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# A generalized nonisospectral Camassa–Holm equation and its multipeakon solutions

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*in*Advances in Mathematics 263:154–177 · October 2014

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Abstract

Motivated by the paper of Beals, Sattinger and Szmigielski (2000) [3], we propose an extension of the Camassa–Holm equation, which also admits the multipeakon solutions. The novel aspect is that our approach is mainly based on classic determinant technique. Furthermore, the proposed equation is shown to possess a nonisospectral Lax pair.

- ... Inspired by the work on the non-isospectral CH equation [5], Chang, Hu and Li [6] recently proposed a new two-component non-isospectral cubic Camassa-Holm (2NSQQ) system via an algebraic method: ...... Moreover, some elaborate estimates are derived to deal with the nonlocal velocity ∂ −1 x ψ by means of the Littlewood-Paley decomposition. The techniques used in this paper may be applied to other nonlocal CH-type models, such as the nonisospectral CH equation in [5]. ...... , where M(t, x) is exactly the slop of the velocity ρ(t, x). It is worth pointing out that the similar nonlocal equation may be derived for the nonisospectral CH equation provided in [5]. With proper sign condition on initial profiles and using the invariant property of solutions (see Lemma 4.1), one can derive the desired inequality. ...
- ... Particularly, F (i,j) k = 0 for k > n follows immediately from the integral representation in Remark 2.5. Such phenomenon has appeared in the Hankel determinants with finite measures for CH peakon problems [2,10,13], and actually the corresponding result some specific ...... has been obtained in [35,48]. In the following, we present a generic result with a proof based on matrix factorizations, which are motivated by those for Hankel determinants [10,13] but much more complicated. ...... □ Clearly, our proof implies that the DP peakon lattice can be seen as an isospectral flow on a manifold cut out by determinant identities. Recall that the CH peakon lattice is a flow on a manifold cut out by determinant identities [10], while the Novikov peakon lattice corresponds to a manifold cut out by Pfaffian identities [12]. The difference arises from the different structures of the corresponding tau functions belonging to different Lie algebra of A,B,C types. ...Article
- Oct 2018
- NONLINEARITY

In this paper, we propose a finite Toda lattice of CKP type (C-Toda) together with a Lax pair. Our motivation is based on the fact that the Camassa–Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows in some sense. As an intriguing analogue to the CH equation, the Degasperis–Procesi (DP) equation also supports the presence of peakon solutions. Noticing that the peakon solution to the DP equation is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equation. The corresponding quartic representation is shown to be a continuum limit of a reduced discrete CKP equation, due to which we call the obtained equation a finite Toda lattice of CKP type. Then, a nonlinear version of the C-Toda lattice together with a Lax pair is derived. As a result, it is shown that the DP peakon lattice and the finite C-Toda lattice form opposite flows under a certain transformation. - ... This is not surprising because the moments as elements are of finite measure. Such phenomenon has appeared in the Hankel determinants with finite measures for CH peakon problems [2,10,13], and actually the corresponding result some specific F ...... has been obtained in [35,48]. In the following, we present a generic result with a proof based on matrix factorizations, which are motivated by those for Hankel determinants [10,13] but much more complicated. Lemma 2.9. ...... Clearly, our proof implies that the DP peakon lattice can be seen as an isospectral flow on a manifold cut out by determinant identities. Recall that the CH peakon lattice is a flow on a manifold cut out by determinant identities [10], while the NV peakon lattice corresponds to a manifold cut out by Pfaffian identities [12]. The difference arises from the different structures of the corresponding tau functions belonging to different Lie algebra of A,B,C types. ...ArticleFull-text available
- Dec 2017

