A view of the peakon world through the lens of approximation theory

DOI10.1016/j.physd.2022.133446zbMath1498.35003arXiv2203.12954OpenAlexW4286256627WikidataQ114141884 ScholiaQ114141884MaRDI QIDQ2083706

Jacek Szmigielski, Hans Lundmark

Publication date: 11 October 2022

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2203.12954

zbMATH Keywords

orthogonal polynomialsPadé approximationspeakonsLax integrabilityCauchy biorthogonal polynomialsCamassa-Holm-type equations

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Soliton equations (35Q51) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02) Soliton solutions (35C08)

Related Items

Global conservative solutions of the nonlocal NLS equation beyond blow-up ⋮ Preface to the special issue: orthogonal polynomials and integrable systems, peakons, Toda lattices and Painlevé transcendents ⋮ An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa-Holm equation ⋮ Similarity reductions of peakon equations: integrable cubic equations

Cites Work

• A nonlinear generalization of the Camassa-Holm equation with peakon solutions
• Bilinear equations and new multi-soliton solution for the modified Camassa-Holm equation
• The inverse spectral problem for indefinite strings
• Multi-component generalization of the Camassa-Holm equation
• A Riemann-Hilbert approach for the Novikov equation
• On an integrable Camassa-Holm type equation with cubic nonlinearity
• Further results on the smooth and nonsmooth solitons of the Novikov equation
• Integrable peakon systems with weak kink and kink-peakon interactional solutions
• Stability of peakons for the Novikov equation
• The Cauchy problem for the integrable Novikov equation
• The limiting behavior of smooth periodic waves for the Degasperis-Procesi equation
• The Cauchy problem for a two-component Novikov equation in the critical Besov space
• On bi-Hamiltonian structure of two-component Novikov equation
• Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions
• The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
• On the peakon inverse problem for the Degasperis-Procesi equation
• Low-regularity solutions of the periodic general Degasperis-Procesi equation
• Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications
• Well-posedness and persistence properties for the Novikov equation
• Blow-up phenomenon for the integrable Novikov equation
• Smooth and peaked solutions of the Camassa-Holm equation and applications
• Long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation
• The Degasperis-Procesi equation as a non-metric Euler equation
• Some geometric investigations on the Degasperis-Procesi shallow water equation
• Integrable hierarchy, \(3\times 3\) constrained systems, and parametric solutions
• The Degasperis-Procesi equation on the half-line
• On the initial boundary value problems for the Degasperis-Procesi equation
• On well-posedness of the Degasperis-Procesi equation
• Non-uniform dependence on initial data for the periodic Degasperis-Procesi equation
• Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function
• On isospectral deformations of an inhomogeneous string
• Exponential decay and symmetry of solitary waves to Degasperis-Procesi equation
• Asymptotic stability of peakons for the Novikov equation
• A new type of bounded waves for Degasperis-Procesi equation
• On the well-posedness of the Degasperis-Procesi equation
• Dissipative solutions for the Camassa-Holm equation
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation
• Global conservative solutions of the Camassa-Holm equation
• Riemann-Hilbert approach for the Camassa-Holm equation on the line
• On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation
• Formation and dynamics of shock waves in the Degasperis-Procesi equation
• Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems
• A remark on the stability of peakons for the Degasperis-Procesi equation
• Periodic conservative solutions of the Camassa-Holm equation
• The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach
• The bifurcation and peakon for Degasperis-Procesi equation
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
• Cauchy biorthogonal polynomials
• Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations
• On a class of physically important integrable equations
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• Well-posedness of modified Camassa-Holm equations
• Stability of multi antipeakon-peakons profile
• Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation
• Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation
• Korteweg - de Vries suoerequation as an Euler equation
• Three integrable Hamiltonian systems connected with isospectral deformations
• On the inverse spectral problem for the Camassa-Holm equation
• Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa-Holm equation
• Acoustic scattering and the extended Korteweg-de Vries hierarchy
• Wave breaking for nonlinear nonlocal shallow water equations
• The geometry of peaked solitons and billiard solutions of a class of integrable PDE's
• On a completely integrable nonlinear hyperbolic variational equation
