Study On Theory And Solving Method For Propagation And Interaction Of Finite-amplitude Waves And Solitary Waves In Fluid And Solid Media

Such characters of medium as the nonlinearity, thermoviscous and relaxation can strongly influence the behaviors of nonlinear wave in fluid and solid media, and their effects are different; either their total or partial influence might lead to the new wave phenomena. From the two aspects, the systematical analysis of properties, interactions of nonlinear wave and proposing, development, applications of new methods, we launched our research. Main achievements in research are reflected in published six pieces of paper. In first part of this paper, we directly and systematically investigated the effects of media by using new analytical, numerical and graphical methods. Further, we explicitly studied the interactions between the shock waves, solitary waves that are formed in the media; also explored the influence of interaction of solitons on the solid medium. (1) Taking as a basic, we studied the essential propagating properties of finite-amplitude wave in the ideal fluid medium, and the result indicated that the waveform aberration, generation of harmonic waves and formation of discontinuous shock wave are basic properties of finite-amplitude wave in the ideal fluid medium. (2) On summarizing former works, furthermore, we studied the propagating behaviors of finite-amplitude wave in thermoviscous fluid medium by using the exact solution of Burgers equation, and the analyzing result indicated that because of the combined effects of nonlinear and dissipative, not the discontinuous shock wave but the continuous shock wave formed in the medium. Next we obtained the multiple shock wave solutions of Burgers equation, using new analytical method—homogeneous balance method. By means of multiple shock wave solutions, we investigated interaction of shock waves, and found that (in the retarded coordinate) when the two shock wave's amplitude are different, after collided face to face, they fused into another shock wave which amplitude is the sum of the former two shock wave's amplitudes and the velocity is the subtract of two shock waves; when the two shock wave's amplitude are equal, after collided face to face, they would fused into a non-propagation shock wave which amplitude is the sum of the former two wave; when their amplitude are different, after collided by catch manner, they would become a shock wave which amplitude is the sum of the former two shock waves and the velocity is the subtract of two shock waves. (3) We numerically stimulated one kind of nonlinear wave equation of relaxation fluid medium by means of finite difference method. From the numerical results can observe the waveform of finite-amplitude wave gradually became steepening and formed discontinuous shock wave, meanwhile the symmetry wave gradually became asymmetry. In the following step, we studied the KdV equation of infinite relaxation fluid medium by means of improved homogeneous balance method, and obtained the two soliton solutions. We explicitly analyzed the interactions between the solitons, singular solitons and between the soliton and singular soliton, the result indicated that (in the retarded coordinate) except the interaction of bell type solitons have the particle elastic scattering characters, the singular solitons also posses this kind of characters. In the case of interacting between soliton and singular soliton, when the quicker bell type soliton catch up the slower singular soliton and collided each other, result in the amplitude of bell type soliton increase dramatically; after the collision their waveform and velocity remain unchanged. But in the reverse condition when the quicker singular soliton catch up the slower bell type soliton and collided each other, all their properties remain unchanged. (4) Firstly, we investigated the propagation problems of strain waves in a kind of nonlinear solid medium. Using new stress—strain relation for solid medium proposed by Hokstad (2004), we obtained one-dimensional nonlinear wave equation, i.e. combined Boussinesq equation for describing nonlinear wave propagation in infinite solid medium. From the combined Boussinesq equation we obtained a non-dispersive nonlinear wave equation, under the neglecting dispersion effect. We solved the non-dispersive nonlinear wave equation, and obtained strain simple wave solutions, by means of characteristic method. This confirmation the fact that the waveform of strain wave also come into being aberration and generate discontinuous shock wave. Secondly, we studied solitary waves formation and propagation in infinite one-dimensional solid medium with nonlinearity and dispersion, and solitary waves interaction and its influence on solid medium. Using improved homogeneous balance method, we solved several Boussinesq equations obtained from combined Boussinesq equation, and obtained two-soliton solutions. With the help of these