of our recent paper on the large-spin expansions of Gubser-Klebanov-Polyakov (GKP) strings to the large-spin and large-winding expansions of finite-size giant magnons and finite-size single spikes. By expressing the energies of long open strings in RxS2 in terms of Lambert's W-function, we compute the leading, subleading and next-to-subleading series of classical exponential corrections to the dispersion relations of Hofman-Maldacena giant magnons and infinite-winding single spikes. We also compute the corresponding expansions in the doubled regions of giant magnons and single spikes that are respectively obtained when their angular and linear velocities become smaller or greater than unity.Comment: 43 pages, 13 figures; Matches published version. Rewritten appendix

On the variational structure of breather solutions

In this paper we give a systematic and simple account that put in evidence that many breather solutions of integrable equations satisfy suitable variational elliptic equations, which also implies that the stability problem reduces in some sense to $(i)$ the study of the spectrum of explicit linear systems (\emph{spectral stability}), and $(ii)$ the understanding of how bad directions (if any) can be controlled using low regularity conservation laws. We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV), Gardner, and sine-Gordon (SG) equations. Then we perform numerical simulations that confirm, at the level of the spectral problem, our previous rigorous results, where we showed that mKdV breathers are $H^2$ and $H^1$ stable, respectively. In a second step, we also discuss the Gardner and the Sine-Gordon cases, where the spectral study of a fourth-order linear matrix system is the key element to show stability. Using numerical methods, we confirm that all spectral assumptions leading to the $H^2\times H^1$ stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime. In a second part, we study the periodic mKdV case, where a periodic breather is known from the work of Kevrekidis et al. We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability. However, we also identify the source of nonlinear instability in the case described in Kevrekidis et al. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.Comment: 55 pages; This paper is an improved version of our previous paper 1309.0625 and hence we replace i

Kink fluctuation asymptotics and zero modes

In this paper we propose a refinement of the heat kernel/zeta function treatment of kink quantum fluctuations in scalar field theory, further analyzing the existence and implications of a zero energy fluctuation mode. Improved understanding of the interplay between zero modes and the kink heat kernel expansion delivers asymptotic estimations of one-loop kink mass shifts with remarkably higher precision than previously obtained by means of the standard Gilkey-DeWitt heat kernel expansion.Comment: 21 pages, 8 figures, to be published in The European Physical Journal

Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.Comment: 60 pages, 18 figure

Symbolic computation of solitary wave solutions and solitons through homogenization of degree

A simplified version of Hirota's method for the computation of solitary waves and solitons of nonlinear PDEs is presented. A change of dependent variable transforms the PDE into an equation that is homogeneous of degree. Solitons are then computed using a perturbation-like scheme involving linear and nonlinear operators in a finite number of steps. The method is applied to a class of fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic homogeneous equations for which the bilinear form might not be known. Furthermore, homogenization of degree allows one to compute solitary wave solutions of nonlinear PDEs that do not have solitons. Examples include the Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When applied to a wave equation with a cubic source term, one gets a bi-soliton solution describing the coalescence of two wavefronts. The method is largely algorithmic and is implemented in Mathematica.Comment: Proceedings Conference on Nonlinear and Modern Mathematical Physics (NMMP-2022) Springer Proceedings in Mathematics and Statistics, 60pp, Springer-Verlag, New York, 202