on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

an asymptotic property of the camassa holm equation abstract asymptotic densities are used to study an asymptotic property of the dispersionless camassa holm equation an asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity for a global solution with non negative compactly supported initial momentum density we show that if the asymptotic density is unique then it is a positive combination of dirac measures supported in a bounded interval in the non negative axis with zero as the only possible accumulation point in other words if the scaled momentum density does not oscillate as time goes to infinity the solution behaves as a combination of peakons moving to the right at different speeds in contrast to many investigations on the topic our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation

convexity of stokes waves of extreme form existence is established of a piecewise convex periodic planar curve s below which is defined a harmonic function which simultaneously satisfies prescribed dirichlet and neumann boundary conditions on s in hydrodynamics this corresponds to the existence of a periodic stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120

global weak solutions to the novikov equation abstract the existence of global weak solutions to the cauchy problem for the novikov equation is established in the space c 0 r l 0 h 1 r without the sign condition on the initial value the limit of viscous approximations for the equation is used to establish the existence of the global weak solution the key elements in our analysis include a new one sided super bound and a new higher norm estimate on the first order derivatives of the solution

on the pressure transfer function for solitary water waves with vorticity in this paper we analyse the role which the pressure function on the sea bed plays in determining solitary waves with vorticity we prove that the pressure function on the flat bed determines a unique real analytic solitary wave solution to the governing equations given a real analytic vorticity distribution in particular the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave

parallel simultaneous stabilization of two systems governed by partial differential equations subject to actuator saturation abstract a parallel simultaneous stabilization of two systems governed by partial differential equation pde with conservation law subject to actuator saturation is considered and a method on the control design is proposed based on the orthogonal decomposition for a port controlled hamiltonian system an abstract approach to the parallel simultaneous stabilization of two systems governed by partial differential equation with conservation law subject to actuator saturation is established the study shows that the parallel simultaneous stabilization controller obtained in this paper is valid in the case of the systems governed by the pdes

convergence of a particle method and global weak solutions of a family of evolutionary pdes the purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations the considered family of pdes is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics we apply a particle method to the studied evolutionary equations and provide a new self contained method for proving its convergence the latter is accomplished by using the concept of space time bounded variation and the associated compactness properties from this result we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established

general variational approach to the interpolation problem abstract the talmi and gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model the connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one two and three dimensional cases they are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

local well posedness and blow up criteria for a two component novikov system in the critical besov space abstract in this paper we mainly investigate the cauchy problem of a two component novikov system we first prove the local well posedness of the system in besov spaces b p r s 1 b p r s with p r 1 s max 1 1 p 3 2 by using the littlewood paley theory and transport equations theory then by virtue of logarithmic interpolation inequalities and the osgood lemma we establish the local well posedness of the system in the critical besov space b 2 1 1 2 b 2 1 3 2 moreover we present two blow up criteria for the system by making use of the conservation laws

multi soliton and multi cuspon solutions of a camassa holm hierarchy and their interactions in this paper we study an integrable camassa holm hierarchy whose high frequency limit is the camassa holm equation by a method associated with the darboux transform we construct the explicit multi soliton and multi cuspon solutions then we study in detail the interactions of soliton cuspon and soliton soliton further an interesting phenomenon is found the soliton with smaller amplitude can travel faster than the one with larger amplitude when they interact at last we investigate in detail the head on collision of one soliton and one cuspon

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

blow up of solutions for the dissipative dullin gottwald holm equation with arbitrary coefficients abstract we study the cauchy problem for the weakly dissipative dullin gottwald holm equation describing unidirectional propagation of surface waves in a shallow water regime u t 2 u x x t c 0 u x 3 u u x u x x x u 2 u x x 2 2 u x u x x u u x x x in the present paper we demonstrate the simple conditions on the initial data that lead to blow up of the solution in finite time

on the low regularity solutions and wave breaking for an equation modeling shallow water waves of moderate amplitude abstract in this paper we consider a nonlinear evolution equation which models the propagation of surface waves of moderate amplitude in shallow water regime we first prove the existence of the low regularity solutions for the cauchy problem of this equation then we discuss the initial profiles under which the solutions of this equation blow up in finite time

