A direct method for producing Hamiltonian structure of nonlinear evolution equations

The equation hierarchy presented in this paper contains the KdV equation and the mKdV equation. By use of the concept of characteristic number, an undetermined-constant method is proposed by us, for which the polynomial Hamiltonian functions are constructed. By employing the method, the Hamiltonian structure of the equation hierarchy is established. The approach presented in the paper shares extensive applications. In addition, four explicit expressions of the travelling wave solutions to the above equation hierarchy are obtained. One of them is regular, the other three are singular.     

Time-domain implementation of nonreflecting boundary-conditions for the nonlinear Euler equations

The present paper studies nonreflecting boundary-conditions for the 2-D unsteady nonlinear Euler equations, applied to the propagation of monochromatic pressure-waves in a uniform mean flow. Various boundary-conditions (1-D nonlinear, approximate linearized 2-D, and exact linearized 2-D) are compared for a wide range of both propagating and decaying waves. An original methodology, based on a moving-averages technique, is developed for the application of the exact linearized boundary-conditions, which requires the computation of 2-D (space–time) Fourier coefficients. It is shown that the exact linearized boundary-conditions yield very low reflection, and also that the approximate conditions may perform poorly in difficult cases. The reflection-coefficient shows some correlation with the group-velocity (direction and Mach-number) of the reflected waves, suggesting that proposed nonreflecting boundary-conditions should always be validated against the entire range of group-velocities.     

Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending damping

In this paper, we are concerned with the asymptotic behavior of solutions to the system of Euler equations with time-depending damping, in particular, include the constant coefficient damping. We rigorously prove that the solutions time-asymptotically converge to the diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which justifies Darcy's law. Compared with previous results about Euler equations with constant coefficient damping obtained by Hsiao and Liu (1992) [2], and Nishihara (1996) [9], we obtain a general result when the initial perturbation belongs to the same space, i.e. $H^3(\mathbb{R})\times H^2(\mathbb{R})$. Our proof is based on the classical energy method.     

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

There are few results on the numerical stability of nonlinear neutral stochastic delay differential equations (NSDDEs). The aim of this paper is to establish some new results on the numerical stability for nonlinear NSDDEs. It is proved that the semi-implicit Euler method is mean-square stable under suitable condition. The theoretical result is also confirmed by a numerical experiment.     

The second Hamiltonian structure for a special case of the Lotka-Volterra equations

A special case of the Lotka-Volterra equations is considered for which it is possible to find the second Hamiltonian structure that is complementary to the known one. The form of the new Hamiltonian makes it possible to solve the equations by quadratures, which is the main feature of the case under examination. As a consequence, the period can also be represented by quadratures. In terms of the new variables, the equations of motion admit a mechanical analogy with the oscillations of a mass on a nonlinear spring.     

Oscillation of solutions of second-order nonlinear differential equations of Euler type

We consider the nonlinear Euler differential equation t²x^″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t²x^″+a(t)g(x)=0.     

A relaxation scheme for the approximation of the pressureless Euler equations

In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δ‐shocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed first‐order scheme but also a classical MUSCL‐type second‐order extension confirm the reliability and robustness of the relaxation approach. The article extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established, and two‐dimensional numerical results are presented. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

Precise regularity results for the Euler equations

It has already been proved, under various assumptions, that no singularity can appear in an initially regular perfect fluid flow, if the L^∞ norm of the velocity's curl does not blow up. Here that result is proved for flows in smooth bounded domains of (d?2) when the regularity is expressed in terms of Besov (or Triebel-Lizorkin) spaces.     

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it has been proposed by Jameson, Schmidt, and Turkel to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed here, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which can move all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping can be effective in accelerating convergence to steady state.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical -matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 264–273, 1982.     

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

^** Email: dlevy{at}math.stanford.edu We derive a second-order, semi-discrete central-upwind schemefor the incompressible 2D Euler equations in the vorticity formulation.The reconstructed velocity field preserves an exact discreteincompressibility relation. We state a local maximum principlefor a fully discrete version of the scheme and prove it usinga convexity argument. We then show how similar convexity argumentscan be used to prove that the scheme maps certain Orlicz spacesinto themselves. The consequences of this result on the convergenceof the scheme are discussed. Numerical simulations support theexpected properties of the scheme.