Stability of peakons
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Publication:4487962
DOI<603::AID-CPA3>3.0.CO;2-L 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-LzbMath1049.35149OpenAlexW2136437647WikidataQ56420348 ScholiaQ56420348MaRDI QIDQ4487962
Adrian Constantin, Walter A. Strauss
Publication date: 25 June 2000
Full work available at URL: https://doi.org/10.1002/(sici)1097-0312(200005)53:5<603::aid-cpa3>3.0.co;2-l
Stability in context of PDEs (35B35) KdV equations (Korteweg-de Vries equations) (35Q53) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems (37K45) Soliton equations (35Q51)
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b-equation ⋮ Well-posedness and persistence property for a shallow water wave equation for waves of large amplitude ⋮ New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons ⋮ Optimal control of the viscous Degasperis-Procesi equation ⋮ New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation ⋮ Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation ⋮ Peakons Arising as Particle Paths Beneath Small-Amplitude Water Waves in Constant Vorticity Flows ⋮ On Solutions to a Two-Component Generalized Camassa-Holm Equation ⋮ Stability in the energy space of the sum of N peakons for a modified Camassa-Holm equation with higher-order nonlinearity ⋮ Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system ⋮ THE EXPLICIT NONLINEAR WAVE SOLUTIONS AND THEIR BIFURCATIONS OF THE GENERALIZED CAMASSA–HOLM EQUATION ⋮ Traveling Wave Solutions of the Camassa-Holm and Korteweg-de Vries Equations ⋮ On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey ⋮ Finite propagation speed for the Camassa–Holm equation ⋮ EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS ⋮ A Note on the Degasperis-Procesi Equation ⋮ About the Explicit Characterization of Hamiltonians of the Camassa-Holm Hierarchy ⋮ A Variational Approach to the Stability of Periodic Peakons ⋮ Blowup phenomena for a new periodic nonlinearly dispersive wave equation ⋮ Well-posedness for the Cauchy problem of the modified Hunter-Saxton equation in the Besov spaces ⋮ Low-regularity solutions of the periodic Fornberg–Whitham equation ⋮ Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation ⋮ COEXISTENCE OF MULTIFARIOUS EXACT NONLINEAR WAVE SOLUTIONS FOR GENERALIZED b-EQUATION ⋮ The properties of solutions to the dissipative 2-component Camassa–Holm system ⋮ The Cauchy problem for a family of generalized Camassa–Holm equations ⋮ A Synthetical Two-Component Model with Peakon Solutions ⋮ Orbital Stability: Analysis Meets Geometry ⋮ On the blow-up phenomena of the periodic Dullin–Gottwald–Holm equation ⋮ Blow-up issues for the weakly dissipative periodic b-family of equations revisited ⋮ The existence of global weak solutions for a generalized Camassa–Holm equation ⋮ Existence of permanent and breaking waves for the periodic Degasperis–Procesi equation with linear dispersion ⋮ Non‐uniform dependence and persistence properties for coupled Camassa–Holm equations ⋮ On the Cauchy problem for the Degasperis-Procesi equation ⋮ ON THE LOW REGULARITY SOLUTIONS FOR A MODIFIED TWO-COMPONENT CAMASSA–HOLM SHALLOW WATER SYSTEM ⋮ A Note on Traveling Wave Solutions to the Two Component Camassa–Holm Equation ⋮ Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa-Holm equation ⋮ COEXISTENCE OF MULTIFARIOUS EXPLICIT NONLINEAR WAVE SOLUTIONS FOR MODIFIED FORMS OF CAMASSA–HOLM AND DEGAPERIS–PROCESI EQUATIONS ⋮ Global well posedness and limit behavior of the solutions to the viscous Degasperis–Procesi equation ⋮ Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons ⋮ Stability of Solitary Waves and Global Existence of a Generalized Two-Component Camassa–Holm System ⋮ Growth of Perturbations to the Peaked Periodic Waves in the Camassa--Holm Equation ⋮ A note on the Cauchy problem of the Novikov equation ⋮ Initial boundary value problem for a coupled Camassa–Holm system with peakons ⋮ Infinite propagation speed for the Degasperis-Procesi equation ⋮ Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system ⋮ On the global weak solutions for a modified two-component Camassa-Holm equation ⋮ Traveling wave solutions of the Camassa-Holm equation ⋮ Time periodic solution of the viscous