skip to main content

Attention:

The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 11:00 PM ET on Thursday, June 13 until 2:00 AM ET on Friday, June 14 due to maintenance. We apologize for the inconvenience.


This content will become publicly available on July 11, 2024

Title: Rogue waves in the massive Thirring model
Abstract

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index‐ and two dimension‐ones are proved to be consistent by only one constraint relation on parameters of tau‐functions of the KP‐Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright‐type, and the higher order rogue waves represent the superposition of fundamental ones in which the nonreducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of orderNbeing times the background. Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii–Vorob'ev polynomial hierarchy through a linear transformation.

 
more » « less
NSF-PAR ID:
10441538
Author(s) / Creator(s):
 ;  ;  
Publisher / Repository:
Wiley-Blackwell
Date Published:
Journal Name:
Studies in Applied Mathematics
ISSN:
0022-2526
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton. 
    more » « less
  2. Abstract

    We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three‐wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii–Vorob'ev hierarchy, a new feature in the present rogue patterns is that the mapping from the root structure of Okamoto‐hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto‐hierarchy root structures, unless the underlying internal parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto‐hierarchy root structures.

     
    more » « less
  3. null (Ed.)
    Abstract General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $2\times 2$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained. 
    more » « less
  4. Abstract

    Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every pointWof Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

     
    more » « less
  5. Abstract

    In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details.

     
    more » « less