† Corresponding author. E-mail:
Project supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA01020304) and the National Natural Science Foundation of China (Grant Nos. 91026005, 11547304, 11275156, 11047010, and 61162017).
A nonlinear Schröodinger equation in one-dimensional bead chain is first obtained and an envelope solitary wave of the system is verified numerically in this system. The reflection and the transmission of an incident envelope solitary wave due to impurities has also been investigated. It is found that the magnitudes of both the reflection and the transmission not only depend on the characters of impurity materials, the wave number, the incident wave amplitude, but also on the impurity number. This can be used to detect the character and the number of the impurity materials in the bead chain by measuring the reflection and the transmission of an incident pulse.
The nonlinear Schrödinger equation (NLSE) serves as a central model in nonlinear science, especially in the nonlinear wave.[1] The one-dimensional (1D) NLSE can show some very exciting and fascinating nonlinear phenomena, such as the modulation instability and envelope solitons.[2,3] The NLSE belongs to the remarkable class of integrable systems[4] and can be solved by using the integration techniques. The phenomenon of the envelope solitary waves and the modulation instability (MI) have been extensively studied in the past years.[5–11] Solitary wave solution of NLSE has been extensively studied in many branches of the science such as the deep water wave,[4] the dynamics of the Duffing oscillator[4] in optics,[12] plasmas,[13] etc.
Granular chains have been an interesting topic in many branches of science.[14] The force displacement law between two grain beads in contact is governed by Hertzian interaction.[14] Granular crystals provide a versatile type of metamaterial for both fundamental physical phenomena and applications.[15,16]
Granular chains have been used to investigate numerous coherent structures such as traveling waves, breathers, and dispersive shock waves.[17–22] It is found that the granular chains support a new type of solitary wave which is qualitatively different from the well-known weakly nonlinear Korteweg–de Vries (KdV) solitary waves when it is without initial pre-stress.[14] These kinds of waves do not support sound waves. Different groups investigated numerically and experimentally the properties of these waves.[17–22] On the other hand, the KdV solitary wave with the initial pre-stress is also found in the bead chain.[23] The reflection and the transmission of these solitary waves at the interface of two different bead chains, from imperfections or a wall, have been thoroughly studied.[23,27]
However, envelope solitary wave described by the NLSE has not been reported until now. This work presents an NLSE in 1D bead chain and then finds the envelope solitary waves in this system. Furthermore, the reflection and the transmission of an incident envelope solitary wave due to impurities are also studied. Interesting phenomena are observed.
The simplest granular systems are 1D chains of elastic spheres. The dynamics of the 1D bead chain is described by N coupled nonlinear equations[14]
If we give an initial pre-stress p0 that leads to an initial average bead overlap
We let the initial displacements of all the beads satisfy Eq. (
The profiles of the envelope solitary waves of both the analytical results and the numerical ones are shown in Fig.
To get more insight into how the analytical results present the real envelope solitary wave, the comparisons between the analytical results and the numerical ones for different parameters of ε and k are shown in Fig.
Nesterenko proposed that the propagation of a perturbation in a chain of beads in the Hertzian contact can evolve into soliton-like pulse.[28] Later, several investigations confirmed the existence of such soliton-like pulses.[17,29–31] Moreover, the interaction between a solitary wave and the boundary has also been studied.[21,26,27,32–34]
We now study the scattering of an envelope solitary wave at an interface, as shown in Fig.
To further study the dependence of the reflection and the transmission on the impurity material the numerical results of the dependence of the amplitude ratios of both the reflected wave to the incident wave (
Dependence of the reflection and the transmission on the impurity number are shown in Fig.
The dependence of the length and the period of the vibration observed in Fig.
For more general case, we calculate the ratios of the kinetic energy of all beads of both transmitted wave to the incident wave and the reflected wave to the incident wave. We give the incident wave as follows:
An NLSE in 1D bead chain has been first given in the present work and an envelope solitary wave solution of the NLSE is verified numerically in this system. Furthermore, the reflection and the transmission of an incident envelope solitary wave due to impurities are also studied. It is found that the magnitudes of both the reflection and the transmission not only depend on the impurity materials, the wave number, the incident wave amplitude, but also on the impurity number. Similar results are obtained for more general incident waves.
The obtained results can be used to detect the impurities in the bead chain by measuring the reflection of a given pulse. The reflection due to an incident pulse contains the information of the material and the number of the impurity. Our results may help to devise a bead protection system. Disintegrating impulse or shock wave into many weak impulses is one of the possible ways. We can then protect something important from different kinds of disastrous external impacts, such as an earthquake, bomb explosion, automobile collision, etc. It seems that it is very important to further investigate the reflection and the transmission by one or several impurities from a given incident wave.
This phenomenon is actually the acoustic diode effect. We can control the reflection or the transmission from an arbitrary incident acoustic waves by changing the numbers of the impurities and adding several identical pieces of impurities in the bead chains. It can be employed to design tunable information transportation lines with the unique possibility to manipulate the signal delay and scrambling of security-related information.
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