206 research outputs found

### An integrable (2+1)-dimensional Camassa-Holm hierarchy with peakon solutions

In this letter, we propose a (2+1)-dimensional generalized Camassa-Holm
(2dgCH) hierarchy with both quadratic and cubic nonlinearity. The Lax
representation and peakon solutions for the 2dgCH system are derived

### Persistence Properties and Unique Continuation for a Dispersionless Two-Component Camassa-Holm System with Peakon and Weak Kink Solutions

In this paper, we study the persistence properties and unique continuation
for a dispersionless two-component system with peakon and weak kink solutions.
These properties guarantee strong solutions of the two-component system decay
at infinity in the spatial variable provided that the initial data satisfies
the condition of decaying at infinity. Furthermore, we give an optimal decaying
index of the momentum for the system and show that the system exhibits unique
continuation if the initial momentum $m_0$ and $n_0$ are non-negative

### On Negative Order KdV Equations

In this paper, based on the regular KdV system, we study negative order KdV
(NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many
conservation laws, and explicit multi-soliton and multi-kink wave solutions
thorough bilinear B\"{a}cklund transformations. The NKdV equations studied in
our paper are differential and actually derived from the first member in the
negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent
to the Camassa-Holm equation through some hodograph transformations, but also
closely related to the Ermakov-Pinney systems, and the Kupershmidt deformation.
The bi-Hamiltonian structures and a Darboux transformation of the NKdV
equations are constructed with the aid of trace identity and their Lax pairs,
respectively. The single and double kink wave and bell soliton solutions are
given in an explicit formula through the Darboux transformation. The 1-kink
wave solution is expressed in the form of $tanh$ while the 1-bell soliton is in
the form of $sech$, and both forms are very standard. The collisions of
2-kink-wave and 2-bell-soliton solutions, are analyzed in details, and this
singular interaction is a big difference from the regular KdV equation.
Multi-dimensional binary Bell polynomials are employed to find bilinear
formulation and B\"{a}cklund transformations, which produce $N$-soliton
solutions. A direct and unifying scheme is proposed for explicitly building up
quasi-periodic wave solutions of the NKdV equations.
Furthermore, the relations between quasi-periodic wave solutions and soliton
solutions are clearly described. Finally, we show the quasi-periodic wave
solution convergent to the soliton solution under some limit conditions.Comment: 61 pages, 4 figure

### A new two-component integrable system with peakon solutions

A new two-component system with cubic nonlinearity and linear dispersion:
\begin{eqnarray*} \left\{\begin{array}{l}
m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\
n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2} n(uv_x-u_xv),
\\m=u-u_{xx},~~ n=v-v_{xx}, \end{array}\right. \end{eqnarray*} where $b$ is an
arbitrary real constant, is proposed in this paper. This system is shown
integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many
conservation laws. Geometrically, this system describes a nontrivial
one-parameter family of pseudo-spherical surfaces. In the case $b=0$, the
peaked soliton (peakon) and multi-peakon solutions to this two-component system
are derived. In particular, the two-peakon dynamical system is explicitly
solved and their interactions are investigated in details. Moreover, a new
integrable cubic nonlinear equation with linear dispersion \begin{eqnarray*}
m_t=bu_{x}+\frac{1}{2}[m(|u|^2-|u_x|^2)]_x-\frac{1}{2}m(uu^\ast_x-u_xu^\ast),
\quad m=u-u_{xx}, \end{eqnarray*} is obtained by imposing the complex conjugate
reduction $v=u^\ast$ to the two-component system. The complex valued $N$-peakon
solution and kink wave solution to this complex equation are also derived

### Alice-Bob Peakon Systems

In this letter, we study the Alice-Bob peakon system generated from an
integrable peakon system through using the strategy of the so-called Alice-Bob
non-local KdV approach [13]. Non-local integrable peakon equations are obtained
and shown to have peakon solutions.Comment: 3 figure

### Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function

In this paper, we study the Cauchy problem for an integrable multi-component
(2N-component) peakon system which is involved in an arbitrary polynomial
function. Based on a generalized Ovsyannikov type theorem, we first prove the
existence and uniqueness of solutions for the system in the Gevrey-Sobolev
spaces with the lower bound of the lifespan. Then we show the continuity of the
data-to-solution map for the system. Furthermore, by introducing a family of
continuous diffeomorphisms of a line and utilizing the fine structure of the
system, we demonstrate the system exhibits unique continuation

