asymptotic spatial patterns on the complex time dependent ginzburg landau equation asymptotic spatial patterns wavefront solutions on time dependent ginzburg landau tdgl equations with complex coefficients are discussed the condition for the existence of a spatial limit cycle solution is found to be given in terms of the propagation velocity of the wavefront solution and the stability criterion for the spatial limit cycle is obtained the amplitude oscillation of an asymptotic spatial pattern on the tdgl equation non linear schrodinger equation with purely imaginary coefficients is expressed in terms of jacobi s elliptic function while an exact solution of the asymptotic spatial pattern on the tdgl equation force free duffing equation with purely real coefficients is aperiodic and has a unique propagation velocity

bifurcation from double eigenvalue for nonlinear equation with third order nondegenerate singularity in this paper the steady state bifurcation from a double eigenvalue for nonlinear equation with singularity being second order fully degenerate and third order nondegenerate is investigated by the normalized lyapunov schmidt reduction method the precise criteria for the existence and nonexistence of bifurcation and the topological properties of regular bifurcated branches are obtained

effects of boundary conditions on spatially periodic states abstract we investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension for all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary type i solution a linear stability analysis of the uniform state at threshold reveals that type i solutions are often unstable then there exists in the full eckhaus stable band a static solution where the amplitude rises above its bulk value near the boundary type ii solution or a limit cycle solution where the amplitude near the boundary oscillates these solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold

two global several parameter bifurcation theorems and their applications abstract in this paper we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u l u h u u r m x where m is a positive integer x is a banach space l x x is a positively homogeneous compact operator and h r m x x is compact with h o u near u 0 uniformly on bounded sets we obtain some results involving unilateral global bifurcation two examples of p laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied

on the eckhaus instability for spatially periodic patterns abstract the range of stable wavevectors is near the threshold for appearance of periodic patterns in quasi one dimensional systems limited by the long wavelength eckhaus instability at this instability saddle point solutions characterizing the wavelength changing processes inside the stable range merge with the periodic solutions we first analyse this bifurcation near threshold using the amplitude expansion in lowest order then a nonlinear equation for the evolution of slow modulations of the periodic pattern far from threshold but near the eckhaus instability is derived and used to analyse the universal properties of the eckhaus bifurcation more detailed information concerning the spatial symmetry of saddle point solutions is obtained by numerical integration of simple model systems

domain structures in fourth order phase and ginzburg landau equations abstract in pattern forming systems competition between patterns with different wave numbers can lead to domain structures which consist of regions with differing wave numbers separated by domain walls for domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments close to threshold a fourth order ginzburg landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers the existence and stability regime of domain structures is found to be very intricate due to interactions with other modes in contrast to the phase equation the ginzburg landau equation allows a spatially oscillatory interaction of the domain walls thus close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold and can be stable even without phase conservation we study their regime of stability as a function of their quantized length domain structures are related to zig zags in two dimensional systems the latter are therefore expected to be stable only when quenched far enough beyond the zig zag instability

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

multiparameter local bifurcation based on the linear part abstract in this paper we give a result in multiparameter local bifurcation theory this result is a generalization of the hopf bifurcation theorem and of a previous result by j k hale our result can be applied to fredholm operators with arbitrary index

a new kind of periodic stationary solution of the cubic ginzburg landau equation abstract we analyze the long scale dynamics of the modulationally unstable plane wave in the cubic complex ginzburg landau equation without a diffusion term our numerical studies reveal that when the dissipative effects dominate over the conservative effects a new kind of temporally stationary but spatially periodic solution can be found

nonequilibrium phenomena outlines and bibliographies of a workshop the following is a set of outlines and bibliographies for lectures presented at a summer workshop on nonequilibrium phenomena held from june 22 to july 3 1981 at the institute for theoretical physics in santa barbara these outlines were distributed to the participants in lieu of formal proceedings and they are being presented for publication in the same form in the belief that the information they contain will be useful to a wider audience it should be clearly stated however that the compilation is an informal one which does not claim to be a complete survey of the subject

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

a class of generalized ginzburg landau equations with random switching abstract this paper focuses on a class of generalized ginzburg landau equations with random switching in our formulation the nonlinear term is allowed to have higher polynomial growth rate than the usual cubic polynomials the random switching is modeled by a continuous time markov chain with a finite state space first an explicit solution is obtained then properties such as stochastic ultimate boundedness and permanence of the solution processes are investigated finally two time scale models are examined leading to a reduction of complexity

stable and chaotic solutions of the complex ginzburg landau equation with periodic boundary conditions abstract we study analytically and numerically the dynamical behavior of the solutions of the complex ginzburg landau equation with diffraction but without diffusion which governs the spatial evolution of the field in an active nonlinear laser cavity accordingly the solutions are subject to periodic boundary conditions the analysis reveals regions of stable stationary solutions in the model s parameter space and a wide range of oscillatory and chaotic behaviors close to the first bifurcation destabilizing the spatially uniform solution a stationary single humped solution is found in an asymptotic analytical form which turns out to be in very good agreement with the numerical results simulations reveal a series of stable stationary multi humped solutions

explicit peakon and solitary wave solutions for the modified fornberg whitham equation in this paper the modified fornberg whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis in some parametric conditions some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained

