stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

wandering domains for composition of entire functions abstract c bishop in 10 theorem 17 1 constructs an example of an entire function f in class b with at least two grand orbits of oscillating wandering domains in this paper we show that his example has exactly two such orbits that is f has no unexpected wandering domains we apply this result to the classical problem of relating the julia sets of composite functions with the julia set of its members more precisely we show the existence of two entire maps f and g in class b such that the fatou set of f g has a wandering domain while all fatou components of f or g are preperiodic this complements a result of a singh in 22 theorem 4 and results of w bergweiler and a hinkkanen in 6 related to this problem

exact solutions for the evolution of a bubble in stokes flow a cauchy transform approach a cauchy transform approach to the problem of determining the free surface evolution of a single bubble in stokes flow is developed a number of exact solutions to a class of problems have been derived in the literature using conformal mapping theory and these solutions are retrieved and further generalized using the new formulation certain quantities which are conserved by the dynamics are also identified the existence of which had not previously been pointed out a principal purpose of this paper is to use the new formulation to understand when it is possible to externally specify the evolution of the bubble area in such classes of exact solution it is found to be possible only for certain types of far field boundary conditions

solitary shock waves and other travelling waves in a general compressible hyperelastic rod in the literature it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods recently it has been shown that this conjecture is true one might guess that when compressibility is taken into account such a wave which is both a solitary wave and a shock wave can still arise one of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description it is difficult to directly tackle the fully nonlinear rod equations here by using a non dimensionalization process and the reductive perturbation technique we derive a new type of nonlinear dispersive equation as the model equation we then focus on the travelling wave solutions of this new equation as a result we obtain a system of ordinary differential equations an important feature of this system is that there is a vertical singular line in the phase plane which leads to the appearance of shock waves by considering the equilibrium points and their relative positions to the singular line we are able to determine all qualitatively different phase planes those paths in phase planes which represent physically acceptable solutions are discussed one by one it turns out that there is a variety of travelling waves including solitary shock waves solitary waves periodic shock waves etc analytical expressions for all these waves are obtained a new phenomenon is also found a solitary wave can suddenly change into a periodic wave with finite period in dynamical systems this represents a homoclinic orbit suddenly changing into a closed orbit to the authors9 knowledge such a bifurcation has not been found in any other dynamical systems

baker s conjecture for functions with real zeros baker s conjecture states that a transcendental entire function of order less than 1 2 has no unbounded fatou components it is known that for such functions there are no unbounded periodic fatou components and so it remains to show that they can also have no unbounded wandering domains here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1 this raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1 key ingredients of our proofs are new results in classical complex analysis with wider applications these new results concern the winding properties of the images of certain curves proved using extremal length arguments growth estimates for entire functions and the distribution of the zeros of entire functions of order less than 1

finger competition dynamics in rotating hele shaw cells we report analytical results for the development of interfacial instabilities in rotating hele shaw cells we execute a mode coupling approach to the problem and examine the morphological features of the fluid fluid interface at the onset of nonlinear effects the impact of normal stresses is accounted for through a modified pressure jump boundary condition a differential equation describing the early nonlinear evolution of the interface is derived being conveniently written in terms of three relevant dimensionless parameters viscosity contrast a surface tension b and gap spacing b we focus our study on the influence of these parameters on finger competition dynamics it is deduced that the link between finger competition and a b and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations our results show good agreement with existing experimental and numerical investigations of the problem both in low and high a 0 a 0 lead to enhanced competition of outward inward fingers within this context the role of b and b in determining different finger competition behaviors is also discussed

on the cauchy problem for the generalized camassa holm equation in this paper we consider the cauchy problem for the generalized camassa holm equation by using the littlewood paley decomposition and nonhomogeneous besov spaces we prove that the cauchy problem for the generalized camassa holm equation is locally well posed in the besov space b p r s with 1 max 1 1 p 3 3 which generalizes the local well posedness result in 1 we also establish the ill posedness result of the generalized camassa holm equation and give a blow up criterion

dynamics of viscous fingers in rotating hele shaw cells with coriolis effects a growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating hele shaw cells however only a few of them consider the influence of coriolis forces the studies including coriolis effects are mostly restricted to the high viscosity contrast limit and rely on either purely linear stability analyses or intensive numerical simulations we approach the problem analytically and use a modified darcy s law including the exact form of the coriolis effects to execute a mode coupling analysis of the system by imposing no restrictions on the viscosity contrast a dimensionless viscosity difference we go beyond linear stages and examine the onset of nonlinearities our results indicate that when coriolis effects are taken into account an interesting interplay between the reynolds number re and a arises this leads to important changes in the stability and morphological features of the emerging interfacial patterns we contrast our mode coupling approach with previous theoretical models proposed in the literature

