Tsutomu Kawata

Inverse Method for the Mixed Nonlinear Schrödinger Equation and Soliton Solutions

A mixed nonlinear Schrodinger equation which has a usual cubic nonlinear term and a derivative cubic term is exactly solved by the inverse scattering method under the nonvanishing boundary condition. We obtain the soliton solution which generally pulsates. But it becomes stationary which is algebraic in some special cases. A peculiar structure of spiky modulation is also shown.

Simulation of a Collision between Shock Waves and a Magnetic Flux Tube: Excitation of Surface Alfvén Waves and Body Alfvén Waves

To explain the observed dynamics of the small-scale magnetic flux tubes in the quiet photospheric network, Furusawa & Sakai presented simulation results on the collision of two flux tubes. They found that shock waves appear during the collision of two magnetic flux tubes, when two magnetic flux tubes with weak electric current collide with each other. The shock waves so generated can subsequently collide with another flux tube, and we investigate here the interaction process of the shock with the flux tube. It is found that during the collision of a shock wave with a magnetic flux tube with weak electric current, surface Alfven waves can be generated and propagate along the flux tube. However, when the shock wave collides with a magnetic flux tube with strong current, body Alfven waves can be generated and propagate along the flux tube. It is also shown that, when we take into account the effect of a background density inhomogeneity due to gravity, there occurs a strong upward plasma jet along the flux tube, as well as surface Alfven waves. The energy conversion rate from the shock wave energy to the upward MHD waves, as well as upward plasma flows, is about 40% and thus is very efficient. We apply our results to the problem of solar coronal heating.

Nonlinear Torsional and Compressional Waves in a Magnetic Flux Tube with Electric Current near the Quiet Solar Photospheric Network

Recent high-resolution observations from photospheric magnetograms made with the SOHO/Michelson Doppler Imager instrument and the Swedish Vacuum Solar Telescope on La Palma showed that magnetic flux tubes in the quiet photospheric network of the solar photosphere are highly dynamic objects with small-scale substructures. We investigate nonlinear waves propagating along a magnetic flux tube in weakly ionized plasmas with high plasma beta (β 1) by using three-dimensional neutral MHD equations. Recently Sakai et al. investigated nonlinear wave propagation along a magnetic flux tube with a weak current for the two cases of uniform density along the flux tube and density inhomogeneity due to solar gravity. They showed that shear Alfven waves are excited by localized, predominantly rotational perturbations and that excited waves with a strong upflow of wave energy can propagate only upward along the flux tube when density inhomogeneity due to gravity is taken into account. In this paper we extend this work by investigating nonlinear torsional and compressional waves in a magnetic flux tube with a strong electric current, i.e., a twisted magnetic field, near the quiet solar photospheric network. If gravity is neglected, the torsional waves are found to propagate in a direction such as to decrease the twist of the magnetic field, while the compressional waves propagate symmetrically. We have found that solar gravity results in the important effect that wave energies excited by both torsional and compressional disturbances can be transferred upward in both untwisted and highly twisted flux tubes and eventually contribute to coronal heating.

Matrix Piemann Problem of the AKNS Equation

The AKNS equation is reformulated by means of the matrix Riemann problem. The projection operator arising for solving the Riemann problem is connected to the scattering data. By this fact we can completely separate the procedure for solving the problem to the determination of projection operator and to the regular problem consisting only with continuous scattering data. As an application of this we construct the way of potential proliferation.

Simulation of Nonlinear Waves in a Magnetic Flux Tube near the Quiet Solar Photospheric Network

Recent high-resolution observations from photospheric magnetograms made with the SOHO/Michelson Doppler Imager instrument and Swedish Vacuum Solar Telescope on La Palma showed that magnetic flux tubes in the quiet photospheric network of the solar photosphere are highly dynamic objects with small-scale substructures. We investigate nonlinear waves propagating along a magnetic flux tube in weakly ionized plasmas with high plasma beta ( β 1) by using three-dimensional neutral-MHD equations. We investigate the wave propagation along a magnetic flux tube with weak current for the two cases of uniform density along the flux tube and density inhomogeneity due to solar gravity. It is shown that shear Alfven waves are excited due to localized predominantly rotational perturbations, which might be induced in the quiet photospheric network boundaries. Excited waves with strong upflow of wave energy can propagate only upward along the flux tube when the density inhomogeneity due to the gravity is taken into account. We apply the simulation results to the problem of coronal heating from the quiet photospheric network of the solar photosphere.

2×2-Matrix Riemann-Hilbert Transform and Its Connection to the Continuous Scattering Data

A basic 2×2-matrix Riemann-Hilbert problem is studied and its complete connection to the associated eigenvalue problem is made clear. A potential formula independent on the spectral parameter λ is derived by considering a transformation contributed from the continuous data. As a by-product we get a mapping from the variation of scattering data to the one of potentials.

Propagation Speed of a Magnetic Flux-tube Soliton with Electric Current

We present some results of a magnetic flux-tube soliton propagating along a current loop surrounded by a weakly ionized plasma, by using a 3-D Neutral-MHD simulation code. When the velocity of mass flows outside the current loop exceeds about 0.6υA, the magnetic pulse behaves as an isolated string wave which is called a curved soliton, propagating with a velocity less than that one of exterior mass flow. The propagation speed of the magnetic flux-tube soliton is studied by changing the intensity of the electric current along the flux tube, which usually cannot be observed directly. It is found that the soliton speed decreases proportionally to the increment of the electric current, and the speed is independent of the direction of the electric current. We can estimate the current intensity inside a magnetic flux-tube soliton by observations of the soliton speed and the external plasma flow velocity. These results should be compared with recent high-resolution observations of moving magnetic features (MMFs) observed near sunspots.

Current-Vortex Filament and Soliton-Like Waves in Weakly Ionized Plasmas

We present results of magnetic flux-tube waves propagating along a current loop in a weakly ionized plasma, by means of a 3-D Neutral-MHD simulation code. After an impulsive velocity perturbation, the magnetic wave takes a tube structure and behaves as an isolated and stable string, to say a magnetic flux-tube soliton. Its velocity is associated with the electric current of the loop and a vortex induced inside the tube. There exists an interval of the electric current, where the velocity of the soliton is nearly proportional to the current. While on the stronger current region the velocity of the soliton is almost constant. The characteristics of the magnetic flux-tube soliton observed in this simulation can be explained for a simple MHD model, in which we give necessary treatments to reduce the equation of current-vortex filament to a frame of integrable system.

Derivation of Bäcklund Transformation for the AKNS-Integrable System by Means of the Riemann-Hilbert Problem

Using a direct proliferation technique based on the Riemann-Hilbert problem, we obtained two systematic ways for deriving Backlund transformations. The first is directly given by the potential proliferation formula, while the second is obtained from singular transformations with fractional forms of Riccati equations. Both produce the same result.

Integration of Linear-Partial-Differential Equations Arising from a N×N-Matrix Spectral Problem

We study the partial-differential system (LPDE) obtained from linearization of a primitive nonlinear equation (NLPDE) integrable by virture of the N × N -matrix spectral problem. Solutions of the LPDE is exactly related with “squared eigenstates” appropriately defined in the spectral equation. The completeness of squared eigenstates is derived by applying the “Riemann-Hilbert” transformation (RHT) and triangular factorization procedure (TFP) for matrices. Finally the LPDE is integrated by using a Green function uniquely defined for the LPDE.