effects of boundaries on one dimensional reaction diffusion equations near threshold a simple set of two reaction diffusion equations is analyzed near the threshold for the appearance of spatially periodic solutions in one dimension exact amplitude equations are derived to second order in the deviation from the threshold wavevector selection is studied analytically for parameters varying slowly in space from supercritical to subcritical conditions the nonuniversality of the selection process is demonstrated explicitly for nonpotential cases the effect of boundary conditions on restricting the available band of wavevectors is studied for solutions which fall below their bulk value near the boundary type i the existence of another class of static solutions type ii remaining finite at the boundary even near threshold is demonstrated a linear stability analysis of the uniform state reveals that type i solutions are often unstable immediately above threshold then one has type ii states or oscillatory solutions some numerical results are presented which confirm the analytic calculations

homoclinic explosions in the complex ginzburg landau equation a bifurcation sequence of the complex ginzburg landau cgl equation in an even subspace of spatially periodic solutions is analyzed with local three dimensional bifurcation theory near homoclinic orbits bifurcation to chaos occurs via homoclinic explosions similar to bifurcation to chaos in the lorenz model and a similar global bifurcation picture is developed an analysis via one dimensional maps is developed to predict attractors it is shown that the two mode galerkin truncation can mimic some bifurcations of the full cgl equationdeceptively and also that the truncation fails to mimic many features of full pde precisely because the truncation lacks many of the same homoclinic orbits

mean flow effects in the electro hydrodynamic convection in nematic liquid crystals abstract on the basis of the weakly nonlinear analysis a set of two coupled amplitude equations is derived which lead to a better understanding of the electro hydrodynamic instability in nematic liquid crystals near onset it is shown that the normal convective patterns are very sensitive against large scale flow mean flow modes which are systematically included simulation of the equations shows typically a complex spatio temporal behavior defect turbulence very similar to the situation often observed in experiments

symmetry and chaos in the complex ginzburg landau equation ii translational symmetries abstract the complex ginzburg landau cgl equation on a one dimensional domain with periodic boundary conditions has a number of different symmetries and solutions of the equation may or may not be fixed by the action of these symmetries we investigate the stability of chaotic solutions that are spatially periodic with period l with respect to subharmonic perturbations that have a spatial period kl for some integer k 1 this is done by considering the isotypic decomposition of the space of solutions and finding the dominant lyapunov exponent associated with each isotypic component we find a region of parameter space in which there exist chaotic solutions with spatial period l and homogeneous neumann boundary conditions that are stable with respect to perturbations of period 2l on varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2l while remaining chaotic leading to a supercritical subharmonic blowout bifurcation for a large number of parameter values checked chaotic solutions with spatial period l were found to be unstable with respect to perturbations of period 3l we conclude that while periodic boundary conditions are often convenient mathematically we would not expect to see chaotic spatially periodic solutions forming starting with an arbitrary non periodic initial condition

new exact bounded travelling wave solutions for the zhiber shabat equation abstract in this paper the zhiber shabat equation is studied by using the bifurcation theory and the method of phase portraits analysis many smooth solitary wave solutions compacton solutions and smooth periodic wave solutions are obtained some previous results are extended

analytic solutions for the 2 1 dimensional generalized sine gordon equations in nonlinear optics abstract the 2 1 dimensional generalized sine gordon equations can be used to describe the propagation of femtosecond laser pulse in a systems of two level atoms in this paper bilinear forms of the equations have been constructed through variable transformation and bell polynomial based on the bilinear method and truncated painleve expansion multi soliton and quasi periodic peakon solutions for such a system are derived respectively overtaking elastic interactions between the two solitons and among the three solitons have been found it is presented that the soliton with the smaller amplitude can travel faster than the larger one apart from the soliton other type of nonlinear wave propagating in the systems of two level atoms has been displayed namely the quasi periodic peakon the peakon oscillates in both space and time while its left and right derivatives at peak point are relatively large which means the electric field intensity of the femtosecond laser pulse is rather strong

new peakon solitary wave and periodic wave solutions for the modified camassa holm equation abstract in this paper an independent variable transformation is introduced to solve the modified camassa holm equation using the bifurcation theory and the method of phase portrait analysis some peakons solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained

pattern competition and the decay of unstable patterns in quasi one dimensional systems abstract the dynamics of quasi one dimensional pattern forming systems with simple periodic initial conditions and additional noise or with almost periodic initial conditions is studied analytically and numerically using the lowest order amplitude equation in this way the decay of eckhaus unstable patterns and subsequent formation of stable ones as observed recently and the competition of patterns with different wavenumbers is studied

noise and pattern selection in the one dimensional swift hohenberg equation the question of pattern selection in the presence of noise is addressed in the context of the one dimensional swift hohenberg equation a model for the onset of convection we show how noise destroys long range order in the long time patterns so that characterization of the selected pattern in terms of the fourier mode with the maximum spectral power is not always suitable the number of zeros of the configurations turns out to be a better quantity we consider also the decay process after an eckhaus instability it is shown that the selected pattern is close to the one of fastest growth during the linear regime and not to the variationally preferred this mechanism is robust to small noise

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

bifurcation analysis of the eckhaus instability the eckhaus instability 1 is the only purely two dimensional instability occurring in convection 2 taylor couette flow 3 liquid crystals 4 and other pattern forming systems despite its widespread occurrence the eckhaus instability has not been analyzed in bifurcation theoretic terms it can be studied analytically by averaging over the depth 5 leading to the classic one dimensional ginzburg landau equation frac partial a partial t mu a frac partial 2 a partial x 2 left a right 2 a 1

bifurcations of traveling wave solutions for zhiber shabat equation abstract in this paper we investigate the dynamical behavior of traveling wave solutions in the zhiber shabat equation by using the bifurcation theory and the method of phase portraits analysis as a result we obtain the conditions under which smooth and non smooth traveling wave solutions exist and give some exact explicit solutions for some special cases

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation