共查询到20条相似文献,搜索用时 15 毫秒
1.
Xiangpeng Xin Lihua Zhang Yarong Xi Hanze Liu 《Journal of Applied Analysis & Computation》2020,10(6):2669-2681
In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior. 相似文献
2.
Zheng Li Ya-Guang Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(6):1033-1051
This paper is devoted to the study of asymptotic behavior in the propagation and reflection of discontinuous solutions to linear and semilinear thermoelastic equations with second sound in one space variable. When the relaxation parameter goes to zero, we obtain that the jump of temperature vanishes while jumps of elastic waves and heat flux are propagated and reflected with the elastic speeds. Furthermore, it is observed that these jumps decay exponentially when the time goes to infinity, and the decay rates not only depend on the growth rate of the nonlinear source terms and heat conduction coefficient, but also depend on the change rates of speeds of elastic waves. 相似文献
3.
This paper is devoted to the study of asymptotic behavior in the propagation and reflection of discontinuous solutions to
linear and semilinear thermoelastic equations with second sound in one space variable. When the relaxation parameter goes
to zero, we obtain that the jump of temperature vanishes while jumps of elastic waves and heat flux are propagated and reflected
with the elastic speeds. Furthermore, it is observed that these jumps decay exponentially when the time goes to infinity,
and the decay rates not only depend on the growth rate of the nonlinear source terms and heat conduction coefficient, but
also depend on the change rates of speeds of elastic waves. 相似文献
4.
Naoki Tsuge 《Acta Appl Math》2017,147(1):177-186
We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. In addition to this, the scalar conservation law is used in various areas such as fluid dynamics, traffic problems etc.In the present paper, we prove the global existence and stability of entropy solutions to the Cauchy problem. The difficult point is to obtain the bounded estimate of solutions. To solve it, we introduce some functions as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. This idea comes from the generalized invariant region theory for the compressible Euler equation. The method is also applicable to other nonlinear problems involving similar difficulties. Finally, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness. 相似文献
5.
We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data. 相似文献
6.
Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws 下载免费PDF全文
In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions. 相似文献
7.
In this letter, based on the Lax pair of the Ablowitz–Kaup–Newell–Segur (AKNS) system, the nonlocal symmetry is obtained and successfully localized to a Lie point symmetry by introducing an appropriate auxiliary dependent variable. For the closed prolongation, the construction for the one-dimensional optimal system is presented in detail. Furthermore, using the obtained optimal system, we give the reductions and the explicit analytic interaction solutions between cnoidal waves and solitary waves. For some interesting solutions, the figures are given to show their properties. 相似文献
8.
Summary. The paper is devoted to the construction of a higher order Roe-type numerical scheme for the solution of hyperbolic systems
with relaxation source terms. It is important for applications that the numerical scheme handles both stiff and non stiff
source terms with the same accuracy and computational cost and that the relaxation variables are computed accurately in the
stiff case. The method is based on the solution of a Riemann problem for a linear system with constant coefficients: a study
of the behavior of the solutions of both the nonlinear and linearized problems as the relaxation time tends to zero enables
to choose a convenient linearization such that the numerical scheme is consistent with both the hyperbolic system when the
source terms are absent and the correct relaxation system when the relaxation time tends to zero. The method is applied to
the study of the propagation of sound waves in a two-phase medium. The comparison between our numerical scheme, usual fractional
step methods, and numerical simulation of the relaxation system shows the necessity of using the solutions of a fully coupled
hyperbolic system with relaxation terms as the basis of a numerical scheme to obtain accurate solutions regardless of the
stiffness.
Received October 7, 1994 / Revised version received September 27, 1995 相似文献
9.
Mustafa Inc Hamdy I. Abdel‐Gawad Mohammad Tantawy Abdullahi Yusuf 《Mathematical Methods in the Applied Sciences》2019,42(7):2455-2464
In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients. 相似文献
10.
Mai Duc Thanh 《Nonlinear Analysis: Real World Applications》2012,13(2):987-998
We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave. 相似文献
11.
We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior. 相似文献
12.
G. T. Kossioris 《偏微分方程通讯》2013,38(5-6):747-770
In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on the convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which is the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves. 相似文献
13.
The purpose of this paper is to explore a viscous two-phase liquid-gas model relevant for well and pipe flow. Our approach relies on applying suitable modifications of techniques previously used for studying the single-phase isothermal Navier-Stokes equations. A main issue is the introduction of a novel two-phase variant of the potential energy function needed for obtaining fundamental a priori estimates. We derive an existence result for weak solutions in a setting where transition to single-phase flow is guaranteed not to occur when the initial state is a true mixture of both phases. Some numerical examples are also included in order to demonstrate characteristic behavior of solutions. In particular, we illustrate how two-phase flow is genuinely different compared to single-phase flow concerning the behavior of an initial mass discontinuity. 相似文献
14.
15.
The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables. 相似文献
16.
In this paper, we study an asymptotic behavior of a solution to the outflow problem for a two-phase model with magnetic field. Our idea mainly comes from [1] and [2] which investigate the asymptotic stability and convergence rates of stationary solutions to the outflow problem for an isentropic Navier–Stokes equation. For the two-phase model with magnetic field, we also obtain the asymptotic stability and convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method. 相似文献
17.
Lei Wang Yi-Tian Gao Feng-Hua Qi 《Journal of Mathematical Analysis and Applications》2010,372(1):110-4055
For the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water, we consider a variable-coefficient variant Boussinesq (vcvB) model with symbolic computation. We construct the connection between the vcvB model and a variable-coefficient Ablowitz-Kaup-Newell-Segur (vcAKNS) system under certain constraints. Using the N-fold Darboux transformation of the vcAKNS system, we present two sets of multi-solitonic solutions for the vcvB model, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Dynamics of those solutions are analyzed and graphically discussed, such as the parallel solitonic waves, shape-changing collision, head-on collision, fusion-fission behavior and elastic-fusion coupled interaction. 相似文献
18.
Robert A. Van Gorder 《Mathematical Methods in the Applied Sciences》2015,38(4):636-645
In the present paper, we construct exact solutions to a system of partial differential equations iux + v + u | v | 2 = 0, ivt + u + v | u | 2 = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
19.
F.A. Howes 《Journal of Computational and Applied Mathematics》1984,10(2):195-201
Singular perturbation techniques are used to study boundary value problems whose solutions model the behavior of one-dimensional steady shock waves in ducts of constant and variable cross-sectional area. 相似文献
20.
Aiyong Chen Shuangquan Wen Shengqiang Tang Wentao Huang Zhijun Qiao 《Studies in Applied Mathematics》2015,134(1):24-61
In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation. 相似文献