Solitary Waves And Their Stability Analysis In Nonlinear Phononic Crystals

Propagation of acoustic and elastic waves in nonlinear phononic crystals has been received considerable attention because understanding of their dynamic behaviors not only offers the possibility of realizing nonlinear phenomena such as solitons but also provides a powerful tool to manipulate nonlinear waves.As is well known,a soliton is a solitary wave that can propagate stably with the dynamic behavior like a “particle.” A solitary wave was shown to be an ideal method for transferring vibrational excitations.Indeed,the discovery of phononic solitons has been demonstrated to be of fundamental importance in the study of nonlinear phenomena in phononic crystals.In this thesis,precompressed one-dimensional monoatomic or diatomic spherical chains and one-dimensional nonlinear layered phononic crystals are studied.The main contains and conclusions include:? By introducing the space-moving coordinate and slow-time transformation,the long-wavelength equation of a precompressed single-atom sphere chain with small deformation is transformed into Kd V equation,which contains the physical and geometric parameters of the original system.The analytic solutions of single-and multiple-solitary waves are obtained by using Hirota bilinear method and homogenous balanced principle to solve Kd V equation.By studying the geometric structure and dynamic characteristics of the solution,it is found that the multiple solitary waves can be regarded as the interaction(collision)of the single solitary waves,and can be regarded as the superposition of the single solitary waves in some limit areas,which provides a way for using the single solitary wave to excite the multiple solitary waves.Further,according to the idea of perturbation analysis,the stability of single and double solitary wave solutions of Kd V equation is studied by using the split-step Fourier transform method.The results show that both single and double solitary waves are stable.This paper also presents the existence condition of soliton is that the system has stable eigenmodes of double solitary waves.? By introducing the space-moving coordinate and slow-time transformation,the long-wavelength equation of a precompressed monoatomic spherical chain with small deformation is transformed into Kd V equation which contains the physical and geometric parameters of the original system.The Hirota bilinear method and homogenous balanced principle are applied to solve Kd V equation to obtain the analytic solutions of single-and multiple-solitary waves.By studying the geometric structures and dynamic characteristics of the solutions,it is found that the multiple-solitary waves can be regarded as the interaction(i.e.collision)of the single-solitary waves,and can be regarded as the superposition of the single-solitary waves in some limiting regions,which provides a way for us to excite the multiple-solitary waves using the single-solitary wave.Then,based on the idea of perturbation analysis,the stability of single-and double-solitary wave solutions of Kd V equation is examined by using the split-step Fourier transform method.The results show that both single-and double-solitary waves are stable.Finally,it is also indicated that the existence condition of a soliton is that the system has stable eigenmodes of double solitary waves.? Based on the idea of perturbation analysis,the split-step Fourier transform method and Runge-Kutta method are respectively used to numerically simulate the propagation stability of solitary waves in the continuity equation under small deformation and long wavelength,the discrete equation under small deformation and the original equation for the monoatomic chain and the diatomic chain.The results show that the system can support the stable propagation of the single-solitary wave as well as the double-solitary waves formed by the collision of the single-solitary waves moving in opposite directions(termed the first type of double-solitary wave in this paper).No stable double-solitary waves formed by the collision of the single-solitary waves moving in the same direction(called the second type of double-solitary waves)and higher order multiple-solitary waves are found.In addition,the existence conditions of stable multiple-solitary waves are qualitatively analyzed;and it is pointed out that the closer the angle between any two single-solitary waves is to 90?,the easier it is to obtain stable multiple-solitary waves.That may be the reason why only the first type of double-solitary wave is stable,while that the other multiple-solitary waves are not.? Based on the idea of perturbation analysis and the finite volume method,the dynamic evolution of nonlinear waves in nonlinear layered phononic crystals is studied.The results show that the single-solitary wave and the first type of double-solitary waves can propagate stably in the system,while that the other stable multiple-solitary waves cannot.The width,number and velocity of the solitary waves can be tuned by introducing linear layers into the system.The numerical simulation shows that when the solitary wave in a nonlinear layered phononic crystal encounters a low impedance linear homogeneous medium,most of the energy will propagate through the interface with the wave form becoming smoother.But most of the energy is reflected by the interface when the solitary wave encounters a high impedance linear homogeneous medium.Based on this phenomenon,a nonlinear phononic crystal bound cavity is designed to capture the energy carried by nonlinear elastic waves.The present study of elastic waves in nonlinear phononic crystals provides an important theoretical guidance for the control of elastic waves,and provides a theoretical basis for the design of new elastic wave devices.