a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

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

global existence and blow up phenomena for the weakly dissipative novikov equation in this paper we consider the global existence and blow up for the weakly dissipative novikov equation we firstly establish the local well posedness for the weakly dissipative novikov equation by kato s theorem then we present some blow up results finally we present the global existence of strong solutions to the weakly dissipative equation c 2011 elsevier ltd all rights reserved

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi controls this paper investigates the existence and uniqueness of mild solutions for a class of sobolev type fractional nonlocal abstract evolution equations in banach spaces we use fractional calculus semigroup theory a singular version of gronwall inequality and leray schauder fixed point theorem for the main results a new kind of sobolev type appears in terms of two linear operators is introduced to extend previous works in the field an existence result of optimal multi control pairs governed by the presented system is proved finally an example is also given to illustrate the obtained theory

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

dynamic stability analysis of stochastic fractional order memristor fuzzy bam neural networks with delay and leakage terms abstract this paper deals with the uniform stability in mean square of stochastic fractional order memristor fuzzy bam neural networks with delay and leakage terms by employing the ideas of cauchy schwartz inequality burkholder davis gundy inequality analysis technique some sufficient conditions are derived to ensure the uniform stability in mean square of such networks the existence uniqueness and stability of its equilibrium point are also showed two different fractional order derivatives between the u layer and v layer are taken into consideration synchronously with fractional order 1 2

global convergence of the nonmonotone mbfgs method for nonconvex unconstrained minimization in this paper we propose a new nonmonotone armijo type line search and prove that the mbfgs method proposed by li and fukushima with this new line search converges globally for nonconvex minimization some numerical experiments show that this nonmonotone mbfgs method is efficient for the given test problems

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

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

blow up phenomenon for the integrable novikov equation abstract in this paper we investigate a new integrable equation derived recently by v s novikov generalizations of the camassa holm equation j phys a 42 34 2009 342002 14 pp analogous to the camassa holm equation and the degasperis procesi equation this new equation possesses the blow up phenomenon under the special structure of this equation we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time a global existence result is also found

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

global weak solutions for the novikov equation in the paper we mainly study the existence and uniqueness of global weak solutions for the novikov equation we first recall some results and definitions on strong solutions and weak solutions for the equation then we show that the equation has smooth solutions which exist globally in time provided the initial data satisfy certain sign conditions finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions

generalized fractional controller for singular systems of differential equations abstract in this article we consider a class of singular linear systems of first order and introduce a generalized fractional order feedback controller of caputo type the closed loop system in question is a singular system of differential equations having both first and fractional order derivatives we provide a comprehensive theory for the existence and uniqueness of solutions as well as for the stability of the system with inclusion of the fractional order controller an example of a singular system with a fractional order proportional integral controller as well as an example on a 3 bus power system with inclusion of a fractional order damping controller is given to illustrate our theory

fractional variational calculus with classical and combined caputo derivatives abstract we give a proper fractional extension of the classical calculus of variations by considering variational functionals with a lagrangian depending on a combined caputo fractional derivative and the classical derivative euler lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved

on time optimal control of fractional order systems abstract fractional optimal control problem is treated from convex analytical viewpoint sufficient conditions for time optimal control similar to that of pontryagin s maximum principle are obtained for fractional order systems in the sense of riemann liouville and caputo theoretical results are supported by examples

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions we introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a hilbert space is studied the results are obtained by using the fractional power of operators multi valued analysis and sadovskii s fixed point theorem main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable as an example a fractional partial nonlocal control functional differential inclusion is considered

analysis and numerical approximation of tempered fractional calculus of variations problems abstract in this paper we study variational problems where the cost functional involves the tempered caputo fractional derivative several important optimization conditions are derived to find the optimal solution sufficient and necessary conditions are presented for different variational problems for example the cases of integral isoperimetric problem and holonomic constraints are considered as well as problems with high order derivatives a numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples

well posedness and persistence properties for the novikov equation abstract recently novikov found a new integrable equation we call it the novikov equation in this paper which has nonlinear terms that are cubic rather than quadratic and admits peaked soliton solutions peakons firstly we prove that the cauchy problem for the novikov equation is locally well posed in the besov spaces b 2 r s which generalize the sobolev spaces h s with the critical index s 3 2 then well posedness in h s with s 3 2 is also established by applying kato s semigroup theory finally we present two results on the persistence properties of the strong solution for the novikov equation

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

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

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

oscillation induced blow up to the modified camassa holm equation with linear dispersion abstract in this paper we provide a blow up mechanism to the modified camassa holm equation with varying linear dispersion we first consider the case when linear dispersion is absent and derive a finite time blow up result with an initial data having a region of mild oscillation a key feature of the analysis is the development of the burgers type inequalities with focusing property on characteristics which can be deduced from tracing the ratio between solution and its gradient using the continuity and monotonicity of the solutions we then extend this blow up criterion to the case of negative linear dispersion and determine that the finite time blow up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild oscillation region finally we demonstrate that in the case of non negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile in contrast to the camassa holm equation with linear dispersion the effect of linear dispersion of the modified camassa holm equation on the blow up phenomena is rather delicate

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

stability of peakons for an integrable modified camassa holm equation with cubic nonlinearity considered herein is the dynamical stability of the single peaked soliton and periodic peaked soliton for an integrable modified camassa holm equation with cubic nonlinearity the equation is known to admit a single peaked soliton and multi peakon solutions and is shown here to possess a periodic peaked soliton by constructing certain lyapunov functionals it is demonstrated that the shapes of these waves are stable under small perturbations in the energy space

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation