A review is made on recent developments of exact methods for solving soliton equations. Among various exact methods known nowadays, we mainly survey the bilinear transformation method (BTM) initiated by R. Hirota. The essence of the BTM is illustrated concretely by applying it to the KdV equation which is a typical soliton equation.
The relations of the BTM to other exact methods such as the inverse scattering method, the Backlund transformation and the Sato theory (the theory of τ function) are also discussed. Finally, some linearizable nonlinear equations are considered on the basis of the BTM.

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A review is given on the basic idea of the method of renormalized expansions and on the properties of Eulerian and Lagrangian approximations of turbulence derived by the method.

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The analytical methods by which nonlinear interfacial fluid motions can be modelled are introduced. The partial differential equations which completely satisfy the interfacial conditions or the free surface conditions are derived. In particular, the model equation of internal waves involves two indefinite functions and these are related to the difference of velocities and the difference of pressure gradients at the interface respectively.
Furthermore, for irrotational motions the indefinite function can be determined so as to satisfy other boundary conditions such as bottom conditions. Finally, the modelling of the Rayleigh-Taylor problem and the breaking waves in the finite depth channel are discussed by the present method.

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不飽和状態の土壌中の浸透流のランダム・ウォーク法による数値シミュレーションを行った.平衡状態にある土壌に一定強度の降雨が一定時間継続した後止むという状況における土壌中の水分量の時間的変化および水分の移動の様子が求められた.一様な透水係数を持つ土壌と,植生の存在を想定し場所的に透水係数が異なる土壌における浸透流が計算され,既存の実験結果とよい一致が得られた.

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本報は解析と数値解で求めた球周りの流れの形および球が受ける流体力の結果をRey-nolds 数(Re数)0.01～120の範囲で比較し,中間Reynolds数での解析結果を調べる.解析と数値解の基礎式にはそれぞれOseen方程式とNavier-Stokes方程式を使用する.低Reynolds数の範囲ではProudman-Pearsonの方法で求めた球周りの流線はOseen解よりも数値解からそれ,かならずしも彼らの結果がよい結果を与えているとは言えない.又Oseen解と数値解で求めた流れの形が一致するRe数の範囲はRe≦1.0であるが,中間Reynolds数での特性値の比較結果はこの範囲のRe数でもOseen方程式を用いた定性的な議論が可能であることを示す.しかし,定量的にはかなりの違いがあり,例えば,解析と数値解による球の渦度発生の臨界Reynolds数はRe=7.7と21になる.
最後に,Navier-Stokes方程式の逐次近似をOseen解を用いて数値計算で行なった結果を示す.

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A numerical method for computing inviscid transonic flow around an arbitrary aircraft configuration is described in this paper. Basic equations are simultaneous equations for velocity potential and Mach number. These equations are equivalent to the full potential equation. The simultaneous equations are numerically solved satisfying given boundary conditions. Shock waves are captured as discontinuous surfaces where shock waves are expected to occur.
The calculated flow field is a rectangular box, where grid points and boundary points are generated in Cartesian coordinates. The grid points are equidistantly distributed, but their intervals can be varied with the location. External forms of the aircraft and lifting surfaces are shaped by use of numerous small triangular planes. The majority of the boundary points are on these external forms. The basic equations are expressed as difference equations using the grid points and the boundary points. The difference equations are solved in the numerical procedure. Several calculated results of this method are compared with the experimental and other numerical results.

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