Camassa-Holm equation ⋮ Note on the evolution of compactly supported initial data under the Camassa-Holm flow ⋮ Wave Breaking in a Class of Nonlocal Dispersive Wave Equations ⋮ Global analytic solutions and traveling wave solutions of the Cauchy problem for the Novikov equation ⋮ Low-Regularity Solutions of the Periodic Camassa–Holm Equation ⋮ Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation ⋮ On the global weak solution to a generalized two-component Camassa-Holm system ⋮ On the wave breaking phenomena for the generalized periodic two-component Dullin-Gottwald-Holm system ⋮ Blow-up criteria of solutions for a modified two-component hyperelastic rod system ⋮ The Cauchy problem for the generalized Camassa-Holm equation ⋮ A note on the solution map for the periodic Camassa–Holm equation ⋮ On the well-posedness of the Degasperis-Procesi equation ⋮ On the blow-up of solutions for the Fornberg-Whitham equation ⋮ Stability of the Camassa-Holm multi-peakons in the dynamics of a shallow-water-type system ⋮ Local well-posedness and blow-up for a family of \(U(1)\)-invariant peakon equations ⋮ On the Cauchy problem and peakons of a two-component Novikov system ⋮ Equations with peakon solutions in the negative order Camassa-Holm hierarchy ⋮ Weak well-posedness for a modified two-component Camassa-Holm system ⋮ Some properties of solutions to the Camassa-Holm-type equation with higher-order nonlinearities ⋮ Rigorous derivation and propagation speed property for a two-component Degasperis-Procesi system in shallow water regimes ⋮ The global attractor for a viscous weakly dissipative generalized two-component \(\mu\)-Hunter-Saxton system ⋮ Solutions of a Camassa-Holm hierarchy in \(2+1\) dimensions ⋮ Radius of analyticity for the Camassa-Holm equation on the line ⋮ Orbital stability of peakons for a generalized CH equation ⋮ Infinite propagation speed of the Camassa-Holm equation ⋮ On the blow-up of solutions to a nonlinear dispersive rod equation ⋮ Well-posedness and blow-up phenomena for an integrable three-component Camassa-Holm system ⋮ Global weak solutions and blow-up structure for the Degasperis-Procesi equation ⋮ Global well-posedness for the viscous Camassa--Holm equation ⋮ Existence and uniqueness of low regularity solutions for the Dullin-Gottwald-Holm equation ⋮ Compacton-like wave and kink-like wave of GCH equation ⋮ Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics ⋮ Global conservative solutions of the Camassa-Holm equation ⋮ Traveling wave solutions of a highly nonlinear shallow water equation ⋮ Well-posedness, blow-up criteria and Gevrey regularity for a rotation-two-component Camassa-Holm system ⋮ Liouville theorems for periodic two-component shallow water systems ⋮ On the Cauchy problem for a higher-order \(\mu\)-Camassa-Holm equation ⋮ Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation ⋮ Optimal control and optimality condition of the Camassa-Holm equation ⋮ Riemann-Hilbert approach for the Camassa-Holm equation on the line ⋮ Uniqueness of global conservative weak solutions for the modified two-component Camassa-Holm system ⋮ Moment modification, multipeakons, and nonisospectral generalizations ⋮ Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions ⋮ On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation ⋮ Persistence properties of the solutions to a generalized two-component Camassa-Holm shallow water system ⋮ Well-posedness and global existence for a generalized Degasperis-Procesi equation ⋮ Integrable \(U(1)\)-invariant peakon equations from the NLS hierarchy ⋮ Well-posedness and blow-up phenomena for the cross-coupled Camassa-Holm system ⋮ The Cauchy problem and blow-up phenomena of a new integrable two-component Camassa-Holm system ⋮ A remark on the stability of peakons for the Degasperis-Procesi equation ⋮ A note on well-posedness for Camassa-Holm equation. ⋮ 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 weak solutions for a generalized Novikov equation ⋮ Application of the integral bifurcation method for solving modified Camassa-Holm and Degasperis-Procesi equations ⋮ Wave breaking for a periodic shallow water equation. ⋮ Periodic conservative solutions of the Camassa-Holm equation ⋮ Some geometric investigations