### Multi-component generalization of Camassa-Holm equation

In this paper, we propose a multi-component system of Camassa-Holm equation,
denoted by CH($N$,$H$) with 2N components and an arbitrary smooth function $H$.
This system is shown to admit Lax pair and infinitely many conservation laws.
We particularly study the case of N=2 and derive the bi-Hamiltonian structures
and peaked soliton (peakon) solutions for some examples

### Global existence and propagation speed for a generalized Camassa-Holm model with both dissipation and dispersion

In this paper, we study a generalized Camassa-Holm (gCH) model with both
dissipation and dispersion, which has (N + 1)-order nonlinearities and includes
the following three integrable equations: the Camassa-Holm, the
Degasperis-Procesi, and the Novikov equations, as its reductions. We first
present the local well-posedness and a precise blow-up scenario of the Cauchy
problem for the gCH equation. Then we provide several sufficient conditions
that guarantee the global existence of the strong solutions to the gCH
equation. Finally, we investigate the propagation speed for the gCH equation
when the initial data is compactly supported

### The Camassa-Holm hierarchy, related N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold

This paper shows that the Camassa-Holm (CH) spectral problem yields two
different integrable hierarchies of nonlinear evolution equations (NLEEs), one
is of negative order CH hierachy while the other one is of positive order CH
hierarchy. The two CH hierarchies possess the zero curvature representations
through solving a key matrix equation. We find that the well-known CH equation
is included in the negative order CH hierarchy while a Dym type equation is
included in the positive order CH hierarchy. Furthermore, under two constraint
conditions between the eigenfunctions and the potentials, the CH spectral
problem is cast in: (enumerate) a new Neumann-like N-dimensional system when it
is restricted into a symplectic submanifold of $\R^{2N}$ which is proven to be
integrable by using the Dirac-Poisson bracket and the r-matrix process; and a
new Bargmann-like N-dimensional system when it is considered in the whole
$\R^{2N}$ which is proven to be integrable by using the standard Poisson
bracket and the r-matrix process. (enumerate) In the paper, we present two
$4\times4$ instead of $N\times N$ r-matrix structures. One is for the
Neumann-like CH system (not the peaked CH system), while the other one is for
the Bargmann-like CH system. The whole CH hierarchy (both positive and negative
order) is shown to have the parametric solution which obey the constraint
relation. In particular, the CH equation constrained to some symplectic
submanifold, and the Dym type equation have the parametric solutions. Moreover,
we see that the kind of parametric solution of the CH equation is not gauge
equivalent to the peakons. Solving the parametric representation of solution on
the symplectic submanifold gives a class of new algebro-geometric solution of
the CH equation.Comment: 44 pages, 0 figure

### Integrable hierarchy, $3\times 3$ constrained systems, and parametric and peaked stationary solutions

This paper gives a new integrable hierarchy of nonlinear evolution equations.
The DP equation: $m_t+um_x+3mu_x=0, m=u-u_{xx}$, proposed recently by
Desgaperis and Procesi \cite{DP[1999]}, is the first one in the negative
hierarchy while the first one in the positive hierarchy is:\
$m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. The whole
hierarchy is shown Lax-integrable through solving a key matrix equation. To
obtain the parametric solutions for the whole hierarchy, we separatedly discuss
the negative and the positive hierarchies. For the negative hierarchy, its
$3\times3$ Lax pairs and corresponding adjoint representations are
nonlinearized to be Liouville-integrable Hamiltonian canonical systems under
the so-called Dirac-Poisson bracket defined on a symplectic submanifold of
$\R^{6N}$. Based on the integrability of those finite-dimensional canonical
Hamiltonian systems we give the parametric solutions of the all equations in
the negative hierarchy. In particular, we obtain the parametric solution of the
DP equation. Moreover, for the positive hierarchy, we consider the different
constraint and use a similar procudure to the negative case to obtain the
parametric solutions of the positive hierarchy. In particular, we give the
parametric solution of the 5th-order PDE
$m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. Finally, we
discuss the stationary solutions of the 5th-order PDE, and particularly give
its four peaked stationary solutions. The stationary solutions may be included
in the parametric solution, but the peaked stationary solutions not. The
5th-order PDE does not have the cusp soliton although it looks like a higher
order Harry-Dym equation.Comment: 23 pages, 1 figur

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