solutions we found the expressions of maximum strain and maximum stress caused by collision of two small amplitude solitons. We calculated maximum stresses caused by collision of two small amplitude solitons propagating in different rock media, compared these stresses with strength of rocks and then analyzed possible damage to rocks. Our analysis show that maximum stress caused by collision of two solitons of "good"Boussinesq equation may cause damage to ordinary rocks, but maximum stress caused by collision of two solitons of "bad"Boussinesq equation may not cause damage toordinary rocks, under the condition of small amplitude of strain. Even if the amplitudes of strain soliton are very small, maximum tensile stress caused by collision of soliton and anti-soliton of modified Boussinesq equation may cause damage to ordinary rocks, but maximum compressive stress caused by collision of soliton and anti-soliton of modified Boussinesq equation may not cause damage to ordinary rocks. From the above analysis, we concluded that the interaction of strain solitons is a possible mechanism to damage rock medium, under the certain conditions. But this conclusion is a tentative one, so need further verification. In second part of this paper, we conducted innovative investigation of solving methods for nonlinear wave equations. (1) We improved some important processes of the homogeneous balance method proposed recently, thereby the method is more straightforward and more effective for the application in some nonlinear equations (including higher-dimensional equations). Employing the improved method, we studied one-dimensional and two-dimensional shallow water wave equations and found new multiple soliton-like solutions (solitary wave solutions). Using graphical method, we analyzed detailedly interaction between solitary waves, and found some new phenomena of solitary waves interaction. As for the kink solitary wave and bell solitary wave of (1+1)-dimensional dispersive long wave equation, after the interactions, the kink solitary wave and bell solitary wave could fuse, split or form non-propagating kink solitary wave and bell solitary wave. As for the rotational kink solitary wave and bell solitary wave of (2+1)-dimensional dispersive long wave equation, their interactions may cause many complex phenomena. Mainly, after the collision of two variable amplitude line kink solitary waves with different rotation velocity and opposite rotation direction, they fused into a variable amplitude line kink solitary wave, which rotate in the same direction as the line kink solitary wave that had the larger rotation velocity; Whereas those with the same rotation velocity and opposite rotation direction fused into a variable amplitude line kink solitary wave, and stopped at the location of collision. The collision behavior of variable amplitude curve kink solitary waves is same as that of line kink solitary wave. After the collision of variable amplitude line bell solitary wave with larger velocity and anti-bell solitary wave with small velocity and opposite rotation direction, they fused into a line anti-bell solitary wave, which rotate in the same direction as the line bell solitary wave that had the larger rotation velocity; Whereas the collision of variable amplitude line bell solitary wave and anti-bell solitary wave with same rotation velocity and opposite rotation direction, they fused into a equal amplitude kink-like solitary wave, and stopped at the location of collision. After the collision of variable amplitude curve bell solitary wave with larger velocity and anti-bell solitary wave with small velocity and opposite rotation direction, they fused into a curve anti-bell solitary wave, which rotate in the same direction as the curve bell solitary wave thathad the larger rotation velocity; Whereas the collision of variable amplitude curve bell solitary wave and anti-bell solitary wave with the same rotation velocity and opposite rotation direction, they fused into a variable amplitude kink-like solitary wave, and stopped at the location of collision. (2) We proposed a new function transformation method. Using this method solved several KdV like nonlinear wave equations and obtained bell type solitary wave solutions, kink type solitary wave solutions, spiky type solitary wave solutions and singular solitary wave solutions. Practice shows that the method is concise and effective, and suit for solving a class of equations, especially those equations with higher order nonlinear terms, this method has extraordinary advantages. Also, this method offered us effective approach degenerate hyperelliptic integral into elementary integral. (3) We introduced a new approach, and solved a kind of nonlinear dispersive-dissipative equation using this approach. We obtained a series of new exact solutions of physical interest such as solitary wave solutions, triangular function solutions, exponential function solutions and singular solitary wave solutions.