Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.     

Conformal Geometry of Gravitational Plane Waves

By Brinkmann’s theorem the only Ricci flat and nonflat 4-manifolds admitting non-homothetic conformal vector fields are certain pp-waves. It seems that the converse direction was never completely clarified Which metrics really do occur? It is well known that the conformal group of a nonconformally-flat spacetime is atmost seven-dimensional and that seven is attained for certain pp-waves. Here we explicitly determine all solutions with a seven-dimensional conformal group. In other words We determine all Ricci flat Lorentzian manifolds admitting a seven-dimensional conformal group. They come in three particular families of gravitational plane waves. All of them are exact analytic solutions in terms of elementary functions. Furthermore, it turns out that Ricci flat Lorentzian manifolds with a six-dimensional conformal group are not necessarily real analytic.     

Rogue-wave solutions of the Zakharov equation

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these Nth-order rogue-wave solutions explicitly in terms of Nth-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane (x, y) arising from a constant background at t ? 0 and then gradually tending to the constant background for t ? 0. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey–Stewartson equation analytically and graphically.     

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω ² u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.     

Running waves and dynamical chaos in active media: A numerical study

In the framework of a self-similar problem, we numerically analyze the onset of solutions in the form of running waves in excitable and oscillating active media. We show that the passage to the solution in the form of a running wave occurs through cascades of bifurcations responsible for the development of chaotic dynamics in nonlinear dissipative systems of differential equations.     

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU ₁ andU ₂ such that a traveling wave leads fromU ₁ toU ₂ and another leads fromU ₂ toU ₁. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.     

Control of space-time chaos in a system of equations of the FitzHugh–Nagumo type

We perform an analytic and numerical study of a system of partial differential equations that describes the propagation of nerve impulses in the heart muscle. We show that, for fixed parameter values, the system has infinitely many distinct stable wave solutions running along the spatial axis at arbitrary velocities and infinitely many distinct modes of space-time chaos, where the bifurcation parameter is the velocity of running wave propagation along the spatial axis, which does not explicitly occur in the original system of equations. We suggest an algorithm for controlling the space-time chaos in the system, which permits one to stabilize any of its unstable periodic running waves.     

Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

Formation of singularities for viscosity solutions of hamilton—jacobi equations in one space variable

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.     

Accelerating inexact successive quadratic approximation for regularized optimization through manifold identification

For regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, ISQA \(^{+}\), that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, ISQA \(^{+}\) possesses superlinear convergence not only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition that is more general than that in existing literature. We also prove iterate convergence under a sharpness condition for inexact SQA, which is novel for this family of methods that could easily violate the classical relative-error condition frequently used in proving convergence under similar conditions. Experiments on real-world problems support that ISQA \(^{+}\) improves running time over some modern solvers for regularized optimization.

     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.     

On Periodic Wave Solutions to (1+1)-Dimensional Nonlinear Physical Models Using the Sine-Cosine Method

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.     

The Third-Harmonic Resonance for Capillary-Gravity Waves with O(2) Spatial Symmetry

It is well known that the addition of surface-tension effects to the classic Stokes model for water waves results in a countable infinity of values of the surface tension coefficient at which two traveling waves of differing wavelength travel at the same speed. In this paper the third-harmonic resonance (interaction of a one-crested wave with a three-crested wave) with O(2) spatial symmetry is considered. Nayfeh analyzed the third-harmonic resonance for traveling waves and found two classes of solutions. It is shown that there are in fact six classes of periodic solutions when the O(2) symmetry is acknowledged. The additional solutions are standing waves, mixed waves and secondary branches of “Z-waves.” The normal form and symmetry group for each of the solution classes are developed, and the coefficients in the normal form are formally computed using a perturbation method. The physical aspects of the most unusual class of waves (three-mode mixed waves) are illustrated by plotting the wave height as a function of x for discrete values of t.     

Well-posedness,global existence,and blowup phenomena for a periodic quasi-linear hyperbolic equation

We establish the local well-posedness of a recently derived model for small-amplitude, shallow water waves. For a large class of initial data we prove global existence of the corresponding solution. Criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained, and an existence and uniqueness result for a class of global weak solutions is also given. © 1998 John Wiley & Sons, Inc.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Existence of traveling waves in the fractional bistable equation

We construct traveling waves of the fractional bistable equation by approximating the fractional Laplacian ${(D^{2})^{\alpha}, \alpha \in (0, 1)}$ , with operators ${J \ast u - (\int_{R} J)u}$ , where J is nonsingular. Since the resulting approximating equations are known to have traveling waves, the solutions are obtained by passing to the limit. This provides an answer to the statement (about existence and properties) “This construction will be achieved in a future work” before Assumption 2 in Imbert and Souganidis [6]. With a modification of a part of the argument, we also get the existence of traveling waves for the ignition nonlinearity in the case ${\alpha \in (1/2, 1)}$ .     

On the solitary wave dynamics,under slowly varying medium,for nonlinear Schrödinger equations

We consider the problem of a solitary wave propagation, in a slowly varying medium, for a variable-coefficients nonlinear Schrödinger equation. We prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media. Moreover, we describe for all time the behavior of these solutions, which include refracted and reflected solitary waves, depending on the initial energy.     

Semiclassical geometry and integrability of the AdS/CFT correspondence

We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005.