non uniform dependence and well posedness for the rotation camassa holm equation on the torus abstract in this paper we study the periodic cauchy problem for a mathematical model of the equatorial water waves propagating mainly in one direction with the weak coriolis effect due to the earth s rotation which reduces to the camassa holm equation as the coriolis effect vanishes we first prove that this model is well posed in the sense of hadamard when initial data belongs to the sobolev space h s t with s 3 2 then we show that the well posedness is sharp in the sense that the continuity of the data to solution map is not better than continuous whose proof rests upon the method of approximate solutions and the well posedness estimate however if h s t is equipped with a weaker h r topology for 0 r s we demonstrate that the data to solution map is holder continuous with the exponent depending on r and s finally we establish a cauchy kowalevski type theorem for this model

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

qualitative analysis for the new shallow water model with cubic nonlinearity abstract considered in this paper is the new shallow water model with cubic nonlinearity which admits the single peaked solitons and multi peakon solutions and includes both the modified camassa holm equation also called fokas olver rosenau qiao equation and the novikov equation as two special cases the main investigation is the cauchy problem of the new shallow water model with qualitative properties of its solutions we first establish the local well posedness for the cauchy problem of the new shallow water model and then derive a precise blow up scenario and the blow up mechanisms for solutions with certain initial profiles moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally we investigate the persistence properties of the solution to the cauchy problem of the new shallow water model

particle trajectories beneath small amplitude shallow water waves in constant vorticity flows abstract we investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water s free surface within the framework of small amplitude waves we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case and we describe the possible particle trajectories depending on the relation between the initial data and the constant vorticity some particle trajectories are undulating curves to the right or to the left others are loops with forward drift or with backward drift others can follow some peculiar shapes

optimal solution for the viscous nonlinear dispersive wave equation in this paper we study the optimal control problem for the viscous nonlinear dispersive wave equation we first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method furthermore the existence of an optimal solution to the viscous nonlinear dispersive wave equation is proved

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

local in space blow up criteria for a class of nonlinear dispersive wave equations abstract this paper is concerned with blow up phenomena for the nonlinear dispersive wave equation on the real line u t u x x t f u x f u x x x g u f u 2 u x 2 x 0 that includes the camassa holm equation as well as the hyperelastic rod wave equation f u k u 2 2 and g u 3 k u 2 2 as special cases we establish some a local in space blow up criterion i e a criterion involving only the properties of the data u 0 in a neighborhood of a single point simplifying and precising earlier blow up criteria for this equation

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

orbital stability of the train of peakons for an integrable modified camassa holm equation abstract in this paper we consider the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peakons and multi peakons using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm we prove that the sum of n sufficiently decoupled peakons is orbitally stable in the energy space

on the nagumo uniqueness theorem abstract by a convenient reparametrization of the integral curves of a nonlinear ordinary differential equation ode we are able to improve the conclusions of a recent generalization of nagumo s uniqueness theorem in this way we establish a flexible uniqueness criterion for odes without lipschitz like nonlinearities

solitary water waves propagating on an underlying uniform current over a finite depth abstract we investigate the nature of velocity and pressure fields for solitary water waves without any restriction on amplitude propagating over finite depth with uniform underlying currents we show that two cases exist corresponding to the fluid velocities u greater than or less than the solitary wave propagation speed c and these two cases result in different nature of flow fields we prove the conditions for existence of solitary waves in 2d with uniform underlying current over finite depth for these two cases we also derive some bounds on froude number and maximum amplitude of solitary wave height

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

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

on the cauchy problem for the modified novikov equation with peakon solutions abstract a new nonlinear dispersive partial differential equation with cubic nonlinearity which includes the famous novikov equation as special case is investigated we first establish the local well posedness in a range of the besov spaces b p r s p r 1 s max 3 2 1 1 p but s 2 1 p which generalize the sobolev spaces h s well posedness in h s with s 3 2 is also established by applying kato s semigroup theory then we give the precise blow up scenario moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally we prove that peakon solutions to the equation are global weak solutions

global existence and blow up phenomena for a weakly dissipative periodic 2 component camassa holm system we first establish local well posedness for a weakly dissipative periodic 2 component camassa holm system we then present two global existence results for strong solutions to the equation we finally obtain several blow up results and the blow up rate of strong solutions to the equation

