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In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painleve method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.

This paper extends the results of a previous work [Parker A. Cusped solitons of the Camassa–Holm equation. I. Cuspon solitary wave and antipeakon limit. Chaos, Solitons & Fractals 2007;34:730–9]—designated I—in which the solitary cuspon solution of the Camassa–Holm equation and its antipeakon limit were considered. Here, explicit binary cuspon–soliton solutions are obtained in parametric form by exploiting the factorisation procedure that was used in I. The structure and dynamics of these two-wave interactions are investigated and some unanswered questions concerning the waveforms are addressed. In particular, it is shown that, while a cuspon may be ‘swallowed-up’ by a soliton, the converse is never possible regardless of the depth of the cuspon trough. Formulae for the characteristic post-collision phase shifts are obtained and analysed in detail; this permits a reassessment of the previously reported results. Examples of twin-cusped and mixed cuspon–soliton solutions are presented within the different parameter regimes. Coincidentally, we find new two-soliton solutions of the related associated -Camassa–Holm equation. These describe binary interactions of classically smooth elevated solitons and solitary-wave troughs.

We perform a multiple scales analysis on the nonlinear Schrodinger (NLS) equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg–de Vries (KdV) flow equations in the bi-Hamiltonian form with the corresponding Hamiltonian functions.

The equation hierarchy presented in this paper contains the KdV equation and the mKdV equation. By use of the concept of characteristic number, an undetermined-constant method is proposed by us, for which the polynomial Hamiltonian functions are constructed. By employing the method, the Hamiltonian structure of the equation hierarchy is established. The approach presented in the paper shares extensive applications. In addition, four explicit expressions of the travelling wave solutions to the above equation hierarchy are obtained. One of them is regular, the other three are singular.

Solitary wave solutions to the modified form of Camassa–Holm (CH) equation are sought. In this work, the homotopy analysis method (HAM), one of the most effective method, is applied to obtain the soliton wave solutions with and without continuity of first derivatives at crest.

By making use of an isospectral transformation, an isospectral matrix with non-zero trace is turned into a zero-trace one. With the help of the simple loop algebra A 1 and the Tu scheme, a soliton-equation hierarchy is generated under the framework of the zero curvature equation. By employing the trace identity, its Hamiltonian structure is obtained. Then, we enlarge the Lie algebra A 1 into three kinds of Lie algebras, which are devote to investigating integrable couplings. The corresponding three types of integrable couplings are worked out, one of them has Hamiltonian structure which is obtained by the quadratic-form identify. The approach for generating integrable couplings has extensive applications.

The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions.

A detailed investigation of the wave dynamics for multiply peaked solitons of the Camassa–Holm equation is presented. The analysis proceeds in terms of the underlying component “peakons” using entirely elementary methods. The two-wave interactions exhibit intricate and subtle features such as role reversal, soliton absorption and annihilation, wave steepening and monodirectional propagation (for finite time) and a critical (amplitude) ratio. The discussion covers the entirety of these waveforms comprising two-peakon, peakon–antipeakon and two-antipeakon solutions. Their properties transfer to multipeakon dynamics and examples of three-wave interactions are given.

A factorisaton method is used to obtain the cusped soliton of the Camassa–Holm equation in parametric form. It is shown how this piecewise analytic solution arises from an associated smooth solitary wave. The PQ-decomposition of the explicit solution is then used to determine the dispersionless limit ( κ → 0 ) in which the cuspon converges to an antipeakon. The special cuspon solution reported by Kraenkel and Zenchuk [Kraenkel RA, Zenchuk A. Camassa–Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions. J Phys A: Math Gen 1999;32:4733–47] is recovered and examined in the context of the parametric representation. The cusped solitary wave of a short-wave version of the Camassa–Holm model is also deduced from the cuspon in an appropriate limit.

A type of higher-dimensional loop algebra G ∼ is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity. This method presented in the paper also suits for other soliton equations.

The Hamiltonian structure of the integrable coupling of the AKNS hierarchy is obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings.

By using the bifurcation theory of planar dynamical systems to a generalized Camassa–Holm equation m t + c 0 u x + um x + 2 mu x = - γ u xxx with m = u − α2uxx, α ≠ 0, c0, γ are constant, which is called CH-r equation, the existence of peakons and periodic cusp wave solutions is obtained. The analytic expressions of the peakons and periodic cusp wave solutions are given and numerical simulation results show the consistence with the theoretical analysis at the same time.

