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Yi Huo - One of the best experts on this subject based on the ideXlab platform.
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Solitary shock waves and other travelling waves in a general compressible hyperelastic rod
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2000Co-Authors: Hui-hui Dai, Yi HuoAbstract:In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is difficult to directly tackle the fully nonLinear rod equations. Here, by using a non–dimensionalization process and the reductive perturbation technique, we derive a new type of nonLinear dispersive equation as the model equation. We then focus on the travelling–wave solutions of this new equation. As a result, we obtain a system of ordinary differential equations. An important feature of this system is that there is a vertical Singular Line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the Singular Line, we are able to determine all qualitatively different phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with finite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors9 knowledge, such a bifurcation has not been found in any other dynamical systems.
Hui-hui Dai - One of the best experts on this subject based on the ideXlab platform.
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Deltons, peakons and other traveling-wave solutions of a Camassa–Holm hierarchy
Physics Letters A, 2009Co-Authors: Xiaochun Peng, Hui-hui DaiAbstract:Abstract In this letter, we study an integrable Camassa–Holm hierarchy whose high-frequency limit is the Camassa–Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a Singular Line on the phase plane. By considering the properties of the equilibrium points and the relative position of the Singular Line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.
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Solitary shock waves and other travelling waves in a general compressible hyperelastic rod
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2000Co-Authors: Hui-hui Dai, Yi HuoAbstract:In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is difficult to directly tackle the fully nonLinear rod equations. Here, by using a non–dimensionalization process and the reductive perturbation technique, we derive a new type of nonLinear dispersive equation as the model equation. We then focus on the travelling–wave solutions of this new equation. As a result, we obtain a system of ordinary differential equations. An important feature of this system is that there is a vertical Singular Line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the Singular Line, we are able to determine all qualitatively different phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with finite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors9 knowledge, such a bifurcation has not been found in any other dynamical systems.
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Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods
Wave Motion, 1998Co-Authors: Hui-hui DaiAbstract:Abstract In this paper, we study an integrable nonLinear evolution equation which arises in the context of nonLinear dispersive waves in hyperelastic rods. To consider bounded travelling-wave solutions, we conduct a phase plane analysis. A new feature is that there is a vertical Singular Line in the phase plane. By considering equilibrium points and the relative position of the Singular Line, we find that there are in total three types of phase planes. The trajectories which represent bounded travelling-wave solutions are studied one by one. In total, we find there are 12 types of bounded travelling waves, both supersonic and subsonic. While in literature solutions for only two types of travelling waves are known, here we provide explicit solution expressions for all 12 types of travelling waves. Also, it is noted for the first time that peakons can have applications in a real physical problem.
Chaudry Masood Khalique - One of the best experts on this subject based on the ideXlab platform.
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The effects of the Singular Lines on the traveling wave solutions of modified dispersive water wave equations
Nonlinear Analysis: Real World Applications, 2019Co-Authors: Maoan Han, Lijun Zhang, Yue Wang, Chaudry Masood KhaliqueAbstract:Abstract In this paper, the bounded traveling wave solutions of the modified water wave equations of which one dependent variable attains the Singular value 2 c in finite or infinite time are investigated by using the bifurcation theory of planar dynamical systems. The Line V = 2 c is the so-called Singular Line of the associated dynamical system and the results of this paper show that the solutions possess Singularity if and only if their corresponding phase orbits intersect with this Singular Line. There are two types of solutions corresponding to these orbits intersecting with the Singular Line: smooth classical solutions and compact solutions possessing compact support in H l o c 1 ( R ) , which suggests that the existence of Singular Line breaks the uniqueness of solutions in H l o c 1 ( R ) space. There is a significant discovery from the investigation of the modified water wave equations that there are new type of solitary wave solutions approaching the Singular value 2 c as time tends to infinite that correspond to some specific orbits connecting with Singular Lines of the associated traveling wave system, which refreshes and enriches the knowledge of the effects of Singular Lines on the traveling wave solutions to nonLinear wave equations. The explicit bounded smooth traveling wave solutions and compact solutions of the modified water wave equations are presented and simulated numerically.
Maoan Han - One of the best experts on this subject based on the ideXlab platform.
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The effects of the Singular Lines on the traveling wave solutions of modified dispersive water wave equations
Nonlinear Analysis: Real World Applications, 2019Co-Authors: Maoan Han, Lijun Zhang, Yue Wang, Chaudry Masood KhaliqueAbstract:Abstract In this paper, the bounded traveling wave solutions of the modified water wave equations of which one dependent variable attains the Singular value 2 c in finite or infinite time are investigated by using the bifurcation theory of planar dynamical systems. The Line V = 2 c is the so-called Singular Line of the associated dynamical system and the results of this paper show that the solutions possess Singularity if and only if their corresponding phase orbits intersect with this Singular Line. There are two types of solutions corresponding to these orbits intersecting with the Singular Line: smooth classical solutions and compact solutions possessing compact support in H l o c 1 ( R ) , which suggests that the existence of Singular Line breaks the uniqueness of solutions in H l o c 1 ( R ) space. There is a significant discovery from the investigation of the modified water wave equations that there are new type of solitary wave solutions approaching the Singular value 2 c as time tends to infinite that correspond to some specific orbits connecting with Singular Lines of the associated traveling wave system, which refreshes and enriches the knowledge of the effects of Singular Lines on the traveling wave solutions to nonLinear wave equations. The explicit bounded smooth traveling wave solutions and compact solutions of the modified water wave equations are presented and simulated numerically.
Paul Wiegmann - One of the best experts on this subject based on the ideXlab platform.
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Fingering patterns in Hele-Shaw flows are density shock wave solutions of dispersionless KdV hierarchy
Physical Review Letters, 2008Co-Authors: Razvan Teodorescu, Seung-yeop Lee, Paul WiegmannAbstract:We investigate the hydrodynamics of a Hele-Shaw flow as the free boundary evolves from smooth initial conditions into a generic cusp Singularity (of local geometry type x{sup 3} {approx} y{sup 2}), and then into a density shock wave. This novel solution preserves the integrability of the dynamics and, unlike all the weak solutions proposed previously, is not underdetermined. The evolution of the shock is such that the net vorticity remains zero, as before the critical time, and the shock can be interpreted as a Singular Line distribution of fluid deficit.