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

global existence for a new periodic integrable equation we establish the local well posedness for a new periodic integrable equation we show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time

absence of wandering domains for some real entire functions with bounded singular sets let f be a real entire function whose set s f of singular values is real and bounded we show that if f satisfies a certain function theoretic condition the sector condition then f has no wandering domains our result includes all maps of the form z mapsto lambda frac sinh z z a with lambda 0 and a in mathbb r we also show the absence of wandering domains for certain non real entire functions for which s f is bounded and f n s f rightarrow infty uniformly as a special case of our theorem we give a short elementary and non technical proof that the julia set of the exponential map f z e z is the entire complex plane furthermore we apply similar methods to extend a result of bergweiler concerning baker domains of entire functions and their relation to the postsingular set to the case of meromorphic functions

wave breaking for shallow water models with time decaying solutions abstract a family of camassa holm type equations with a linear term and cubic and quartic nonlinearities is considered local well posedness results are established via kato s approach conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time dependent monotonically decreasing function of time and bounded from above by the sobolev norm of the initial data under some conditions the existence of wave breaking phenomenon is investigated and necessary conditions for its existence are obtained in our framework the wave breaking is guaranteed among other conditions when the coefficient of the linear term is sufficiently small which allows us to interpret the equation as a linear perturbation of some recent camassa holm type equations considered in the literature

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

on a class of geometry driven free boundary problems an abstract class of two dimensional geometry driven free boundary problems is considered in which the evolution of the cauchy transform of the time evolving domain satisfies a partial differential equation of conservation type this idea generalizes an association between free boundary problems and the inverse gravitational problem it is shown that this class of free boundary problems admits exact solutions expressible in terms of a finite set of time evolving parameters in addition even when exact solutions are not available a theorem is established which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities two distinct problems of physical interest are shown to fall within this abstract class the evolution of a fluid blob in a rotating hele shaw cell and the evolution of highly viscous fluid under the effects of surface tension various aspects of these two problems are discussed

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation this paper is concerned with the global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation we prove the global existence and uniqueness of solution to the viscous degasperis procesi equation for u0 l2 r the local well posedness for u0 l2 r has been proven based on the energy estimate the strong and weak limit behaviors of solution to the viscous degasperis procesi equation are obtained

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

revisit to wave breaking phenomena for the periodic dullin gottwald holm equation in this paper we mainly devote to investigate the periodic dullin gottwald holm equation by overcoming the difficulties caused by the complicated mixed nonlinear structure a very useful priori estimate is derived in lemma 2 7 based on h 1 conservation and l estimate of solution some new blow up phenomena are derived for the periodic dullin gottwald holm equation under different initial conditions

holder continuity of the solution map for the novikov equation the novikov equation ne has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the camassa holm and degasperis procesi equations which have quadratic nonlinearities ne is well posed in sobolev spaces hs on both the line and the circle for s 3 2 in the sense of hadamard and its data to solution map is continuous but not uniformly continuous this work studies the continuity properties of ne further for initial data in hs s 3 2 it is shown that the solution map for ne is holder continuous in hr topology for all 0 r s with exponent depending on s and r

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

blow up phenomena for a family of burgers like equations abstract by introducing a stress multiplier we derive a family of burgers like equations we investigate the blow up phenomena of the equations both on the real line r and on the circle s to get a comparison with the degasperis procesi equation on the line r we first establish the local well posedness and the blow up scenario then we use conservation laws of the equations to get the estimate for the l norm of the strong solutions by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles analogous results are provided in the periodic case especially we find differences between the burgers like equations and the degasperis procesi equation see remark 4 1

rigidity of escaping dynamics for transcendental entire functions we prove an analog of bottcher s theorem for transcendental entire functions in the eremenko lyubich class mathcal b more precisely let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space i e are quasiconformally equivalent in the sense of eremenko and lyubich then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration furthermore this conjugacy extends to a quasiconformal self map of the plane

on solutions of the camassa holm equation the techniques that have been developed for the application of the inverse scattering transform method to the solution of the camassa holm equation principally by constantin are implemented we use this approach first to represent the known solitary wave solution in a simple parametric form and then second to obtain the general two and three soliton solutions these latter two solutions require rather extensive use of mathematical packages mathematica and maple in order to complete the construction of solutions a number of examples are presented the phase shifts evident after an interaction are found and the special limit that recovers the peakon solutions is discussed

