The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock ...The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.展开更多
In [1], Ding et al. studied the nonhomogeneous Burgers equationut + uuX = μ + 4x. (1.1)This paper will prove that when μ→0 the solution of (1.1) will approach the generalized solution ofut + uux = 4z. (1.2)The auth...In [1], Ding et al. studied the nonhomogeneous Burgers equationut + uuX = μ + 4x. (1.1)This paper will prove that when μ→0 the solution of (1.1) will approach the generalized solution ofut + uux = 4z. (1.2)The authors notice that the equation (1.2) is beyond the scope of investigations by OleinikO. in [2]. The solutions here are unbounded in general.The paper also studies the δ-wave phenomenon when (1.2) is jointed with some otherequation.展开更多
This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the densi...This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.展开更多
This paper is concerned with the exact solutions of Khokhlov-Zabolotskaya（KZ） equation with general perturbation. With the help of appropriate transformationsand assumptions, the wave theory of Hopf equation is appl...This paper is concerned with the exact solutions of Khokhlov-Zabolotskaya（KZ） equation with general perturbation. With the help of appropriate transformationsand assumptions, the wave theory of Hopf equation is applied to get partial exactsolutions. In addition, some examples and numerical simulations are presented toillustrate our analytical results.展开更多
By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.Fo...By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.展开更多
基金the National Science Foundation under Grant DMS05-05975.
文摘The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.
文摘In [1], Ding et al. studied the nonhomogeneous Burgers equationut + uuX = μ + 4x. (1.1)This paper will prove that when μ→0 the solution of (1.1) will approach the generalized solution ofut + uux = 4z. (1.2)The authors notice that the equation (1.2) is beyond the scope of investigations by OleinikO. in [2]. The solutions here are unbounded in general.The paper also studies the δ-wave phenomenon when (1.2) is jointed with some otherequation.
文摘This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.
文摘This paper is concerned with the exact solutions of Khokhlov-Zabolotskaya（KZ） equation with general perturbation. With the help of appropriate transformationsand assumptions, the wave theory of Hopf equation is applied to get partial exactsolutions. In addition, some examples and numerical simulations are presented toillustrate our analytical results.
基金Supported by the National Natural Science Foundation of China Grant under Nos.11435005,11175092,and 11205092Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.