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

a new formulation of the water wave problem for stokes waves of constant vorticity we consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs the problem is transformed into one concerning the vertical velocity a representation formula for the stream function in terms of the surface and the vorticity is presented and we show that the surface can be determined from the vertical velocity

nonuniform dependence for the cauchy problem of the general b equationa this paper is concerned with the nonuniform dependence on initial data for the general b equation we prove that the solution map of the cauchy problem of the b equation is not uniformly continuous in hs r s 3 2

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

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

blow up solutions for the modified b family of equations abstract in this paper we firstly derive the blow up criterion and the lower bound of the maximal time of existence to the modified b family of equations the new blow up mechanism for the solutions with certain initial profiles and various parameters is then described in detail

the local well posedness and existence of weak solutions for a generalized camassa holm equation abstract a generalization of the camassa holm equation a model for shallow water waves is investigated using the pseudoparabolic regularization technique its local well posedness in sobolev space h s r with s 3 2 is established via a limiting procedure in addition a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s with 1 s 3 2 is developed

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

instability of internal equatorial water waves abstract in the following paper we present criteria for the hydrodynamical instability of internal equatorial water waves we show by way of the short wavelength perturbation approach that certain geophysical waves propagating above the equatorial thermocline are linearly unstable when the wave steepness exceeds a given threshold

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

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

n soliton solutions for the 2 1 dimensional hirota maccari equation in fluids plasmas and optical fibers abstract under investigation in this paper is the hirota maccari equation which is a generalized 2 1 dimensional model in fluid dynamics plasma physics and optical fiber communication with the aid of the hirota bilinear method and symbolic computation the corresponding n soliton solutions are given and illustrated the characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision including the bidirectional solitons elastic interactions and inelastic interactions finally a kind of special phenomenon with the parameters varying is investigated which might provide people with useful information on the dynamics of the relevant fields

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

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

non uniform continuity of periodic holm staley b family of equations abstract we consider a family of non evolutionary partial differential equations known as the holm staley b family which includes the integrable camassa holm and degasperis procesi equations we show that the solution map is not uniformly continuous the proof relies on a construction of smooth periodic traveling waves with small amplitude

extrema of the dynamic pressure in an irrotational stokes wave with underlying currents and infinite depth abstract in this paper we investigate the dynamic pressure in an irrotational stokes wave in flows of finite depth with uniform underlying currents and infinite depth without underlying currents respectively using the maximum principles in combination with exploiting some of the physical structures of the problem itself we prove that irrespectively of uniform underlying current infinite depth and wave amplitude the maximum and minimum of the dynamic pressure are attained at the wave crest and at the wave trough respectively

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

global existence and blow up phenomena for an integrable two component camassa holm shallow water system abstract this paper is concerned with global existence and blow up phenomena for an integrable two component camassa holm shallow water system a new global existence result and several new blow up results of strong solutions to the system are presented our obtained results for the system are sharp and improve considerably earlier results

blowup and blowup rate of solutions to a weakly dissipative periodic rod equation in this paper we first establish the local well posedness for a weakly dissipative periodic rod equation we then present a precise blowup scenario and several blowup results of strong solutions to the equation we finally give the blowup rate of strong solutions to the equation when blowup occurs our blowup results for the weakly dissipative periodic rod equation improve considerably earlier results for the periodic rod equation

global weak solutions and breaking waves to the degasperis procesi equation with linear dispersion abstract we study here an initial value problem for the degasperis procesi equation with a strong dispersive term which is an approximation to the incompressible euler equations for shallow water waves we first determine the blow up set of breaking waves to the equation we then prove the existence and uniqueness of global weak solutions to the equation with certain initial profiles

the cauchy problem for a 4 parameter family of equations with peakon traveling waves abstract the initial value problem for a novel 4 parameter family of evolution equations which are nonlinear and nonlocal and possess peakon traveling wave solutions is studied on both the line and the circle it is proved that this family of equations is well posed in the sense of hadamard when the initial data belong to the sobolev spaces h s with s 5 2 also it is shown that the data to solution map is not uniformly continuous however if h s s 5 2 is equipped with a weaker h r norm 0 r s then the solution map becomes holder continuous