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

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 c 1998 john wiley sons inc

the immersed interface method for axis symmetric problems and application to the hele shaw flow many physical application problems are axis symmetric using axis symmetric properties many three dimensional problems can be solved efficiently using two dimensional axis symmetric coordinates in this paper the immersed interface method iim in axis symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient solution or flux a staggered grid is used to overcome the pole singularity other challenges include deriving the jump relations in axis symmetric coordinates designing the numerical algorithm when the interface is close to the pole r 0 computing interface quantities such as the normal and tangential directions surface derivatives curvature information etc the numerical algorithm is based on a finite difference discretization and uniform grid in the axis symmetric coordinates the finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface the method is shown to be second order accurate in the infinity norm the developed new iim is applied to the hele shaw flow and compared with the results from the literature

existence and uniqueness of low regularity solutions for the dullin gottwald holm equation we establish the local existence and uniqueness of solutions for the dullin gottwald holm equation with continuously differentiable periodic initial data the regularity conditions needed for the cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered

limit behavior of the global solutions to the degasperis procesi type equation abstract in this paper we study the degasperis procesi equation with a physically perturbation term a linear dispersion based on the global existence result we show that the solution of the degasperis procesi equation with linear dispersion tends to the solution of the corresponding degasperis procesi equation as the dispersive parameter goes to zero moreover we prove that smooth solutions of the equation have finite propagation speed they will have compact support if its initial data has this property

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

inverse problems of potential theory and flows in porous media with time dependent free boundary abstract it is known that fluid flows in a homogeneous porous medium with time dependent free boundary have the property that integrals over the fluid domain of harmonic functions vary linearly in time 1 this property relates these flows to the classical inverse problem of newtonian potentials 2 it is possible to apply this relation to construction of finite or infinite sets of distinct domains homeomorphic to a ball in r d and having the same potential outside considering flows in a non homogeneous medium one obtains examples of non uniqueness of solutions of the inverse potential problem for arbitrary charge density and space dielectric permeability

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

on the cauchy problem for a generalized degasperis procesi equation we mainly study the cauchy problem of the generalized degasperis procesi equation first we establish the local well posedness for the equation second we give the precise blow up scenario a conservation law and prove the equation has smooth solutions which blow up in finite time finally we investigate the blow up rate for the blow up solutions

singularities in hele shaw flows driven by a multipole we study analytically and numerically singularity formation in an interface flow driven by a multipole for a two dimensional hele shaw cell with surface tension our analysis proves that singularity formation is inevitable in the case of a dipole for a multipole of a higher order we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole it is therefore very likely that this solution will develop finite time singularities our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole in addition it is observed that the interface approaches the multipole from directions which can be predicted beforehand we also estimate as a function of time the distance between the finger tip and multipole and the results are in excellent agreement with numerical computations

continuity properties of the data to solution map for the generalized camassa holm equation abstract this work studies a generalized camassa holm equation with higher order nonlinearities g kbch the camassa holm the degasperis procesi and the novikov equations are integrable members of this family of equations g kbch is well posed in sobolev spaces h s s 3 2 on both the line and the circle and its solution map is continuous but not uniformly continuous in this work it is shown that the solution map is holder continuous in h s equipped with the h r topology for 0 r s and the holder exponent is expressed in terms of s and r

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

on the cauchy problem for a generalized camassa holm equation the local well posedness of a generalized camassa holm equation is established by means of kato s theory for quasilinear evolution equations and two types of results for the blow up of solutions with smooth initial data are given

density of hyperbolicity for classes of real transcendental entire functions and circle maps we prove density of hyperbolicity in spaces of i real transcendental entire functions bounded on the real line whose singular set is finite and real and ii transcendental self maps of the punctured plane which preserve the circle and whose singular set apart from zero and infinity is contained in the circle in particular we prove density of hyperbolicity in the famous arnol d family of circle maps and its generalizations and solve a number of other open problems for these functions including three conjectures by de melo salom ao and vargas we also prove density of real hyperbolicity for certain families as in i but without the boundedness condition our results apply in particular when the functions in question have only finitely many critical points and asymptotic singularities or when there are no asymptotic values and the degree of critical points is uniformly bounded

blow up phenomena and persistence properties for an integrable two component peakon system abstract in this paper we are concerned with an integrable two component peakon system which was proposed by xia qiao and zhou we present a precise blow up scenario and a new blow up result for strong solutions to the system moreover we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically respectively

infinite propagation speed for the degasperis procesi equation abstract we prove that any nontrivial classical solution of the degasperis procesi equation will not have compact support if its initial data has this property

well posedness and analytic solutions of a two component water wave equation abstract we first establish the local well posedness for the cauchy problem of a two component water waves system in a critical nonhomogeneous besov space then we derive a continuation criterion for strong solutions to the system finally we prove the existence of analytic solutions to the system

systematic weakly nonlinear analysis of radial viscous fingering we present a weakly nonlinear analysis of the interface dynamics in a radial hele shaw cell driven by both injection and rotation we extend the systematic expansion introduced in e alvarez lacalle et al phys rev e 64 016302 2001 to the radial geometry and compute explicitly the first nonlinear contributions we also find the necessary and sufficient condition for the uniform convergence of the nonlinear expansion within this region of convergence the analytical predictions at low orders are compared satisfactorily to exact solutions and numerical integration of the problem this is particularly remarkable in configurations with no counterpart in the channel geometry for which the interplay between injection and rotation allows that condition to be satisfied at all times in the case of the purely centrifugal forcing we demonstrate that nonlinear couplings make the interface more unstable for lower viscosity contrast between the fluids

