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In this paper, we investigate the generalized Camassa–Holm equation with k + 1 degree nonlinearities. The global existence is established, provided that 0 < b < k and the initial data y0=(1−∂x2)u0 do not change sign. Long time behavior for the support of momentum density is obtained, and more precisely, we deduce the limit of the support of momentum density as t goes to +∞.

We study the effect of noise on the stochastic modified Camassa–Holm equation. We first derive the stochastic modified Camassa–Holm equation by the stochastic variational principle. Then, we prove the well-posedness of the stochastic modified Camassa–Holm equation by the iterative process. Under the condition of the small noise intensity, we can also get the regularization of the solution from the parabolic term.

This paper deals with a model the equatorial water waves with the Coriolis effect in the rotating fluid, which is called rotation-two-component Camassa–Holm system. The purpose of this work is to utilize the pseudo-parabolic regularization to establish the existence and uniqueness of weak solutions in a lower order Sobolev space Hs(R)×Hs−1(R) with 1

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.

In this paper, the local existence and blow-up for an initial boundary value problem of the Camassa–Holm equation are obtained.

In this paper the ∂-problem has been constructed for a class of multidimensional integrable partial differential equations (PDE) which can be classified as the multidimensional hierarchies of the (1+1)-dimensional systems of integrable PDE. We introduce the nonsymmetric dressing procedure for this purpose. Among the examples we consider the (n+1)-dimensional (n>1) hierarchies of nonlinear Schrodinger, modified Korteveg–de Vries, and Camassa–Holm equations.

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions. Furthermore, these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem. The key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro-differential operator which generates the integrable equation, and the linearization operator of the lowest-order integrable equation in the same hierarchy. We also obtain similar results for the adjoint linearization operator of an integrable equation. Even though our analysis is conducted only for the KdV, NLS and mKdV hierarchies, similar results are expected for other integrable hierarchies as well. We further explicitly present the complete eigenfunctions for the KdV, NLS and mKdV hierarchy equations and give their inner products, thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations.

In this article we are devoted to the construction of integrable and superintegrable two-dimensional Hamiltonian systems with scalar and vector potentials. All integrable systems with a quadratic polar coordinate type integral of motion are found. Classical trajectories are calculated in integrable cases and compared with those for a system that is not integrable.

We present an investigation of the nonlinear partial differential equations (PDE) which are asymptotically representable as a linear combination of the equations from the Camassa–Holm hierarchy. For this purpose we use the infinitesimal transformations of dependent and independent variables of the original PDE. This approach is helpful for the analysis of the systems of the PDE which can be asymptotically represented as the evolution equations of polynomial structure.

After embedding the Kadomtsev–Petviashvili equation in higher dimensions and extending the Painleve analysis approach to a new form such that the coefficients of the expansion around the singular manifold possess conformal invariance and contain explicit new space variables, we can get infinitely many Painleve integrable models in (3+1)-dimensions and higher dimensions. Some concrete higher dimensional modified Korteweg–de Vries type of extensions are given. Whether the models are Lax integrable or integrable under other meanings remain still open.

The nonlinearization approach is extended to the Hirota–Satsuma coupled Kortweg–de Vries (KdV) equation associated with a 4×4 matrix spectral problem, from which two new finite-dimensional integrable Hamiltonian systems are obtained in the Liouville sense. It is shown that the solutions of this coupled KdV equation are reduced to solving a compatible system of ordinary differential equations. The reductions of the two Hamiltonian systems and their separability are discussed. An interesting connection between the two reduced Hamiltonian systems in the case of one-parameter and known two-dimensional integrable systems is revealed. As application, some explicit solutions of the Hirota–Satsuma coupled KdV equation are derived.

A higher‐order nonlocal evolution equation describing internal waves in a deep fluid is shown to be asymptotically integrable only if the coefficients of the higher‐order terms satisfy certain constraints. In this case, the nonlocal equation can be transformed to the integrable Benjamin–Ono equation. The asymptotic integrability of the reductions of the higher‐order evolution equation to a complex Burgers equation, to an envelope‐wave equation, and to a finite‐dimensional dynamical system is also considered.

The Hamiltonian theory for the two‐component AKNS hierarchy and Yajima–Oikawa hierarchy is considered from the viewpoint of reduction. We show that the second Hamiltonian structures of the former is a Dirac reduction of the sl(3) current algebra, while the latter is related to the classical W(3)4 algebra.

In this paper we give a simple formula to obtain hereditary symmetries related to the isospectral eigenvalue problem of integrable equations. Using this formula, (i) we prove that eigenfunctions of hereditary strong symmetry are symmetries for the whole hierarchy, which improves the result of Fokas and Anderson [J. Math. Phys. 23, 1066 (1982)]; (ii) we find new integrable equations; and (iii) we give strong symmetries of these new integrable equations and various equations for eigenfunctions of the isospectral eigenvalue problems.

The first Hamiltonian structure is constructed for the Korteweg–de Vries (KdV) hierarchy with sources and for the respective modified systems. Applying the appropriate Miura map the second Hamiltonian structure for the KdV hierarchy with sources is derived. Stationary projection of considered system gives so‐called restricted flows of the KdV hierarchy as well as its bi‐Hamiltonian structure.

