流体与固体介质中有限振幅波、孤立波的传播和相互作用的理论与求解方法研究
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摘要
流体与固体介质中的非线性波动现象与介质的非线性、粘热性及弛豫性密切相关。这些性质对非线性波在介质中传播时产生的效应不同,它们的共同作用或它们中部分的作用可产生一些新的波动现象。本文从非线性波基本传播特性、相互作用的系统分析到新求解方法的提出、发展与应用等两大方面开展研究。
第一部分中,我们采用新的解析方法、数值方法以及图形分析方法,对非线性、粘热性及弛豫性等性质产生的效应进行了进一步直观、系统的分析。特别是,由它们中部分性质的竞争作用而产生的冲击波、孤立波的相互作用进行了详细的分析,进而发现了相互作用产生的合并、不传播冲击波以及孤立子与奇异孤立子碰撞时产生的“幅度倍增”等新的现象。同时,对固体介质中传播的应变孤子的相互作用对固体介质造成的影响进行了试探性的研究,初步得到在一定条件下应变孤立子的相互作用是造成岩石介质破裂的一种可能的机制。
第二部分中,对非线性波动方程的解析求解方法进行了一些创新性的研究。首先,我们对新提出的齐次平衡法的一些关键的步骤进行了改进,使该方法对某些方程组(包括高维方程组)的求解变的更简便、更有效。用改进后的方法研究了一维和二维的浅水波方程,得到了它们新的类多孤子解。根据得到的解分析孤立波之间的相互作用,从而发现了相互作用产生的合并、分裂、不旋转孤立波、波形的转换以及“性质”的改变等一些新的现象。其次,我们还提出了一种新的函数变换方法。用该方法研究了一些重要的非线性波动方程,得到了它们新的孤立波解。在应用中发现该方法对高次非线性波动方程的求解具有独特的优点。同时,该方法还提供了把超椭圆积分退化成初积分的一种有效的途径。最后,我们介绍了一种新提出的初积分方法,并用该方法研究一种非线性频散——耗散方程,得到了该方程的一系列新的精确行波解。
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 to
ordinary 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 that
had 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.
第一部分中,我们采用新的解析方法、数值方法以及图形分析方法,对非线性、粘热性及弛豫性等性质产生的效应进行了进一步直观、系统的分析。特别是,由它们中部分性质的竞争作用而产生的冲击波、孤立波的相互作用进行了详细的分析,进而发现了相互作用产生的合并、不传播冲击波以及孤立子与奇异孤立子碰撞时产生的“幅度倍增”等新的现象。同时,对固体介质中传播的应变孤子的相互作用对固体介质造成的影响进行了试探性的研究,初步得到在一定条件下应变孤立子的相互作用是造成岩石介质破裂的一种可能的机制。
第二部分中,对非线性波动方程的解析求解方法进行了一些创新性的研究。首先,我们对新提出的齐次平衡法的一些关键的步骤进行了改进,使该方法对某些方程组(包括高维方程组)的求解变的更简便、更有效。用改进后的方法研究了一维和二维的浅水波方程,得到了它们新的类多孤子解。根据得到的解分析孤立波之间的相互作用,从而发现了相互作用产生的合并、分裂、不旋转孤立波、波形的转换以及“性质”的改变等一些新的现象。其次,我们还提出了一种新的函数变换方法。用该方法研究了一些重要的非线性波动方程,得到了它们新的孤立波解。在应用中发现该方法对高次非线性波动方程的求解具有独特的优点。同时,该方法还提供了把超椭圆积分退化成初积分的一种有效的途径。最后,我们介绍了一种新提出的初积分方法,并用该方法研究一种非线性频散——耗散方程,得到了该方程的一系列新的精确行波解。
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 to
ordinary 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 that
had 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.
引文
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Zhang J.F., 1999, Multiple soliton solutions of the dispersive long-wave equations, Chin. Phys. Lett., 16(1): 4-5
Zhang J.F., 2000, Backlund transformation and multisoliton-like solutions for (2+1) dimensional dispersive long wave equations, Commun. Theor. Phys., 33(4): 577-580
Zhang J.F., 2002, Exotic localized coherent structures of the (2+1)-dimensional dispersive long-wave equation, Commun. Theor. Phys., 37(3): 277-282
Zhang S.Y. and Zhuang W., 1987, The strain solitary waves in a nonlinear elastic rod, Acta Mechanica Sinica, 3(1): 3–72
Zheng H.S., Zhang Z.J. and Yang B.J., 2004, A numerical study of 1-D nonlinear P-wave propagation in solid, Acta Seismologica Sinica, 17(1): 80-86
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Zhou S.Q.,and Shui, Y.A., 1992, Nonlinear reflection of bulk acoustic waves at an interface, J. Appl. Phys. 72 (11): 5070-5080
戴世强, 1982, 两层流体界面上的孤立波, 应用数学和力学,3: 721-728
杜功焕,2001,声学基础,南京:南京大学出版社
范恩贵,张鸿庆,1998, 非线性孤子方程的齐次平衡法,物理学报,47(3):353-361
谷超豪, 1990,孤立子理论与应用,杭州:浙江科学技术出版社
黄国翔,颜家壬,戴显熹,1990,矩形谐振器中非传播重力—表面张力孤波的理论研究,物理学报,39(8):1234-1241
姜文华,水永安,毛一葳,杜功焕,1996,有限振幅声波在固体中传播的非线性特性,物理学进展,16 (3-4): 303-312
缪国庆,王本仁,1996,槽中水波和非传播孤立子,物理学进展,16(3-4):273-285
李志斌,张善卿,1997, 非线性波方程准确孤立波解的符号计算,数学物理学报,17(1): 81-89
刘财,冯晅,何敏,杨宝俊,赵建国,2000,非线性波动理论在地震勘探中的应用综述,地球物理学进展,15(4):68-78
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