Interactions between defects and propagating hydrodynamic solitons

Abstract

This paper studies the hydrodynamic solitons propagating along a long trough with a defective bed. The slight deviation from the plane in the bed serves as the depth defects. Based on the perturbation method, it finds that the free surface wave is governed by a Korteweg-de Vries (KdV) equation with a defect term (KdVD). The numerical calculations show that, for a single-convexity localized defect, the propagating soliton is decelerated as it comes into the defect region, and it is accelerated back to its initial velocity as it leaves, which has a dipole effect. As a result, its displacement is lagged in contrast to the uninfluenced one. And an up-step defect makes the propagating soliton decelerate simply. The opposite influence will occur for a single-concavity localized defect and a down-step one. The defect-induced influence on propagating hydrodynamic solitons depends on the polarity of defects, which agrees with that on non-propagating ones. However, the involved dipole effect of the single localized defect is not displayed in non-propagating cases.

Influence of impurities on solitons in the nonlinearLCtransmission line

“…This ISI mechanism has been well served in many other nonlinear systems, such as the FK model [3][4][5][6] and hydrodynamics [7,8]. In fact, in the discussion here, the impurity-induced influence near the linear capacitance and inductance impurities depending on the impurity polarities is similar to that in other soliton systems such as the FK model [3][4][5][6] or hydrodynamics [7,8].…”

“…In fact, in the discussion here, the impurity-induced influence near the linear capacitance and inductance impurities depending on the impurity polarities is similar to that in other soliton systems such as the FK model [3][4][5][6] or hydrodynamics [7,8]. Here, the linear capacitance and inductance impurities are essentially equivalent to the pendulum length impurity, which determines the intrinsic frequency in the coupled pendulum chain [3][4][5], namely,…”

“…It seems that the ISIs in the NLCTLs depend much on the impurities being inductance or capacitance. Furthermore, the ISIs in the NLCTLs [15,16] seem to be different from those in the FK model [3][4][5][6] and hydrodynamics [7,8].…”

“…Up to now, both theoretical [3,4] and experimental [5,6] investigations in the FK model found that an impurity can attract or repel nonpropagating solitons depending on its polarity. Moreover, the impurity-induced influence on hydrodynamic solitons including both nonpropagating [7] and propagating (Korteweg-de Vries) [8] ones was also performed. The studies showed that, for the nonpropagating case, the ISIs will show the same results as those in the FK model, while for the propagating case, a soliton passing through an impurity will change its propagating speed, accelerating or decelerating depending on the impurity polarity.…”

“…III, we explain the ISI observations in the NLCTLs within the physical picture of linear uncoupled energy absorption, and following this, we unify the ISIs in NLCTLs, which have been seen to be different before [15,16], for both inductance and capacitance impurities. At the same time, by comparing the ISIs in the NLCTLs with those in other soliton systems such as the FK model [3][4][5][6] and hydrodynamics [7,8], we can further integrate the ISIs in the NLCTLs with those in other different soliton systems. In Sec.…”