In the literature, it has been revealed that the Camassa-Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows. As an intriguing analogue to the CH equation, the Degasperis-Procesi (DP) equation also supports the presence of peakon solutions. A natural question arises: does there exist a corresponding lattice of Toda type for the DP peakon lattice as the CH peakon and Toda lattices do? In this paper, our aim is to give an answer to this question. Noticing that the tau function of the DP peakon lattice is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equations. By introducing appropriate nonlinear variables, a novel Toda lattice of CKP type together with a Lax pair is obtained. As a result, we give a unified picture for the CH peakon and Toda, Novikov peakon and B-Toda, DP peakon and C-Toda lattices. - ... Recall that Hankel determinant appears in the expressions of multipeakons of the CH equation [4]. If one plugs this solution into the CH peakon ODEs and computes the derivatives of the corresponding determinants, the CH peakon ODEs are nothing but determinant identities, which was claimed in [11]. In other words, the CH peakon ODEs can be regarded as a flow on the manifold cut out by determinant identities. ...The Novikov equation is an integrable analogue of the Camassa-Holm equation,with cubic (rather than quadratic) nonlinear term. They both support some special weak solutions called multipeakon solutions. In this paper, Pfaffian technique is employed to study multipeakons of the Novikov equation. First, we notice that the Novikov peakon ODEs describe a flow on the manifold cut out by Pfaffian identities. Then, a bridge between the Novikov peakon and the finite Toda lattice of BKP type (B-Toda lattice) is established with the help of Pfaffian technique. Finally, generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the Pfaffian is introduced in the study of peakon problem.
- ... Remark: The positivity of H l k for l ≥ 0, 0 ≤ k ≤ K immediately follows from this lemma. Similar formulae have appeared in [2, 7]. Proof. ...Article
- Jun 2016
- Adv Math

A spectral and the inverse spectral problem are studied for the two-component modified Camassa-Holm type for measures associated to interlacing peaks. It is shown that the spectral problem is equivalent to an inhomogenous string problem with Dirichlet/Neumann boundary conditions. The inverse problem is solved by Stieltjes' continued fraction expansion, leading to an explicit construction of peakon solutions. Sufficient conditions for the global existence in $t$ are given. The large time asymptotics reveals that, asymptotically, peakons break into two-peakon bound-states moving with constant speeds. The peakon flow is shown to project to one of the isospectral flows of the finite Kac-van Moerbeke lattice. - Preprint
- Jun 2018

In this paper, we consider the explicit wave-breaking mechanism and its dynamical behavior near this singularity for the generalized b-equation. This generalized b-equation arises from the shallow water theory, which includes the Camassa-Holm equation, the Degasperis-Procesi equation, the Fornberg-Whitham equation, the Korteweg-de Vires equation and the classical b-equation. More precisely, we find that there exists an explicit self-similar blowup solution for the generalized b-equation. Meanwhile, this self-similar blowup solution is asymptotic stability in a parameters domain, but instability in other parameters domain. - Article
- May 2018
- J DIFFER EQUATIONS

Firstly, a formal correspondence is established between the Camassa–Holm (CH) equation and a two-component modified CH (or called SQQ) equation according to the method of moment modification for multipeakon formulae. Secondly, based on the generalized nonisospectral CH equation in Chang et al. (2014) [14] and the interlacing multipeakons of the two-component modified CH equation in Chang et al. (2016) [15], we propose a new generalized two-component modified CH equation with two parameters, which possesses a nonisospectral Lax pair. The proposed equation still admits multipeakon solutions of explicit and closed form. Sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented. Last of all, as a by-product, a generalized nonisospectral modified CH equation is deduced, together with its Lax pair. - ArticleFull-text available
- Feb 2018

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N -order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders -3 and -4 are obtained. Indeed, for N≤-2 , the peakons of N -order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for N -order CH equation when N is odd are much different from the CH peakons; we present the case N=-3 as an example.

