Tetsu Yajima

Soliton Solution and Its Property of Unstable Nonlinear Schrödinger Equation

A nonlinear evolution equation i u x + u t t +κ| u | 2 u =0, κ>0, is derived from unstable Sine-Gordon equation u t t }- u x x - m 2 sin u =0. The former nonlinear evolution equation, which we call unstable nonlinear Schrodinger equation, is solved by the inverse scattering method. It is shown that the position shifts for fast (slow) solitons, arising from their mutual collisions, are negative (positive). This is a property common to both nonlinear evolution equations.

Local state probabilities for solvable restricted solid-on-solid models:An,Dn,D n (1) , andA n (1)

The local state probabilities (LSPs) are exactly computed for four hierarchies of solvable lattice models. They are restricted solid-on-solid (RSOS) models whose local states and their adjacent conditions are specified by Dinkin diagrams of typesAn,Dn,Dn(1) andAn(1). The LSPs are expressed in terms of modular functions characterized by branching identities among the theta functions. Their automorphic properties are used to study the critical behaviors. Some fine structures are found in the spectrum of the critical exponents.

Solitons in an Unstable Medium

Taking a model equation u t t - u x x - m 2 sin u =0, the behavior and the roles of solitons in unstable media are studied. The properties of solitons and their mutual collisions are discussed based on the exact solutions derived by the inverse scattering method. Furthermore, an initial boundary value problem of the model equation is studied numerically. It is concluded that creation of solitons serves to stabilize the system.

The theory and applications of the unstable nonlinear Schrödinger equation

Abstract Recent works on the unstable nonlinear Schrodinger (UNS) equation, iqx+qn+2|q|2q=0, are reviewed. It is shown that initial value problems can be solved by the inverse scattering method. Exact N soliton solutions gives an information on the soliton interaction in unstable media. Namely, the position shifts due to the collisions have opposite signs compared with the conventional nonlinear Schrodinger equation, iqt+qxx + 2|q|2q = 0. As applications of the UNS equation, two physical systems, the Rayleigh-Taylor instability problem and electron beam plasma, are discussed. In both systems, the dispersion relation has a critical wave number below which frequency becomes a complex number. It is shown that near the critical wave number the wave amplitude obeys the UNS equation. It is concluded that the UNS equation is a canonical equation which describes nonlinear modulations of wave amplitude in unstable media.

Local state probabilities for an infinite sequence of solvable lattice models

The authors present a new infinite sequence of solvable lattice models. They contrast strikingly with the eight-vertex solid-on-solid models and admit extra degrees of freedom for the local fluctuation variables. The exact one-point functions are obtained. The result is neatly described in terms of theta-function identities. Using their modular invariance, critical behaviour is studied and exponents evaluated.

Numerical Studies on Stability of Dromion and Its Collisions

Stability of the dromion solution of the Davey-Stewartson equations is investigated numerically. Propagation of dromions is found to be stable in a Lyapunov sense. Collisions of two one-dromions are also examined. It is observed that the collision process is inelastic and the initial two dromions break into four pulses after the collision.

Soliton phenomena in unstable media

Abstract As a canonical equation for soliton phenomena in unstable media, the unstable nonlinear Schrodinger (UNS) equation, i q x + q tt +2| q | 2 q =0 is proposed. The UNS equation is derived for two physical systems, Rayleigh-Taylor instability and electron beam plasma. It is shown that the initial value problem can be solved by the inverse scattering method. The exact N -soliton solution gives information on the soliton interaction: the faster (slower) soliton has negative (positive) position shift. It is thus concluded that soliton interaction in unstable media is attractive. Further, it is reported that the UNS equation explains a recent observation on propagations of localized structures in electron beam plasma.

The unstable nonlinear Schrödinger equation and Dark solitons

Initial value problem of the unstable nonlinear Schrodinger (UNS) equation i u x + u t t -2| u |^2u=0 w i t h b o u n d a r y c o n d i t i o n |u|→1 (x→±∞), i s s o l v e d b y t h e i n v e r s e s c a t t e r i n g m e t h o d . A s a n a p p l i c a t i o n , d a r k s o l i t o n s o l u t i o n s a r e o b t a i n e d . P o s i t i o n s h i f t d u e t o a c o l l i s i o n o f t w o s o l i t o n s i s s h o w n t o b e a t t r a c t i v e ( r e s . r e p u l s i v e ) w h e n t h e d i r e c t i o n s o f t h e i r v e l o c i t i e s a r e o p p o s i t e ( r e s . s a m e ). I t i s a l s o s h o w n t h a t t h e U N S e q u a t i o n h a s a n i n f i n i t e n u m b e r o f c o n s e r v e d q u a n t i t i e s .

Time evolution of Gaussian-type initial conditions associated with the Davey - Stewartson equations

The Cauchy problem of the Davey - Stewartson equations with non-trivial boundaries is studied. Initial conditions for the equations are chosen to have Gaussian-type envelope shapes, and the time evolution is investigated both theoretically and numerically. It is found that an initial packet grows and oscillates radiating ripples, then a localized structure called a dromion appears asymptotically. It is also observed that the ripples run away mainly along the mean flows. The results of numerical simulations and the analysis by the inverse scattering transform show good agreement with each other.

Exact Solutions and Flow–Density Relations for a Cellular Automaton Variant of the Optimal Velocity Model with the Slow-to-Start Effect

A set of exact solutions for a cellular automaton, which is a hybrid of the optimal velocity and the slow-to-start models, is presented. The solutions allow coexistence of free flows and jamming or...