We investigate the Painleve analysis for a (2+1) dimensional Camassa-Holm equation. Our results show that it admits only weak Painleve expansions. This then confirms the limitations of the Painleve test as a test for complete integrability when applied to non-semilinear partial differential equations., Comment: Chaos, Solitons and Fractals (Accepted for publication)

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.

Explicit travelling-wave solutions of the Camassa–Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis–Procesi equation are given in an appendix.

The dynamical behavior of travelling wave solutions in the Generalized Camassa–Holm equation u t + 2 ku x − u xxt + au m u x = 2 u x u xx + uu xxx is analyzed by using the bifurcation theory and the method of phase portraits analysis. The condition under which compactons and cusp waves appear are also given. In addition, the reason for solitary cusp wave and periodic cusp wave to exist is highlighted.

An alternate approach is proposed for obtaining periodic wave and peaked solitary wave solutions of the following nonlinear generalized Camassa–Holm equation u t + 2 ku x - u xxt + au m u x = 2 u x u xx + uu xxx For m = 1, 2, 3 we give the explicit expressions for the peakons. The ansatz, introduced in this paper, will also demonstrate the existence of a new class of discontinuous soliton solutions with infinite spikes.

We explore numerically different aspects of periodic traveling-wave solutions of the Camassa–Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied.

In this paper, we consider generalized Camassa–Holm equations and the generalized weakly dissipative Camassa–Holm equations and derive some new exact peaked solitary wave solutions. For m=3, where m is representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa–Holm equations.

Designing a new isospectral problem from a loop algebra A 1 . It follows that a type of new integrable hierarchy of evolution equations, possessing bi-Hamiltonian structure, is obtained. As a reduction case, the well-known TC hierarchy is presented. Then a new seven-dimensional loop algebra G is constructed. From which, expanding integrable model of the TC hierarchy is obtained.

The analytic expressions of peaked solitary wave solutions and peaked periodic wave solutions of Camassa–Holm equation are obtained by using bifurcation method of planar dynamical systems. The convergence of the peaked periodic wave solutions is proved. Numerical simulation results show the consistence with the theoretical analysis. The results in this paper are wider than those already known.

A new subalgebra of loop algebra A 2 is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. As its reduction cases, the well-known standard Schrodinger equation and MKdV equation are presented, respectively. Furthermore, by making use of bi-symmetry constraints, generalized Hamiltonian regular representations for the hierarchy are obtained. At last, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra G. In particular, the expanding integrable models of Schrodinger equation and MKdV equation are presented, respectively.

In this paper we consider an extended Kaup–Newell (EKN) isospectral problem with an arbitrary smooth function and the corresponding two kinds of Lax integrable hierarchies by introducing two types of auxiliary spectral problems. The Hamiltonian structure of the second hierarchy is established. It is shown that the Hamiltonian system are integrable in Liouville’s sense and the set of Hamiltonian functions is the conserved densities of the second hierarchy, as well as they are in involutive in pairs under the Poisson bracket.

Two central and significative but difficult subjects in soliton theory and nonlinear integrable dynamic systems are to seek new Lax integrable hierarchies and their Hamiltonian structure (bi-Hamiltonian structure). In this paper, an new isospectral problem with an arbitrary function and the associated Lax integrable hierarchy of evolution equations are presented by using zero curvature equation. As a result, a representative system of the generalized derivative nonlinear Schrodinger equations with an arbitrary function in the hierarchy is obtained. Bi-Hamiltonian structure is established for the whole hierarchy based upon a pair of Hamiltonian operators and it is shown that the hierarchy is Liouville integrable. In addition, infinitely many commuting symmetries of the hierarchy are given.

We study the peakons and the periodic cusp wave solutions of the following equation: u t +2ku x −u xxt +auu x =2u x u xx +uu xxx with a,k∈ R , which we will call the generalized Camassa–Holm equation, or simply the GCH equation, for when a =3 it was the so-called CH equation given by R. Camassa and D.D. Holm [Phys. Rev. Lett. 71 (11) (1993) 1661–1664]. They showed that the CH equation has a class of new solitary wave solutions called “peakons”. J.P. Boyd [Appl. Math. Comput. 81 (2–3) (1997) 173–187] studied another class of new periodic wave solutions called “coshoidal waves”. Using the bifurcation method of the phase plane, we first construct peakons and show that a =3 is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter a or the wave speed c varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.

A classical r -matrix in conjunction with a zero-curvature equation is used to generate a new nonlinear integrable system known as a four-wave interaction equation. On reduction it yields a new integrable Klein-Gordon system. These equations turn out to be bi-Hamiltonian. Finally, it is indicated how the modified equations can be obtained by the zero-curvature equation. Our system is very similar to those studied by Tsuchida and Wadati.