- A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.
- Jan 1983
- 52-53

- R A Brualdi
- H Schneider

R.A. Brualdi, H. Schneider, Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Linear Algebra Appl. 52-53 (1983) 769–791.- Article
- Jan 1999

The multiscale expansion is shown to be a convenient tool to define asymptotic integrability up to order N of 1+1 dispersive nonlinear wave equations. Its connection with complete integrability, an algorithmic test and few examples are discussed. Approximate Lax pairs for asymptotically integrable PDEs are also provided. - We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a 'discrete cubic string' type -- a nonselfadjoint generalization of a classical inhomogeneous string -- but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher-Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein's solution of the inverse problem for the Stieltjes string.
- ArticleFull-text available
- Aug 2009

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. - 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.
- Article
- Dec 1991
- SIAM J APPL MATH

An asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed. The equation is used to study a nonlinear instability in the director field of a nematic liquid crystal. It is shown that smooth solutions of the asymptotic equation break down in finite time. Also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up. - The non-isospectral problem (Lax pair) associated with a hierarchy in 2+1 dimensions that generalizes the well known Camassa-Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose $1+1$ Lax pair is also non-isospectral.
- The nonlinear partial differential equation was proposed by Hunter and Saxton as an asymptotic model equation for nematic liquid crystals. Hunter and Zheng showed that it is a member of the Harry Dym hierarchy of integrable flows, and solved the equation explicitly for a family of finite dimensional, piecewise linear functions in the case when ux has compact support. In this note, the associated inverse scattering problem is used to obtain the explicit solutions of the finite dimensional flows in both the compact and non-compact case.
- Article
- Aug 2002
- J NONLINEAR SCI

We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances. - Article
- Jan 2009
- NONLINEARITY

A generalization of integrable peakon equations with cubic nonlinearity and the Degasperis-Procesi equation with peakon solutions is proposed, which is associated with a 3×3 matrix spectral problem with two potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The generalization is exactly a negative flow in the hierarchy and admits exact solutions with N-peakons and an infinite sequence of conserved quantities. Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed. - Article
- May 2000
- Comm Pure Appl Math

The peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods. We give a very simple proof of the orbital stability of the peakons in the H1 norm. © 2000 John Wiley & Sons, Inc. - An explicit reciprocal transformation between a two-component generalization of the Camassa–Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established. This transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented
- Article
- Jul 1999
- ADV MATH

Classical results of Stieltjes are used to obtain explicit formulas for the peakon–antipeakon solutions of the Camassa–Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon–antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly. - Article
- Oct 1981
- PHYSICA D

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 Bäcklund 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. - Article
- Jul 1983
- LINEAR ALGEBRA APPL

We give a common, concise derivation of some important determinantal identities attributed to the mathematicians in the title. We also give a formal treatment of determinantal identities of the minors of a matrix. - Article
- Jan 2011
- ADV MATH

A three-component generalization of Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3×3 matrix spectral problem with three potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The three-component generalization of Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N-peakons and an infinite sequence of conserved quantities. - Article
- Jan 1990

We have already noticed that a linear map ø: k → k is completely determined by its value on one element of k, e.g., on 1. Also we noted that a linear map $$ \varphi :k^2 \to k $$ is determined by φ(1,0) and φ(0,1). We Will now see how a linear map $$ \varphi :V \to W $$ (V, W being vector over k of dimensions n, m) is determined by nm elements of k. - Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.
- We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak. Comment: LaTeX file. Figure available from authors upon request
- We use an inverse scattering approach to study multipeakon solutions of the Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm (CH) shallow water equation. The spectral problem associated to the DP equation is equivalent under a change of variables to what we call the cubic string problem, which is a third-order nonselfadjoint generalization of the well-known equation describing the vibrational modes of an inhomogeneous string attached at its ends. We give two proofs that the eigenvalues of the cubic string are positive and simple; one using scattering properties of DP peakons, and another using the Gantmacher-Krein theory of oscillatory kernels. For the discrete cubic string (analogous to a string consisting of n point masses) we solve explicitly the inverse spectral problem of reconstructing the mass distribution from suitable spectral data, and this leads to explicit formulas for the general n-peakon solution of the DP equation. Central to our study of the inverse problem is a peculiar type of simultaneous rational approximation of the two Weyl functions of the cubic string, similar to classical Padé-Hermite approximation but with lower order of approximation and an additional symmetry condition instead. The results obtained are intriguing and nontrivial generalizations of classical facts from the theory of Stieltjes continued fractions and orthogonal polynomials.
- We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation. Comment: Updated version (minor errors corrected), 5 pages, uses iopart.cls