arXiv subjects

Yoshimasa Matsuno

Publications and source records attributed to Yoshimasa Matsuno.

26 records · Page 2Linked to original sources

arXiv2010

A direct method for solving the generalized sine-Gordon equation II

The generalized sine-Gordon (sG) equation $u_{tx}=(1+ν\partial_x^2)\sin\,u$ was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with $ν=-1$. Here, we address the equation with $ν=1$. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. Last, we provide a recipe for deriving an infinite number of conservation laws by using a novel Bäcklund transformation connecting solutions of the sG and generalized sG equations.

nlin.SI↗

arXiv2010

A direct method for solving the generalized sine-Gordon equation

The generalized sine-Gordon (sG) equation was derived as an integrable generalization of the sG equation. In this paper, we develop a direct method for solving the generalized sG equation without recourse to the inverse scattering method. In particular, we construct multisoliton solutions in the form of parametric representation. We obtain a variety of solutions which include kinks, loop solitons and breathers. The properties of these olutions are investigated in detail. We find a novel type of solitons with a peculiar structure that the smaller soliton travels faster than the larger soliton. We also show that the short pulse equation describing the propagation of ultra-short pulses is reduced from the generalized sG equation in an appropriate scaling limit. Subsequently, the reduction to the sG equation is briefly discussed.

nlin.PS↗

arXiv2009

Soliton and Periodic solutions of the Short Pulse Model Equation

The short pulse (SP) equation is a novel model equation describing the propagation of ultra-short optical pulses in nonlinear media. This article reviews some recent results about the SP equation. In particular, we focus our attention on its exact solutions. By using a newly developed method of solution, we derive multisoliton solutions as well as 1-and 2-phase periodic solutions and investigate their properties.

nlin.SI↗

arXiv2008

Periodic solutions of a resistive model for nonlocal Josephson dynamics

A novel method is developed for constructing periodic solutions of a model equation describing nonlocal Josephson electrodynamics. This method consists of reducing the equation to a system of linear ordinary differential equations through a sequence of nonlinear transformations. The periodic solutions are then obtained by a standard procedure which are represented in terms of trigonometric functions. It is found that the large time asymptotic of the solution exhibits a steady profile which does not depend on initial conditions.

nlin.SI↗

arXiv2007

The Peakon Limit of the N-Soliton Solution of the Camassa-Holm Equation

We show that the analytic N-soliton solution of the Camassa-Holm (CH) shallow-water model equation converges to the nonanalytic N-peakon solution of the dispersionless CH equation when the dispersion parameter tends to zero. To demonstrate this, we develop a novel limiting procedure and apply it to the parametric representation for the N-soliton solution of the CH equaiton. In the process, we use Jacobi's formula for determinants as well as various identities among the Hankel determinants to facilitate the asymptotic analysis. We also provide a new representation of the N-peakon solution in terms of the Hankel determinants.

nlin.SI↗

arXiv2006

The Lyapunov stability of the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation

The Lyapunov stability is established for the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono (BO) equation. We characterize the N-soliton profiles as critical points of certain Lyapunov functional. By using several results derived by the inverse scattering transform of the BO equation, we demonstarate the convexity of the Lyapunov functional when evaluated at the N-soliton profiles. From this fact, we deduce that the N-soliton solutions are energetically stable.

nlin.SI↗

arXiv2005

The N-soliton solution of the Degasperis-Procesi equation

This paper extends the results of the previous paper designated I hereafter in which the one- and two-soiton solutions of the Degasperis-Procesi(DP) equation were obtained and their peakon limit was considered. Here, we present the general N-soliton solution of the DP equation and investigate its property. We show that it has a novel structure expressed by a parametric representation in terms of the BKP tau-functions. A purely algebraic proof of the solution is given by establishing various identities among the tau-functions. The large time asymptotic of the solution recovers the formula for the phase shift which was derived in I by a different method. Finally, the structure of the N-soliton solution is discussed in comparison with that of the Camassa-Holm shallow water wave equation.

nlin.SI↗

arXiv2005

Parametric Representation for the Multisoliton Solution of the Camassa-Holm Equation

The parametric representation is given to the multisoliton solution of the Camassa-Holm equation. It has a simple structure expressed in terms of determinants. The proof of the solution is carried out by an elementary theory of determinanats. The large time asymptotic of the solution is derived with the fomula for the phase shift. The latter reveals a new feature when compared with the one for the typical soliton solutions. The peakon limit of the phase shift ia also considered, showing that it reproduces the known result.

nlin.SI↗