Energy For Hamiltonian Systems Research Articles

Exact periodic wave solutions of the cubic-quintic Zakharov equation and their evolution with Hamilton energy

In this paper, we study the exact periodic wave solutions of the Zakharov equation with cubic and quintic nonlinear terms, and their evolution with the energy of Hamiltonian system corresponding to the amplitudes. Based on the theory of plane dynamical system, we first make a detailed qualitative analysis to the plane dynamical system corresponding to the amplitudes of traveling wave solutions of the studied equation, then by applying the analysis method based on the first integral and several appropriate transformations, all seven families of elliptic function periodic wave solutions of the Zakharov equation are obtained. In addition, by studying the evolution limit of periodic wave solutions with respect to Hamilton energy and using the analysis method based on the first integral, all ten pairs of solitary wave solutions of the studied equations are also given under various parameter conditions. From the evolution analysis to the periodic wave solutions with respect to Hamilton energy, it can be seen that it is the energy H of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solution of this equation appear as periodic wave solution or solitary wave solution.

Role of shift constant in Energy Shifted SAV for Hamiltonian Systems

In recent years, considerable work has been devoted to the design of energy-stable numerical methods, a class of geometrical integration technique in which the discrete energy or pseudo-energy remains conserved. This property finds practical applications in systems for which the energy is bounded from below since the growth of the solutions is then itself bounded, yielding a form of stability. Such a property may be further exploited to transform the energy overall into a quadratic form, representing the core of recent numerical techniques such as the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. These methods have been applied to a large class of problems due to their remarkable efficiency. Yet, several aspects of such techniques have seen little investigation. In this work, the role of the “shift” constant in the expression of the potential energy in Hamiltonian systems is investigated. This is a positive gauge constant that may be used to augment the expression of the energy. Since Hamilton’s equations are given in terms of gradients of the Hamiltonian, such a constant has no influence on the resulting dynamics in the continuous system. However, empirical evidence has suggested that the convergence properties of the associated SAV approaches are affected by the magnitude of the shift factor. In this work, the behaviour of the relative global error of SAV schemes in the cases of the harmonic and Duffing oscillators is numerically investigated. The results reveal an optimal shift factor that increases the convergence rate. Using the optimal shift factor, the proposed method displays variable order accuracy ranging from second to twelfth order.

Infinitely many non-constant periodic solutions with negative fixed energy for Hamiltonian systems

ABSTRACTIn this paper, we obtain the existence of infinitely many non-constant periodic solutions with negative fixed energy for a class of second-order Hamiltonian systems, which greatly improves the existing results such as Zhang [Periodic solutions for some second order Hamiltonian systems. Nonlinearity. 2009;22(9):2141–2150, Theorem 1.5]. Moreover, we exhibit two simple and instructive examples to make our result more clear, which have not been solved by known results.

Dynamical behaviors and exact traveling wave solutions for a modified Broer-Kaup system

In this paper, we focus on smooth periodic wave, periodic blow-up wave, solitary wave, blow-up wave, kink and anti-kink wave solutions for a modified Broer-Kaup system. Based on the approach of dynamical systems, the existence of above traveling waves is proved and their all possible exact parametric representations are presented. When the energy of Hamiltonian system corresponding to this system varies, we also show the convergence of the periodic wave solutions. Along with the details of the analysis, the analytical results lastly are numerical simulated.

EXPLICIT EXACT PERIODIC WAVE SOLUTIONS AND THEIR LIMIT FORMS FOR A LONG WAVES-SHORT WAVES MODEL

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively

Periodic wave solutions and solitary wave solutions of generalized modified Boussinesq equation and evolution relationship between both solutions

In this paper, we focus on studying the exact solitary wave solutions and periodic wave solutions of the generalized modified Boussinesq equation utt−δuttxx−(a1u+a2u2+a3u3)xx=0, as well as the evolution relationship between these solutions. Detailed qualitative analysis is conducted on traveling wave solutions of this equation, and global phase portraits in various parameter conditions are proposed. Various significant results about the existence of both solutions, including three forms of solitary wave solutions and four exact bounded periodic wave solutions in different conditions are obtained. Then, we further discuss the relationship between energy of Hamiltonian system corresponding to this equation and the periodic wave solutions and solitary wave solutions. It is concluded that the essential reason of periodic wave solutions and solitary wave solutions is the different values for the energy of Hamiltonian system corresponding to this equation. In addition, the limited relations of periodic wave solutions and solitary wave solutions with the energy of Hamiltonian system are proposed, and the schematic diagram of evolution from periodic wave solutions to solitary wave solutions is drawn.

Multiplicity problems of brake orbits with prescribed energy in Hamiltonian systems

Multiplicity and stability of brake orbits are important problems in the study of Hamiltonian systems, while the Maslov-type index theory and its iteration theory are the powerful tools to study such problems. In this survey paper, we briefly introduce the brake orbit boundary valued Maslov-type index theory and its iteration theory. As an application, we prove that Seifert conjecture holds in the even convex case. At the end of this paper we introduce the minimal period problem and stability problems related to brake orbits.

Achieving Brouwer’s law with implicit Runge–Kutta methods

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times.

On stabilization of energy for Hamiltonian systems

Abstract We study post-stabilization of numerical integrators for an autonomous Hamiltonian system in the form H = H 1 + H 2 , where H 1 and H 2 are two constants of motion. It is better to stabilize the two energies H 1 and H 2 , respectively. As a particular case, the effectiveness of post-stabilization by the total energy is equivalent to one by the two independent energies when the splitting Hamiltonian becomes isotropic.

A geometric approach to a result of Benci and Giannoni for Hamiltonian systems with singular potentials

By using the least action principle of Maupertuis-Jacobi, we find that a geodesic convex condition plays an important role in proving the result of Benci and Giannoni on existence of periodic solutions of prescribed energy for Hamiltonian systems with singular potentials.

2. Morse theory of completely integrable Hamiltonian systems. Topology of the level surfaces of constant energy of Hamiltonian systems, obstructions to integrability, classification of rearrangements of general position of Liouville tori in a neighborhood of a bifurcation diagram

2. Morse theory of completely integrable Hamiltonian systems. Topology of the level surfaces of constant energy of Hamiltonian systems, obstructions to integrability, classification of rearrangements of general position of Liouville tori in a neighborhood of a bifurcation diagram

Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems

We introduce a noncanonical ("new-time") transformation which exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable systems and, for example, explain the observed duality between the H\'enon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak- and full-Painlev\'e Hamiltonians. We also discuss quantum integrability and find the origin of the deformation $\ensuremath{-}\frac{5}{72}{\ensuremath{\hbar}}^{2}{x}^{\ensuremath{-}2}$.