Exact solutions to the KdV-Burgers'' equation

This paper presents two different methods for the construction of exact solutions to the KdVB equation. The first is a direct one based on a combination of solutions to the KdV equation and Burgers' equation. In this approach a number of unknown constants are involved, and it is shown that the equations leading to their determination are properly determined and are capable of solution.

The second method involves a series, and is essentially an extension of Hirota's method. This approach is capable of solving the KdVB equation exactly, and also of generalization to higher order equations with a KdVB-type nonlinearity.     

Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

In this paper, the Lie symmetry analysis and the dynamical system method are performed on an integrable evolution equation for surface waves in deep water

$$\begin{aligned} 2\sqrt{\frac{k}{g}}u_{xxt}=k^2u_x-\frac{3}{2}k(uu_x)_{xx}. \end{aligned}$$

All of the geometric vector fields of the equation are presented, as well as some exact similarity solutions with an arbitrary function of t are obtained by using a special symmetry reduction and the dynamical system method. Different kinds of traveling wave solutions also be found by selecting the function appropriately.

     

Exact solutions to the Korteweg-de Vries-Burgers equation

We introduce a transormation which reduces the Korteweg-de Vries-Burgers (KdVB) equation, u_t + 2auu_x + 5bu_xx + cu_xxx = 0, to a quadratic form involving a new dependent variable and its partial derivatives. Exact solutions of the KdVB equation can be obtained by solving this equation. The exact form of the travelling wave solution to the KdVB equation is obtained in this paper, and its nature depends on the direction of propagation of the wave.     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Exact solutions of a dipolar fluid flow

Exact solutions for three canonical flow problems of a dipolar fluid are obtained: (i) The flow of a dipolar fluid due to a suddenly accelerated plate, (ii) The flow generated by periodic oscillation of a plate, (iii) The flow due to plate oscillation in the presence of a transverse magnetic field. The solutions of some interesting flows caused by an arbitrary velocity of the plate and of certain special oscillations are also obtained.     

EXACT SOLUTIONS OF NEAR CRACK LINE FIELDS FOR MODE I CRACK UNDER PLANE STRESS CONDITION IN AN ELASTIC－PERFECTLY PLASTIC SOLID

ＥＸＡＣＴＳＯＬＵＴＩＯＮＳＯＦＮＥＡＲＣＲＡＣＫＬＩＮＥＦＩＥＬＤＳＦＯＲＭＯＤＥＩＣＲＡＣＫＵＮＤＥＲＰＬＡＮＥＳＴＲＥＳＳＣＯＮＤＩＴＩＯＮＩＮＡＮＥＬＡＳＴＩＣ－ＰＥＲＦＥＣＴＬＹＰＬＡＳＴＩＣＳＯＬＩＤＥＸＡＣＴＳＯＬＵＴＩＯＮＳＯＦＮＥＡ...     

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods

In this paper, we study an integrable nonlinear evolution equation which arises in the context of nonlinear dispersive waves in hyperelastic rods. To consider bounded travelling-wave solutions, we conduct a phase plane analysis. A new feature is that there is a vertical singular line in the phase plane. By considering equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. The trajectories which represent bounded travelling-wave solutions are studied one by one. In total, we find there are 12 types of bounded travelling waves, both supersonic and subsonic. While in literature solutions for only two types of travelling waves are known, here we provide explicit solution expressions for all 12 types of travelling waves. Also, it is noted for the first time that peakons can have applications in a real physical problem.     

Complete plane problems of plastic flow

The present paper is devoted to a complete plane problem of a plastic flow with mixed boundary conditions. A method allowing to carry out a joint construction of stresses and velocities fields has been worked out.     

Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack

Based on the complex potential approach, the two-dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied. The explicit, closed-form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole. When the elliptic hole degenerates into a crack, the field intensity factors are obtained. It is shown that the stress intensity factors are the same as that of isotropic material, while the electric displacement intensity factor depends on both the material properties and the mechanical loads, but not on the electric loads. In other words, the uniform electric loads have no influence on the field singularities. It is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials.     

Exact multi-wave solutions for the KdV equation

By means of the known Darboux transformation, starting from some stationary wave solutions, we construct a large number of new explicit multiple waves solutions to the Korteweg–de Vries equation, including stationary periodic–soliton solutions, stationary soliton–periodic solutions, doubly periodic solutions, triply periodic solutions, as well as two- and three-soliton solutions.     

On elastic strains and a viscoplastic flow in a heavy layer placed on an inclined plane

We present a closed solution of the boundary value problem of the theory of large elastoplastic strains concerned with the straight motions of the medium which is a layer of heavy material placed on the inclined plane with a load on its free surface. Under the conditions that the medium is incompressible, we study the conditions of the flow onset, development, and retardation. Special attention is paid to the repeated loading under which the elastoplastic boundary may enter the region with accumulated plastic strains, and the latter may decrease because of the further motion of this surface.     

Exact solutions of the one-dimensional Russo-Smereka kinetic equation

We obtain new classes of invariant solutions of the integrodifferential equations describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid. For all the solutions obtained, trajectories of particle motion in phase space are calculated. The stability of some flows is studied in a linear approximation. In several cases, the construction of solutions reduces to an integrodifferential equation of the second kind, which can be solved by the iteration method. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 21–32, July–August, 2000.     

Saint-Venant flow in a thin layer under plastic compression

The flow of perfectly rigid-plastic material in a thin layer subjected to a load was considered in numerous studies, including the classical ones [1–12]. In the case of rigid plates coinciding with the layer face surfaces and approaching each other in a prescribed way, we deal with the Prandtl problem, which was considered in [1], or with its numerous generalizations (e.g., see [13, 14]). Traditionally, the process of obtaining the Prandtl solution is based on the hypothesis that the tangential stress is linear in thickness and hence the tangential stress attains its maximum absolute value on the surfaces of the rough plates.     

Some plane solutions of the relativistic navier-stokes equation

Summary We discuss the plane laminar flow of a viscous relativistic fluid, first in the incompressible and thermally non-conducting case, then for a compressible fluid with a non-vanishing heat flow such that the model is completely stationary. Some typically relativistic effects are discussed.

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Sommario Si considera il flusso laminare piano di un fluido viscoso relativistico, sia in condizioni di perfetta incomprimibilità ed adiabaticità, sia nel caso di un fluido comprimibile ed in presenza di un flusso termico tale da assicurare condizioni di completa stazionarietà. Si discutono alcuni effetti tipicamente relativistici.

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This work was in part carried out under the auspices of the Mathematical Research Group no. 115.1032.0.5172 of the Consiglio Nazionale delle Ricerche, and was sponsored in part by the Aerospace Research Laboratories, Office of Aerospace Research, U. S. Air Force, contract F 33615-68-C-1675.