Nonlinear String Equation Research Articles

Global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law

AbstractIn this paper, we consider the global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law. First, we will give the representation of the traveling wave solution to the recast hyperbolic system. Then, by using the weighted function method, we obtain the global stability of longitudinal wave solution for nonlinear elastic string equation.

Blow-up solitons at the nonlinear stage of the two-stream instability in quantum plasmas

The nonlinear evolution of the quantum two-stream instability in a plasma with counter-streaming electron beams is studied. It is shown that in the long-wave limit the nonlinear stage of the instability can be described by the elliptic nonlinear string equation. We present two types of nonlinear solutions. The first one is an unstable nonlinear mode that is continuously related with the growing linear solution and the second one is a pulsating soliton. We show that both of these solutions blow up in a finite time.

A Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Solutions of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

Abstract A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation

In this work, we propose a fractional extension of the one-dimensional nonlinear vibration problem on an elastic string. The fractional problem is governed by a hyperbolic partial differential equation that considers a nonlinear function of spatial derivatives of the Riesz type and constant damping. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of the real line. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant in the absence of damping, and it is non-increasing otherwise. Some boundedness properties of the solutions are established mathematically. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved in the absence of damping, and dissipated when the damping coefficient is positive. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model using a discrete form of the energy method. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservative/dissipative properties.

Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs

In the first part of the article we study Hamiltonian diffeomorphisms of \mathbb R^{2n} which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo’s symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the nonlinear string equation and treat it as an infinite-dimensional Hamiltonian system. In this context we are able to apply Kuksin’s approximation by finite dimensional Hamiltonian flows and prove a PDE version of the rigidity result for coisotropic cylinders. As a particular example, this result can be applied to the sine-Gordon equation.

On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential〈p,q〉=∫Rp(x)q(x)e−x4+2tx2dx+Mp(0)q(0).We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given.

Nonlinear Evolution Equations with Infinitely Many Derivatives

We study the nonlinear evolutionary euclidean bosonic string equation $$\begin{aligned} u_t = \Delta e^{-c \Delta }\,u + U(t,u) , \quad c > 0 \; \end{aligned}$$ on the Euclidean space \({\mathbb {R}}^n\). We interpret the nonlocal operator \(\Delta e^{-c\,\Delta }\) using entire vectors of \(\Delta \) in \(L^2({\mathbb {R}}^n)\). We prove that it generates a bounded holomorphic \(C_0\)-semigroup on \(L^2({\mathbb {R}}^n)\) (so that it also satisfies maximal \(L^p\) regularity) and we show the well-posedness of the corresponding nonlinear Cauchy problem.

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

Existence and Stability of Almost Periodic Solutions of Nonlinear Damped Equations of a Suspended String

In this paper we shall show the existence and the stability of almost periodic solutions of the boundary value problem to a nonlinear suspended string equation with a linear damping term and an almost periodic weakly nonlinear forcing term. We treat both weak solutions and strong solutions. Also we show the existence of time global solutions of the initial boundary value problem to the equation.

Aeroelastic Analysis of Membrane Microair Vehicles—Part II: Computational Study of a Plunging Membrane Airfoil

In the second paper of the two part study of membrane microair vehicles, computations are performed for a plunging membrane airfoil. The computational model uses a sixth-order finite difference solution of the Navier–Stokes equations coupled to a finite element solution of a set of nonlinear string equations. The effect, on the structural and fluid response, of plunging Strouhal number, reduced frequency, and static angle of attack is examined. Qualitatively, the flow field is found to be very complex with interactions of vortices shed from various locations along the chord of the airfoil. At a low angle of attack and a low Strouhal number, increasing reduced frequency results in a decrease and an increase in the mean sectional lift and drag coefficients, respectively. Also, at a low angle of attack, increasing the Strouhal number has minimal effect at high and low values of reduced frequencies, but a significant effect is found at an intermediate value of reduced frequency. When the effect of angle of attack is studied for fixed values of Strouhal number and reduced frequency, it is found that the act of plunging gives improved mean sectional lift when compared with the case of a fixed flexible airfoil. The improvement does not increase monotonically with the angle of attack but instead is maximum at an intermediate value. Finally, increasing the value of the membrane prestrain, which stiffens the airfoil, results in a reduced value of the sectional lift coefficient for a given Strouhal number, reduced frequency, and angle of attack.