on the Degasperis-Procesi shallow water equation ⋮ Global existence and blow-up phenomena for a periodic 2-component Camassa-Holm equation ⋮ Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy ⋮ Conformal properties and Bäcklund transform for the associated Camassa-Holm equation ⋮ Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis-Procesi equation ⋮ The integral bifurcation method and its application for solving a family of third-order dispersive PDEs ⋮ Controllability and asymptotic stabilization of the Camassa-Holm equation ⋮ Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system ⋮ The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation ⋮ Linear stability of weak-compacton solutions to the nonlinear dispersive Ostrovsky equation ⋮ Optimal control of the viscous generalized Camassa-Holm equation ⋮ Well-posedness and blow-up phenomena for the generalized Degasperis-Procesi equation ⋮ Orbital stability of peakons for the Degasperis-Procesi equation with strong dispersion ⋮ Orbital stability of floating periodic peakons for the Camassa-Holm equation ⋮ Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval ⋮ On the initial boundary value problems for the Degasperis-Procesi equation ⋮ Global weak solutions for a modified two-component Camassa-Holm equation ⋮ Non-uniform dependence on initial data for the periodic Degasperis-Procesi equation ⋮ Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons ⋮ Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion ⋮ Blow-up of solutions to the DGH equation ⋮ Orbital stability and dynamical behaviors of solitary waves for the Camassa-Holm equation with quartic nonlinearity ⋮ On the Cauchy problem of the modified Hunter-Saxton equation ⋮ Wave breaking and persistent decay of solution to a shallow water wave equation ⋮ Stability of a class of solitary waves in compressible elastic rods ⋮ Blow-up phenomena and persistence property for the modified b-family of equations ⋮ Transfer of energy in Camassa-Holm and related models by use of nonunique characteristics ⋮ Blow-up solutions for the modified b-family of equations ⋮ On the inverse scattering approach and action-angle variables for the Dullin-Gottwald-Holm equation ⋮ The global attractor of the viscous Fornberg-Whitham equation ⋮ Global weak solutions and wave breaking phenomena to the periodic Degasperis-Procesi equation with strong dispersion ⋮ An operator splitting method for the Degasperis-Procesi equation ⋮ The attractor on viscosity Degasperis-Procesi equation ⋮ Initial boundary value problems for nonlinear dispersive wave equations ⋮ Persistence properties for a family of nonlinear partial differential equations ⋮ The attractor of a parabolic-elliptic system ⋮ Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs ⋮ Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III) ⋮ Well-posedness of modified Camassa-Holm equations ⋮ Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation ⋮ Parallel simultaneous stabilization of two systems governed by partial differential equations subject to actuator saturation ⋮ An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations ⋮ Time-periodic solution of the weakly dissipative Camassa-Holm equation ⋮ Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation ⋮ Equations of the Camassa-Holm hierarchy ⋮ Global weak solutions and breaking waves to the Degasperis-Procesi equation with linear dispersion ⋮ Periodic and solitary wave solutions to the Fornberg-Whitham equation ⋮ The global Gevrey regularity of the rotation two-component Camassa-Holm system ⋮ Asymptotic stability of peakons for the Novikov equation ⋮ Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces ⋮ Optimal control of the viscous weakly dispersive Degasperis-Procesi equation ⋮ Bifurcations and exact bounded travelling wave solutions for a partial differential equation ⋮ Optimal control of the viscous Dullin-Gottwalld-Holm equation ⋮ Blow-up of solutions to a nonlinear dispersive rod equation ⋮ Hybrid solitary wave solutions of the Camassa-Holm equation ⋮ Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation ⋮ Ill-posedness of the Cauchy problem for the \(\mu\)-Camassa-Holm-type equations ⋮ Complete classification of local conservation