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

global weak solutions to the weakly dissipative degasperis procesi equation abstract the global weak solutions for the degasperis procesi equation with weakly dissipative term are investigated provided that u 0 h 1 r y 0 1 x 2 u 0 m r supp y 0 x 0 and supp y 0 x 0 the existence and uniqueness of global weak solutions for the equation are shown to be true in the space w l o c 1 r r l l o c r h 1 r

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

well posedness and blow up phenomena for the generalized degasperis procesi equation abstract in this paper we mainly study the cauchy problem of the generalized degasperis procesi equation we establish the local well posedness and give the precise blow up scenario for the equation then we show that the equation has smooth solutions which blow up in finite time

singularities in hele shaw flows we study analytically and numerically singularities in an interface flow driven by surface tension and a sink in a two dimensional hele shaw cell our analysis proves that singularity formation is inevitable in generic situations our numerical simulations suggest that the hele shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out a corner is formed right at the tip of the interface that touches the sink

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

blow up phenomena and local well posedness for a generalized camassa holm equation in the critical besov space abstract in this paper we mainly study the cauchy problem for a generalized camassa holm equation in a critical besov space first by using the littlewood paley decomposition transport equations theory logarithmic interpolation inequalities and osgood s lemma we establish the local well posedness for the cauchy problem of the equation in the critical besov space b 2 1 1 2 then we give a new blow up criterion for the cauchy problem of the equation finally we present a new blow up result and the exact blow up rate of strong solutions to the equation by making use of the conservation law and the obtained blow up criterion

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

invariants and wave breaking analysis of a camassa holm type equation with quadratic and cubic non linearities abstract a non local evolution equation of the camassa holm type with dissipation is considered the local well posedness of the solutions of the cauchy problem involving the equation is established via kato s approach and the wave breaking scenario is also described to prove such result we firstly construct conserved currents for the equation and from them conserved quantities

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

a cauchy integral approach to hele shaw problems with a free boundary the case of zero surface tension in this paper we study a nonlinear and nonlocal free boundary dynamics the hele shaw problem without surface tension when the fluid domain is either bounded or unbounded the key idea is to use a global quantity the cauchy integral of the free boundary to capture the motion of the boundary this cauchy integral is shown to be linear in time the free boundary at a fixed time is then recovered from its cauchy integral at that time the main tool in our analysis ischerednichenko s theorem concerning the inverse properties of the cauchy integrals as products of our approach we establish the short time existence and uniqueness of classical solutions for analytic initial boundaries we also show the non existence of classical solutions for all smooth but non analytic initial boundaries when there is a sink at either a finite point or at infinity when the fluid domain is bounded all solutions except the circular one break down before all the fluid is sucked out from the sink regularity results are also obtained when there is a source at a finite point or at infinity

optimal control of a class of nonlocal dispersive equations abstract in this paper the optimal control problem for a class of nonlocal dispersive equations is considered we introduce a kind of the definition of a weak solution to this equation and prove the existence of a unique weak solution in a special hilbert space s 0 t we also discuss the existence of optimal control solution to a class of nonlocal dispersive equations with the cost functional adopting the dubovitskii milyutin functional analytical approach we prove the necessary optimality condition of optimal control local maximum principle for the investigative system in fixed final horizon case

exponential decay and symmetry of solitary waves to degasperis procesi equation abstract we improve the decay argument by bona and li 1997 5 for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady degasperis procesi equation in the nonlocal formulation in addition we give a method which confirms the symmetry of solitary waves including those of the maximum height finally we discover how the symmetric structure is connected to the steady structure of solutions to the degasperis procesi equation and give a more intuitive proof for symmetric solutions to be traveling waves the improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations

dynamic rays of bounded type entire functions we construct an entire function in the eremenko lyubich class b whose julia set has only bounded path components this answers a question of eremenko from 1989 in the negative on the other hand we show that for many functions in b in particular those of nite order every escaping point can be connected to 1 by a curve of escaping points this gives a partial positive answer to the aforementioned question of eremenko and answers a question of fatou from 1926

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

constructing entire functions by quasiconformal folding we give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f among other applications we construct a function f with bounded singular set whose fatou set contains a wandering domain