• The Hamiltonian structure of the Camassa-Holm equation
• On the spectral problem for the periodic Camassa-Holm equation
• Soliton interactions for the Camassa-Holm equation
• On the Cauchy problem for the periodic Camassa-Holm equation
• A shallow water equation as a geodesic flow on the Bott-Virasoro group
• The Camassa-Holm hierarchy, \(N\)-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold
• Global existence for a new periodic integrable equation.
• On the Cauchy problem for an integrable equation with peakon solutions
• Algebro-geometric solutions of the Camassa-Holm hierarchy
• On asymptotically equivalent shallow water wave equations
• Breakdown of a shallow water equation
• Multipeakons and the classical moment problem
• Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions
• The classical moment problem and generalized indefinite strings
• Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation
• Lipschitz metric for the Novikov equation
• Isospectral flows for the inhomogeneous string density problem
• Moment modification, multipeakons, and nonisospectral generalizations
• Integrable \(U(1)\)-invariant peakon equations from the NLS hierarchy
• Lax integrability and the peakon problem for the modified Camassa-Holm equation
• On the integrability of Degasperis-Procesi equation: control of the Sobolev norms and Birkhoff resonances
• Global dissipative solutions of the Novikov equation
• Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy
• Peakon solutions of the Novikov equation and properties of the data-to-solution map
• Traveling wave solutions of the Degasperis-Procesi equation
• An application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type
• On a new two-component \(b\)-family peakon system with cubic nonlinearity
• The Camassa-Holm equation: a loop group approach
• On the inverse scattering approach for an integrable shallow water wave equation
• Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE
• Classical solutions of the periodic Camassa-Holm equation.
• Characteristics and the initial value problem of a completely integrable shallow water equation
• Stability of the Camassa-Holm solitons
• Periodic and solitary-wave solutions of the Degasperis-Procesi equation
• Global weak solutions for a new periodic integrable equation with peakon solutions
• The Korteweg-de Vries equation and beyond
• An extension of integrable peakon equations with cubic nonlinearity
• GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• The inverse spectral problem for the discrete cubic string
• Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation
• GLOBAL CONSERVATIVE MULTIPEAKON SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• A new integrable equation with cuspons and W/M-shape-peaks solitons
• PERSISTENCE PROPERTIES FOR THE DEGASPERIS–PROCESI EQUATION
• Integrable peakon equations with cubic nonlinearity
• New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons
• Stability of peakons for the Degasperis-Procesi equation
• Global Dissipative Multipeakon Solutions of the Camassa–Holm Equation
• Variational derivation of the Camassa-Holm shallow water equation
• Cubic string boundary value problems and Cauchy biorthogonal polynomials
• Generalizations of the Camassa–Holm equation
• Dynamics of Director Fields
• On the spectral functions of the string
• Finitely many mass points on the line under the influence of an exponential potential -- an integrable system
• Multi-peakons and a theorem of Stieltjes
• Global weak solutions for a shallow water equation
• A shallow water equation on the circle
• Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs
• Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves
• On the Inverse Scattering Approach to the Camassa-Holm Equation
• Perturbative symmetry approach
• Multi-peakon solutions of the Degasperis–Procesi equation
• Stability of peakons
• An integrable shallow water equation with peaked solitons
• Asymptotic Integrability of Water Waves
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• Construction of 2-Peakon Solutions and Ill-Posedness for the Novikov Equation
• Degasperis–Procesi peakon dynamical system and finite Toda lattice of CKP type
• Multipeakons viewed as geodesics
• The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type
• On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey
• Conservation laws of the Camassa–Holm equation
• PERIODIC PEAKONS AND CALOGERO–FRANÇOISE FLOWS
• A Variational Approach to the Stability of Periodic Peakons
• Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation
• Prolongation algebras and Hamiltonian operators for peakon equations
• Global solutions to a new integrable equation with peakons
• Factorization and Painleve analysis of a class of nonlinear third-order partial differential equations
• Well-posedness and global existence for the Novikov equation
• On the τ-functions of the Degasperis–Procesi equation
• A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations
• Geometric aspects of two- and threepeakons
• A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions
• The Inverse Spectral Problem for Periodic Conservative Multi-peakon Solutions of the Camassa–Holm Equation
• On matrix Cauchy biorthogonal polynomials
• A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method
• Non-interlacing peakon solutions of the Geng–Xue equation
• The Calogero-Franc¸oise Integrable System: Algebraic Geometry, Higgs Fields, and the Inverse Problem
• A reciprocal transformation for the Geng-Xue equation
• The Structure-Preserving Methods for the Degasperis-Procesi Equation
• A High Order Operator Splitting Method for the Degasperis–Procesi Equation
• Persistence properties and wave‐breaking criteria for the Geng‐Xue system
• New solutions with peakon creation in the Camassa–Holm and Novikov equations
• A three-component Camassa-Holm system with cubic nonlinearity and peakons
• Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity
• Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons
• Liouville Integrability of Conservative Peakons for a Modified CH equation
• A new integrable two-component system with cubic nonlinearity
• An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation
• A completely integrable Hamiltonian system
• Zero curvature formulations of dual hierarchies
• A Riemann–Hilbert approach for the Degasperis–Procesi equation
• New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation
• The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations
• Colliding peakons and the formation of shocks in the Degasperis–Procesi equation
• A new two-component integrable system with peakon solutions
• Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model
• Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity
• Hölder continuity of the solution map for the Novikov equation
• Smooth and singular multisoliton solutions of a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion
• Peakon-antipeakon interactions in the Degasperis-Procesi equation
• On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions
• On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions
• Factorization problem on the Hilbert‐Schmidt group and the Camassa‐Holm equation
• Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View
• On the Blow-up Phenomena for the Degasperis-Procesi Equation
• Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation
• Degasperis-Procesi peakons and the discrete cubic string
• A CONTINUOUS INTERPOLATION BETWEEN CONSERVATIVE AND DISSIPATIVE SOLUTIONS FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM
• Parametric Representation for the Multisoliton Solution of the Camassa–Holm Equation
• Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit
• The modified Camassa–Holm equation: Bäcklund transformation and nonlinear superposition formula
• The Cauchy problem for the Novikov equation
• The Cauchy problem for the Novikov equation
• Peakon solutions of the shallow water equation
• The Cauchy two-matrix model
• Integrable evolution equations on spaces of tensor densities and their peakon solutions
• Analysis on the blow-up of solutions to a class of integrable peakon equations
• Integrable generalization of the associated Camassa-Holm equation
• Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy
• Continued fraction expansions of Herglotz–Nevanlinna functions and generalized indefinite strings of Stieltjes type
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation
• A note on the prolongation structure of the cubically nonlinear integrable Camassa-Holm type equation
• The existence of global strong and weak solutions for the Novikov equation
• On the isospectral problem of the dispersionless Camassa-Holm equation
• Wave-breaking and peakons for a modified Camassa-Holm equation
• On the Cauchy problem for the two-component Novikov equation
• The Canada day theorem
• The Camassa Holm equation: Conserved quantities and the initial value problem
• Existence of permanent and breaking waves for a shallow water equation: a geometric approach
• On the finite time singularities for a class of Degasperis-Procesi equations
• A new Liouville transformation for the Geng-Xue system
• Large time behavior of momentum support for a Novikov type equation
• Persistence property and estimate on momentum support for the integrable Degasperis-Procesi equation
• Asymptotic profile of solutions to the Degasperis-Procesi equation
• Orbital and asymptotic stability of a train of peakons for the Novikov equation
• Novikov equation: Bäcklund transformation and applications
• A rigidity property for the Novikov equation and the asymptotic stability of peakons
• A note on the Cauchy problem for the two-component Novikov system
• \(W^{1,\infty}\) instability of \(H^1\)-stable peakons in the Novikov equation
• Asymptotic stability of the Degasperis-Procesi antipeakon-peakon profile
• The two-component Novikov-type systems with peaked solutions and \(H^1\)-conservation law
• Hermite-Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices
• On the orbital stability of the Degasperis-Procesi antipeakon-peakon profile
• Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials
• Multisoliton solutions of the Degasperis-Procesi equation and its shortwave limit: Darboux transformation approach