laws for generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation ⋮ On the Cauchy problem of the two-component Novikov-type system with peaked solutions and H1-conservation law ⋮ Orbital stability of the sum of \(N\) peakons for the generalized modified Camassa-Holm equation ⋮ Stability of traveling wave solutions for the second-order Camassa-Holm equation ⋮ The traveling wave solutions for a two-component b-family equations ⋮ Energy conservation and well-posedness of the Camassa-Holm equation in Sobolev spaces ⋮ Local and global analyticity for theμ-Novikov equation ⋮ Wave-breaking phenomena for the generalized Camassa-Holm equation with dual-power nonlinearities ⋮ Stability of smooth periodic travelling waves in the Camassa–Holm equation ⋮ On a generalized Camassa‐Holm type equation with ()‐degree nonlinearities ⋮ A note on the initial value problem for a higher‐order Camassa–Holm equation ⋮ A highly nonlinear shallow-water model Arising from the full water waves with the Coriolis effect ⋮ Some properties of the solutions of the \(N\)-component Camassa-Holm system with peakons ⋮ Multisoliton solutions of the two-component Camassa-Holm equation and its reductions ⋮ On the Cauchy problem for a generalized Camassa-Holm equation ⋮ Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space ⋮ On the Cauchy problem for the generalized Camassa-Holm equation ⋮ Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy ⋮ Blow-up phenomenon for a periodic rod equation ⋮ Stability of peakons of the Camassa–Holm equation beyond wave breaking ⋮ Qualitative analysis for a two-component peakon system with cubic nonlinearity ⋮ Orbital stability of periodic peakons for a new higher-order μ-Camassa–Holm equation ⋮ Orbital stability of periodic peakons for a generalized Camassa–Holm equation ⋮ Orbital stability of smooth solitary waves for the Degasperis-Procesi equation ⋮ A general family of multi-peakon equations and their properties ⋮ The stability of the b-family of peakon equations ⋮ The Initial-Value Problem to the Modified Two-Component Euler--Poincaré Equations ⋮ Local Well-Posedness and Orbital Stability of Solitary Wave Solutions for the Generalized Camassa–Holm Equation ⋮ Local Well-Posedness and Orbital Stability of Solitary Wave Solutions for the Generalized Camassa–Holm Equation ⋮ Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation ⋮ Global Dissipative Solutions of the Generalized Camassa-Holm Equation ⋮ Spectral Instability of Peakons in the $b$-Family of the Camassa--Holm Equations ⋮ A note on the Cauchy problem for the periodic two-component Novikov system ⋮ A note on wave-breaking criteria for the Fornberg-Whitham equation ⋮ Orbital stability of pseudo-peakons for the fifth-order Camassa-Holm type equation ⋮ Orbital stability of peakons and multi-peakons for a generalized cubic-quintic Camassa-Holm type equation ⋮ Globally conservative weak solutions for a class of nonlinear dispersive wave equations beyond wave breaking ⋮ Blow-up analysis for the \(ab\)-family of equations ⋮ Continuity properties of the solution map for the four-component Novikov system with peakon solutions ⋮ On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations ⋮ The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: long-time and Painlevé asymptotics ⋮ On the Cauchy problem for a generalized Degasperis-Procesi equation ⋮ Local well-posedness and blow-up criteria for a three-component Camassa–Holm type equation ⋮ A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions ⋮ On the Cauchy problem for a four‐component Novikov system with peaked solutions ⋮ Local well-posedness and decay for some generalized shallow water equations ⋮ The well-posedness for the Camassa-Holm type equations in critical Besov spaces \(B_{p , 1}^{1 + \frac{ 1}{ p}}\) with \(1 \leq p < + \infty \) ⋮ Solutions of the Camassa-Holm equation near the soliton ⋮ Zero-filter limit for the Camassa-Holm equation in Sobolev spaces ⋮ Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities ⋮ Orbital stability of two-component peakons ⋮ Persistence properties and blow-up phenomena for a generalized Camassa-Holm equation ⋮ The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities ⋮ Some new decay properties of