In this article the existence of generalized solutions to the modified Korteweg–de Vries equation ut−6σu2ux+uxxx=0 is studied. The solutions are found in certain algebras of new generalized functions containing spaces of distributions.

In this paper, the inverse scattering transform associated with a Riemann-Hilbert problem is formulated for the FQXL model: a generalized Camassa-Holm equation mt=12k1[m(u2−ux2)]x+12k2(2mux+mxu),m=u−uxx, which was originally included in the work of Fokas [Physica D 87, 145 (1995)] and was recently shown to be integrable in the sense of Lax pair, bi-Hamilton structure, and conservation laws by Qiao, Xia, and Li [e-print arXiv:1205.2028v2 (2012)]. We have discussed the following properties: direct scattering problems and Jost solutions, asymptotical and analytical behavior of Jost solutions, the scattering equations in a Riemann-Hilbert problem, and the multi-soliton solutions of the FQXL model. Then, one-soliton and two-soliton solutions are presented in a parametric form as a special case of multi-soliton solutions.

A modified Korteweg–de Vries (mKdV)‐type bilinear equation that has been shown to pass the three‐soliton solution and the Painleve tests [see, J. Math. Phys. 2094, 28 (1987); 2572, 31 (1990)] is considered herein. In this article, a Backlund transformation and a nonlinear superposition formula are proven rigorously. As an application of the obtained results, N‐soliton solutions are obtained. Therefore, the integrability of this mKdV‐type bilinear equation is confirmed.

In this paper all the potential constraints (with or without first‐order partial derivatives) of the KP system, from which the associated linear problem can be restricted into a (1+1)‐dimensional Hamiltonian equation, are obtained by using the sufficient and necessary condition for a nonlinear equation to be a Hamiltonian system. Some well‐known integrable systems, such as (1+1)‐dimensional AKNS system, generalized NS system, and several new Hamiltonian equations, are deduced as particular examples.

In this article, a system of finite‐dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg–de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: vt+1/4vxxx−3/2v2vx=0 and the 5th mKdV equation vt−1/16vxxxxx+5/8v2vxxx +5/2vvxvxx +5/8v3x−3/40v4vx=0.

A detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q22. Their connection with fifth‐order one‐component soliton equations is discussed. Lax representations are constructed, and the bi‐Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

A multicomponent KdV system, Kupershmidt’s coupled KdV equation, is considered. The system is shown to be a modification of some system, and it can be modified.

In this paper, persistence properties of solutions are investigated for a generalized Camassa-Holm equation (g-k bCH) having (k+1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations. The persistence properties will imply that strong solutions of the g-k bCH equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k. Finally, existence of global solutions is established when b = k+1.

The present paper is devoted to the study of persistence properties, infinite propagation, and the traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity. We first show that persistence properties of the solution to the equation provided the initial datum is exponential decay and the initial potential satisfies a certain sign condition. Next, we get the infinite propagation if the initial datum satisfies certain compact conditions, while the solution to Eq. (1.1) instantly loses compactly supported, the solution has exponential decay as |x| goes to infinity. Finally, we prove Eq. (1.1) has a family traveling wave solutions.

In this paper we study the large time behavior for the support of momentum density of the Camassa-Holm equation. More precisely, we deduce the limit of the support of momentum density as t goes to +∞ in some sense.

Starting from an existed Lie algebra introduces a new Lie algebra A1 = {e1, e2, e3} so that two isospectral Lax matrices are established. By employing the Tu scheme an integrable equation hierarchy denoted by IEH is obtained from which a few reduced evolution equations are presented. One of them is the mKdV equation. The elliptic variable solutions and three kinds of Darboux transformations for one coupled equation which is from the IEH are worked out, respectively. Finally, we take use of the Lie algebra A1 to generate eight higher-dimensional Lie algebras from which the linear integrable couplings, the nonlinear integrable couplings, and the bi-integrable couplings of the IEH are engendered, whose Hamiltonian structures are also obtained by the variational identity. Then further reduce one coupled integrable equation to get a nonlinear generalized mKdV equation.

In this paper, using the Morales-Ramis theory, we will give some new non-integrability results for a class of famous Hamiltonian models.

In this paper, from a given Kuper-CH spectral problem, we propose two kinds of super integrable hierarchies. One is the Kuper-CH hierarchy, the other is the generalized Kuper-Harry-Dym hierarchy. Moreover, we construct their zero curvature representations and super-bi-Hamiltonian structures.

By introducing a new set of independent variables, we first transform a modified two-component Camassa–Holm shallow water system into a semilinear system. To obtain a dissipative solution, we modify the corresponding system into a discontinuous system. Then we map the solution of system to the dissipative solution of original equation. Moreover we prove that these global dissipative solutions construct a semigroup.

In this paper, we consider several problems on the Camassa–Holm equation with a weak dissipation term. First, we establish the local well-posedness result and then we present several new blow-up results and discuss the global existence of the solution for the periodic case. Finally, we examine the persistence of decay properties for the weakly dissipative Camassa–Holm equation, and we prove that any nontrivial classical solution of the equation will not have compact support if its initial data have this property.