Harmonic coordinates in the string and membrane equations

In this note, we first show that the solutions to Cauchy problems for two versions of relativistic string and membrane equations are diffeomorphic. Then we investigate the coordinates transformation presented in D. X. Kong and Q. Zhang [Physica D 238, 902 (2009)] [see (2.20)] which plays an important role in the study on the dynamics of the motion of string in Minkowski space. This kind of transformed coordinates is harmonic coordinates, and the nonlinear relativistic string equations can be straightforwardly simplified into linear wave equations under this transformation.

An energy-preserving nonlinear system modeling a string with input and output on the boundary

We study an energy conserving distributed parameter system described by a nonlinear string equation with the input and output at the boundary. We prove the existence of global smooth solutions to this quasilinear hyperbolic system if the initial data and the boundary input are small. If, moreover, the input function becomes zero after some finite time, then the state trajectories decay exponentially.

Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency $\o$ for a large Lebesgue measure set of $\o$ close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = \phi(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional

Global smooth solutions to a class of one-dimensional hyperbolic problems with boundary damping

In this paper an initial--boundary-value problem for a nonlinear string (or wave) equation with nonclassical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the (positive) damping is generated. Existence, uniqueness, and regularity of solutions to this problem are investigated.

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillations of flexible structures such as suspension bridges or overhead transmission lines in a wind field. A multiple time-scales perturbation method will be used to construct formal asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter

Energy Confinement for a Relativistic Magnetic Flux Tube in the Ergosphere of a Kerr Black Hole

In the MHD description of plasma phenomena the concept of magnetic field lines frozen into the plasma turns out to be very useful. We present here a method of introducing Lagrangian coordinates into relativistic MHD equations in general relativity, which enables a convenient mathematical formulation for the behaviour of flux tubes. With the introduction of these Lagrangian, so-called “frozen-in” coordinates, the relativistic MHD equations reduce to a set of nonlinear 1D string equations, and the plasma may therefore be regarded as a gas of nonlinear strings corresponding to flux tubes. Numerical simulation shows that if such a tube/string falls into a Kerr black hole, then the leading portion loses angular momentum and energy as the string brakes, and to compensate for this loss, momentum and energy is radiated to infinity to conserve energy and momentum for the tube. Inside the ergosphere the energy of the leading part turns out to be negative after some time, and the rest of the tube then gets energy from the hole. In our simulations most of the compensated positive energy is also localized inside the ergosphere because the inward speed of the plasma is approximately equal to the velocity of the MHD wave which transports energy outside. Therefore, an additional physical process has to be included which can remove energy from the ergophere. Magnetic reconnection seems to fill this role releasing Maxwellian stresses and producing a relativistic jet.

Normal form and exponential stability for some nonlinear string equations

We study small amplitude solutions of the nonlinear wave equation¶¶\(u_{tt}-u_{xx}=\psi(u) \qquad u(0,t)=0=u(\pi,t)\)(0.1)¶¶ with an analytic nonlinearity of the type \(\psi(u)=\pm u^{2k-1} + {\cal O}(u^{2k})$, $k\geq2\). For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the “simplified system”, namely a Hamiltonian system that we will prove to approximate (0.1) up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.

Families of Periodic Solutions of Resonant PDEs

We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation utt-uxx± u3=0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the nth family converge to 2π/n when the amplitude tends to zero.

Variational principle and a perturbative solution of non-linear string equations in curved space

String dynamics in a curved space-time is studied on the basis of an action functional including the small parameter of rescaled tension ε = γ/ α′, where γ is a metric parametrizing constant. A rescaled slow world-sheet time T = ετ is introduced, and general covariant non-linear string equations are derived. It is shown that in the first order of an ε-expansion these equations are reduced to the known equation for geodesic deviation but complemented by a string oscillatory term. These equations are solved for the de Sitter and Friedmann-Robertson-Walker spaces. The primary string constraints are found to be split into a chain of perturbative constraints and their conservation and consistency are proved. It is established that in the proposed realization of the perturbative approach the string dynamics in the de Sitter space is stable for a large Hubble constant H( α′ H 2 ⪢ 1).