J. D. Gibbon

Low-dimensional behaviour in the complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined as the number of Fourier modes required to span the universal attractor. The authors use concepts from the theory of inertial manifolds to determine rigorous upper bounds on the Fourier spanning dimension, which also establishes the finite dimensionality of the universal attractor. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for some parameter values, computing exactly) the Lyapunov dimension and invoking a recent theorem that asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula with the universal (global) Lyapunov exponents, is an upper bound on the Hausdorff dimension. This study of low dimensionality in the complex Ginzburg-Landau equation allows for an examination of the current techniques used in the rigorous investigation of finite-dimensional behaviour. Contact is made with some recent results for fluid turbulence models, and the authors discuss some unexplored directions in the area of low-dimensional behaviour in the complex Ginzburg-Landau equation.

A New Hierarchy of Korteweg-De Vries Equations

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the standard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various properties associated with it such as Bäcklund transformations. The most interesting of the new K. de V. equations is (unx ≡ ∂nu/∂xn) ( u4x + 30uu2x + 60u3)x + ut = 0. We have proved that this equation has N-soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N-soliton solutions, (uxx + 6u2)xx + uxx – utt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

THE COMPLEX LORENZ EQUATIONS

We have undertaken a study of the complex Lorenz equations x = −σx + σy . y = (r − z)x − ay . z = −bz + 12(x∗y + xy∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r1 + ir2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r2 = 0. We have been able to determine analytically two critical values of r1, namely r1c and r1c . The origin is a stable fixed point for 0 r1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ + 1 then this limit is only stable in the region r1c rlc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

Weak and strong solutions of the complex Ginzburg-Landau equation

Abstract The generalized complex Ginzburg-Landau equation, ∂ t A=RA+(1+iv) δA-(1+iμ)¦A¦ 2σ A , has been proposed and studied as a model for “turbulent” dynamics in nonlinear partial differential equations. It is a particularly interesting model in this respect because it is a dissipative version of the Hamiltonian nonlinear Schrodinger equation possessing solutions that form localized singularities in finite time. In this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions. Appropriately defined weak solutions are established globally in time, and unique strong solutions are found locally. A new collection of a priori estimates are presented, and we discuss the relationship of our results for the complex Ginzburg-Landau equation to analogous issues for fluid turbulence described by the incompressible Navier-Stokes equations.

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

Abstract We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: A t =RA+(1+iν)δA−(1+iμ)A¦A¦ 2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ϵ=1+μν 2n: F n =∫(¦∇ n−1 A¦ 2 +α n ¦A¦ 2n )dx , for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fns, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn‖A‖2∞)Fn−bnF2n/Fn−1, for cn, bn > 0 (F0≡1). Estimates for the “bottom rung” F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and ‖A‖2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A¦A¦2q, the critical case qD=2 gives parallel results.

The real and complex Lorenz equations in rotating fluids and lasers

Abstract The Lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added. This derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation. The addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of Lorenz equations. Both sets of equations are derived in examples in laser optics and baroclinic instability.

Solitons in nonlinear optics. I. A more accurate description of the 2π pulse in self-induced transparency

A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 1018 atoms cm-3. At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.

Numerical study of singularity formation in a class of Euler and Navier–Stokes flows

We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier–Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.

The real and complex Lorenz equations and their relevance to physical systems

Abstract We summarize some recently obtained results on real and complex Lorenz equations and discuss their possible significance in relation to real fluid dynamical processes.

Dynamically stretched vortices as solutions of the 3D Navier—Stokes equations

Abstract A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u=(u1(x,y,t),u2(x,y,t),γ(x,y,t)z+W(x,y,t)) where u1,u2,γ and W are functions of x,y and t but not z. It turns out that the equations for the third component of vorticity ω3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u such that u = −(γ/2)x−(γ/2)y,γz + −ψ y ,ψ x ,W . The strain rate, γ(t), is solely a function of time and is related to the pressure via a Riccati equation γ +γ 2 +p zz (t)=0 . A constraint on pzz(t) is that it must be spatially uniform. The decoupling of ω3 and W allows the equation for ω3 to be mapped to the usual general 2D problem through the use of Lundgren’s transformation, while that for W can be mapped to the equation of a 2D passive scalar. When ω3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2π-periodic θ-dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω = r −1 W θ ,−W r ,ω 3 has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when θ passes through multiples of 2π.