• Bäcklund transformations for the Degasperis-Procesi equation
• Liouville correspondences between multicomponent integrable hierarchies
• Non-uniform dependence for the Novikov equation in Besov spaces
• Spectral stability of smooth solitary waves for the Degasperis-Procesi equation
• A smooth soliton solution and a periodic cuspon solution of the Novikov equation
• Solitons for the modified Camassa-Holm equation and their interactions via dressing method
• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• Stability of multipeakons
• Accelerating dynamical peakons and their behaviour
• A rigidity result for the Holm-Staley \(b\)-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon
• Continuity for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities
• Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
• Asymptotic of Cauchy biorthogonal polynomials
• Instability of \(H^1\)-stable peakons in the Camassa-Holm equation
• Adaptive moving knots meshless method for Degasperis-Procesi equation with conservation laws
• A note on the convergence of the solution of the Novikov equation
• Weak solution of the Novikov equation and optimal control
• Periodic solutions of the Degasperis-Procesi equation: well-posedness and asymptotics
• Stability in the energy space of the sum of \(N\) peakons for the Degasperis-Procesi equation
• On the wave length of smooth periodic traveling waves of the Camassa-Holm equation
• Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation
• On solutions to the Degasperis-Procesi equation
• Global weak solutions to the Novikov equation
• Cauchy-Laguerre two-matrix model and the Meijer-\(G\) random point field
• Global existence and blow-up phenomena for the Degasperis-Procesi equation
• Wave dynamics for peaked solitons of the Camassa-Holm equation
• Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition
• Calogero-Françoise flows and periodic peakons
• Local well-posedness of a new integrable equation
• A generalized nonisospectral Camassa-Holm equation and its multipeakon solutions
• Group analysis of the Novikov equation
• Traveling wave solutions of the Camassa-Holm equation
• The Camassa-Holm equation on the half-line
• Liouville correspondences between integrable hierarchies
• Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice
• Ghostpeakons and characteristic curves for the Camassa-Holm, Degasperis-Procesi and Novikov equations
• A numerical study of variational discretizations of the Camassa-Holm equation
• Smooth multisoliton solutions of a 2-component peakon system with cubic nonlinearity
• Peakons, strings, and the finite Toda lattice
• Peakon-Antipeakon Interaction
• Cusp Soliton of a New Integrable Nonlinear Evolution Equation
• The Lie Algebra Structure of Nonlinear Evolution Equations Admitting Infinite Dimensional Abelian Symmetry Groups
• Inverse moment problem for elementary co-adjoint orbits
• Global well-posedness for the periodic Novikov equation with cubic nonlinearity
• A Synthetical Two-Component Model with Peakon Solutions
• The general peakon–antipeakon solution for the Camassa–Holm equation
• A SEMIGROUP OF SOLUTIONS FOR THE DEGASPERIS-PROCESI EQUATION
• Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa-Holm equation
• A note on the Cauchy problem of the Novikov equation
• Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy
• Multi-loop soliton solutions and their interaction in the Degasperis–Procesi equation
• Smooth multisoliton solutions and their peakon limit of Novikov’s Camassa–Holm type equation with cubic nonlinearity
• The dual modified Korteweg-de Vries–Fokas–Qiao equation: Geometry and local analysis
• Dark solitons of the Qiao's hierarchy
• Wave-breaking phenomena, decay properties and limit behaviour of solutions of the Degasperis—Procesi equation
• Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation
• A Liouville theorem for the Degasperis-Procesi equation
• Dressing Method for the Degasperis-Procesi Equation
• The Modified Camassa-Holm Equation
• On the Camassa-Holm equation and a direct method of solution I. Bilinear form and solitary waves
• Smooth and peaked solitons of the CH equation
• Comment on “A new integrable equation with cuspons and W/M-shape-peaks solitons” [J. Math. Phys. 47, 112701 (2006)]
• Global weak solutions for the Novikov equation
• On the Cauchy problem for the Degasperis-Procesi equation
• The Periodic Cauchy Problem for Novikov's Equation
• Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation
• Global analytic solutions and traveling wave solutions of the Cauchy problem for the Novikov equation
• Inverse scattering transform for the Degasperis–Procesi equation
• The initial value problem for a Novikov system
• Exact Cuspon and Compactons of the Novikov Equation
• A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations
• On an inverse problem associated with an integrable equation of Camassa–Holm type: explicit formulas on the real axis
• On the Integrability of a Class of Nonlinear Dispersive Wave Equations
• The N -soliton solution of the Degasperis–Procesi equation