This paper studies the existence of global solutions to higher-order Camassa–Holm equations. Global solution is constructed by the small viscosity method for the frequency localized equation, especially global solution is energy conservative for given finite band initial data.

In this paper, we prove that the modified Camassa–Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.

We introduce three kinds of column-vector Lie algebras Ls(s=1,2,3). By making invertible linear transformations we get the corresponding three induced Lie algebras. According to the defined loop algebras Ls of the Lie algebras Ls(s=1,2,3), we establish three various isospectral problems. Then by applying Tu scheme, we obtain three different coupling integrable couplings of the Korteweg–de Vries (KdV) hierarchy and further reduce them to three kinds of explicit coupling integrable couplings of the KdV equation. One of the coupling integrable couplings of the KdV hierarchy of evolution equations possesses Hamiltonian structure obtained by using the quadratic-form identity and it is Liouville integrable.

We investigate the integrable (2+1)-dimensional Manakov–Santini equation from the Lax–Sato form. Several particular two- and three-component reductions are considered so that the Manakov–Santini equation can be reduced to systems of hydrodynamic type. Then one can construct infinitely many exact solutions of the equation by the hodograph method.

The Dullin–Gottwald–Holm equation can be described as the unidirectional propagation of surface waves in a shallow water regime. In this paper, we study the orbital stability problem of the peaked solitons (peakons) for the periodic Dullin–Gottwald–Holm equation on the line. By constructing a function only depending on three important conservative laws, we prove that the shapes of peakons are stable under small perturbations. We also demonstrate that the solutions of the Cauchy problem for this equation converge to those of the corresponding periodic Camassa–Holm equation as the linear dispersive parameter converges to zero.

We consider an integrable generalization of the sine-Gordon (sG) equation that was earlier derived by one of the authors using bi-Hamiltonian methods. This equation is related to the sG equation in the same way that the Camassa–Holm equation is related to the Korteweg–de Vries equation. In this paper we (a) derive a Lax pair, (b) use the Lax pair to solve the initial-value problem on the line, (c) analyze solitons, (d) show that the generalized sG and sG equations are related by a Liouville transformation, (e) derive conservation laws, and (f) analyze traveling-wave solutions.

A method of descent for constructing integrable Hamiltonian systems is introduced. The derived periodic and nonperiodic lattices possess Lax representations with spectral parameter and have plenty of first integrals. Examples of Liouville-integrable four-dimensional Hamiltonian Lotka–Volterra systems are presented.

A new Camassa–Holm system with two-component admitting peakon solitons is proposed. The local well posedness for the system is established. A criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained, and an existence result for a class of local weak solution is also given.

In this paper, we consider the weakly dissipative Camassa–Holm equation. First, we try to improve other authors’ results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa–Holm equation.

A procedure of nonlinearization of spectral problem that allows to impose reality conditions or restriction conditions on potentials is presented. As applications, integrable decompositions of the nonlinear Schrodinger equation and the real-valued modified Korteweg–de Vries equation are obtained.

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.

In this paper, we mainly study several problems on the weakly dissipative periodic Camassa-Holm equation. At first, the local well-posedness of the equation is obtained by Kato’s theorem, a necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are established. Then, the blow-up rate of the solution is discussed. Moreover, we prove that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign.

We construct the local and nonlocal conserved densities for the Camassa-Holm equation by solving a suitable Riccati equation. We also define a Kadomtsev-Petviashvili extension for the local Camassa-Holm hierarchy. (C) 2005 American Institute of Physics

We prove that any classical solution of the Camassa–Holm equation will have compact support if its initial data has this property.

Seeking new integrable couplings of the known integrable hierarchies of evolution equations is a quite new interesting aspect and is of great significance in soliton theory. In this paper, a loop algebra P and its basis are first presented. Second, a new isospectral problem with four potentials in the loop algebra P is considered to construct integrable couplings of the well-known TC hierarchy by the zero-curvature equation such that a family of new integrable couplings including two arbitrary functions are obtained. In particular, when we set two arbitrary functions to be zero, a special integrable couplings of the generalized Korteweg–de Vries equation is also given.

A new approach to Hamiltonian structures of integrable systems is proposed by making use of a trace identity. For a variety of isospectral problems that can be unified to one model ψx=Uψ, it is shown that both the related hierarchy of evolution equations and the Hamiltonian structure can be obtained from the same solution of the equation Vx=[U,V].

A new proof of a theorem of Williamson on the complete integrability of time‐independent, real, linear Hamiltonian differential equations with quadratic integrals is given. The sets where these integrals are functionally dependent are explicitly found.

For evolution equations which can be written in Hamiltonian form two ways, there exists a relation between two functions Q(1) and Q(2), both of which are gradients of conserved functionals. The relation can be extended to define (recursively) functions Q(n). It is shown that the Q(n) corresponding to the general evolution equation associated with the Zakharov–Shabat eigenvalue problem are all gradients of conserved functionals. This in turn implies all these functionals are in involution.

We present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation (PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak‐Painleve property’’).