On Peakon and Kink-peakon Solutions to a (2 + 1) Dimensional Generalized Camassa-Holm Equation

In this paper, we study a (2 + 1)-dimensional generalized Camassa-Holm (2dgCH) equation with both quadratic and cubic nonlinearity. We derive a peaked soliton (peakon) solution, double-peakon solutions, and kink-peakon solutions. In particular, weak kink - peakon solution is the first time to address in the 2 + 1-dimensional integrable system.     

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.     

Dynamics of conservative peakons in a system of Popowicz

We consider a two-component Hamiltonian system of partial differential equations with quadratic nonlinearities introduced by Popowicz, which has the form of a coupling between the Camassa–Holm and Degasperis–Procesi equations. Despite having reductions to these two integrable partial differential equations, the Popowicz system itself is not integrable. Nevertheless, as one of the authors showed with Irle, it admits distributional solutions of peaked soliton (peakon) type, with the dynamics of N peakons being determined by a Hamiltonian system on a phase space of dimension 3N. As well as the trivial case of a single peakon ($N=1$), the case $N=2$ is Liouville integrable. We present the explicit solution for the two-peakon dynamics, and describe some of the novel features of the interaction of peakons in the Popowicz system.     

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.     

Partial differential equations possessing Frobenius integrable decompositions

Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.     

Polaron on a one-dimensional lattice: II. A moving polaron

Continuum-limit equations for moving polarons on a one-dimensional lattice with a harmonic interaction potential between adjacent particles and a simple nonlinear potential with a cubic nonlinearity are derived for the first time; for some particular cases, their solutions are obtained. For a harmonic lattice in the continuum limit, a system of integrable nonlinear partial differential equations is derived. A one-soliton solution to this system describes a polaron moving with a constant velocity. The speed of this polaron is uniquely related to its amplitude, with its values ranging from zero to the speed of sound. For a nonlinear lattice, the resulting system of differential equations is integrable at a certain ratio of the problem parameters. The one-soliton solution to this system, as in the harmonic case, describes a polaron moving with a constant velocity. At arbitrary values of the lattice parameters, the nonlinear lattice was studied by numerical methods. It turned out that, in the entire range of parameters, the nonlinear lattice gives rise to moving polarons, with the speed of the polaron being determined by the competition between the electron-photon interaction parameter α and the nonlinearity parameter β. At α ? β, the behavior of the polaron is very close to the dynamics on the harmonic lattice. In the opposite case, the dynamic nonlinearity begins to dominate, giving rise to dynamics inherent to solitons, so that speed of the polaron can exceed the speed of sound. In a certain range of α and β, numerical calculations revealed a family of polaron-type stable solutions, the envelope of which can have several peaks. The numerical and exact analytical solutions are in very good agreement for a sufficiently large radius of the polaron, when the system of equations obtained in the continuum approximation has a solution.     

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.     

An exact solution for polaron in the nonlinear lattice in the Su–Schrieffer–Heeger approximation

The exact solution for a polaron in a lattice with cubic nonlinearity is obtained. The electron–phonon interaction is taken into account in the Su–Schrieffer–Heeger approximation. The system of nonlinear differential equations in partial derivatives, obtained in the continuum approximation, is exactly integrable at a certain ratio between the parameters of nonlinearity, α, and the electron–phonon interaction, χ. An approximate solution is obtained for arbitrary values of α and χ. A good agreement between the analytical results with the numerical simulation is observed at not too large values of parameters α and χ, where the continuum approximation is valid. Stable solutions also exist at higher values of these parameters.     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

Generation and interaction of large-amplitude solitons

The results of a numerical simulation of the interaction and generation of solitons in nonlinear integrable systems which admit the existence of very-large-amplitude solitons are reported. The nonlinear integrable system chosen for study is the Gardner equation, particular examples of which are the Korteweg-de Vries equation (for quadratic nonlinearity) and a modified Korteweg-de Vries equation (for cubic nonlinearity).It is shown that during the evolution process solitons of opposite polarity appear on the crest of the maximum soliton. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 9, 628–633 (10 May 1998)     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.     

Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media

We propose an integrable system of coupled nonlinear Schr?dinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultrashort optical soliton pulse propagation in non-Kerr media. Lax pairs, conserved quantities and exact soliton solutions for the proposed integrable model are given. The explicit form of two solitons are used to study soliton interaction showing many intriguing features including inelastic (shape changing or intensity redistribution) scattering. Another system of coupled equations with fifth-degree nonlinearity is derived, which represents vector generalization of the known chiral-soliton bearing system.     

Dynamics of large-amplitude solitons

The interaction and generation of solitons in nonlinear integrable systems which allow the existence of a soliton of limiting amplitude are considered. The integrable system considered is the Gardner equation, which includes the Korteweg-de Vries equation (for quadratic nonlinearity) and the modified Korteweg-de Vries equation (for cubic nonlinearity) as special cases. A two-soliton solution of the Gardner equation is derived, and a criterion, which distinguishes between different scenarios for the interaction of two solitons, is determined. The evolution of an initial pulsed disturbance is considered. It is shown, in particular, that solitons of opposite polarity appear during such evolution on the crest of a limiting soliton. Zh. éksp. Teor. Fiz. 116, 318–335 (July 1999)     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Liouville Integrability of Conservative Peakons for a Modified CH equation

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev H¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the H¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].     

Non-autonomous bright matter wave solitons in spinor Bose–Einstein condensates

We investigate the dynamics of bright matter wave solitons in spin-1 Bose–Einstein condensates with time modulated nonlinearities. We obtain soliton solutions of an integrable autonomous three-coupled Gross–Pitaevskii (3-GP) equations using Hirota?s method involving a non-standard bilinearization. The similarity transformations are developed to construct the soliton solutions of non-autonomous 3-GP system. The non-autonomous solitons admit different density profiles. An interesting phenomenon of soliton compression is identified for kink-like nonlinearity coefficient with Hermite–Gaussian-like potential strength. Our study shows that these non-autonomous solitons undergo non-trivial collisions involving condensate switching.     

Orbital stability of peakons with nonvanishing boundary for CH and CH-γ equations

In this Letter, we consider the general expressions of peaked traveling wave solutions for CH and CH-γ equations. The orbital stability of these peakons are directly proved in the H¹ norm. Some previous results are extended.     

The bifurcation and peakons for the special C (3, 2, 2) equation

In this paper, we investigate a special C(3, 2, 2) equation $$\begin{array}{@{}rcl@{}} u_{t}+ku_{x}-u_{xxt}+3(u^{3})_{x}=u_{x}(u^{2})_{xx}+u(u^{2})_{xxx}. \end{array} $$ The bifurcation and some new exact representations of peakons, bell-shaped solitary wave solutions and periodic cusp wave solutions for the equation are obtained using the qualitative theory of dynamical systems. It is shown that the peakons are actually the limit of bell-shaped solitary waves and periodic cusp waves. Moreover, a new characteristic of non-smooth solutions, two peakons coexisting for the same wave speed, is found. Some previous results are extended.     

An Integrable Matrix Camassa-Holm Equation

We present an integrable sl(2)-matrix Camassa-Holm(CH) equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.     

Dynamics of solitons in periodic systems with different nonlinear media

To analyze pulse dynamics in an optical system consisting of a periodic sequence of nonlinear media, a composite model is used. It includes a model of the resonance interaction of an ultrashort light pulse with the energy transition of the medium with allowance made for an upper level pump and an almost integrable model that describes the propagation of the light field in the other medium with a cubic nonlinearity and dispersion. Additional allowance is made for losses and other kinds of interaction by introducing perturbation terms. On the bases of the inverse scattering transform and perturbation theory, a simple method for analyzing specific features of soliton evolution in periodic systems of this kind is developed. It is used to describe various modes of soliton evolution in such a system, including chaotic dynamics.