solutions for a modified Camassa–Holm equation with cubic nonlinearity ⋮ Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces: revisited ⋮ Long-time asymptotics of the Hunter-Saxton equation on the line ⋮ Instability of \(H^1\)-stable periodic peakons for the \(\mu\)-Camassa-Holm equation ⋮ Stability of periodic peakons for a nonlinear quartic \(\mu\)-Camassa-Holm equation ⋮ Numerical study of fractional Camassa-Holm equations ⋮ Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains ⋮ Bifurcations of Solitary Waves of a Simple Equation ⋮ Global weak solutions for a two-component Camassa–Holm system with an arbitrary smooth function ⋮ Nonuniform dependence and well-posedness for the generalized Camassa–Holm equation ⋮ Local well-posedness and blow-up phenomena of the generalized short pulse equation ⋮ The local and global existence of solutions for a generalized Camassa-Holm equation ⋮ On the Cauchy problem for a generalized Camassa-Holm equation ⋮ On the weak solutions for the rotation-two-component Camassa–Holm equation ⋮ Compactly Supported Solutions of the Camassa-Holm Equation ⋮ Revisit to wave breaking phenomena for the periodic Dullin–Gottwald–Holm equation ⋮ Multi-soliton solutions of a two-component Camassa–Holm system: Darboux transformation approach ⋮ Unnamed Item ⋮ PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION ⋮ Blow-up phenomena and peakons for the gFQXL/gCH-mCH equation ⋮ Globally conservative solutions for the modified Camassa–Holm (MOCH) equation ⋮ On the Cauchy problem for the generalized Camassa-Holm equation ⋮ The Cauchy problem of the rotation Camassa–Holm equation in equatorial water waves ⋮ A Splitting Method for the Degasperis--Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method ⋮ Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space ⋮ A three‐level linearized finite difference scheme for the camassa–holm equation ⋮ The local and global existence of solutions for a generalized Camassa-Holm equation ⋮ On local-in-space blow-up scenarios for a weakly dissipative rotation-Camassa–Holm equation ⋮ Blow-up for a weakly dissipative modified two-component Camassa–Holm system ⋮ The Cauchy problem for an integrable two-component model with peakon solutions ⋮ The Cauchy problem for the generalized hyperelastic rod wave equation ⋮ Two-component generalizations of the Novikov equation ⋮ Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System ⋮ Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View ⋮ Bifurcations and Exact Solutions of Generalized Two-Component Peakon Type Dual Systems ⋮ Analytical Cartesian solutions of the multi-component Camassa-Holm equations ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system ⋮ Initial Boundary Value Problems of the Camassa–Holm Equation ⋮ Persistence properties for the two-component Novikov equation in weighted Lp spaces ⋮ Bifurcations and Exact Solutions in a Nonlinear Wave Equation ⋮ Bi-Hamiltonian Structure of a Three-Component Camassa-Holm Type Equation ⋮ On a spectral analysis of scattering data for the Camassa-Holm equation ⋮ A three-component Camassa-Holm system with cubic nonlinearity and peakons ⋮ Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations ⋮ Orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equation ⋮ Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons ⋮ On Peakon and Kink-peakon Solutions to a (2 + 1) Dimensional Generalized Camassa-Holm Equation ⋮ Global existence and long time behavior of the generalized Camassa–Holm equation with k + 1 degree nonlinearities ⋮ On a dissipative form of Camassa–Holm equation ⋮ Orbital stability of the sum of N solitary waves for the hyperelastic-rod wave equation ⋮ Stability of peakons and linear dispersion limit for the periodic Dullin–Gottwald–Holm equation ⋮ Stability of blow-up solution for the two component Camassa–Holm equations ⋮ On a novel integrable generalization of the sine-Gordon equation ⋮ Stability of negative solitary waves for an integrable modified Camassa–Holm equation ⋮ Optimal solution for the viscous nonlinear dispersive wave equation ⋮ On the Cauchy problem for a generalized Degasperis–Procesi equation ⋮ Blow-up scenario for a generalized Camassa–Holm equation with both quadratic and cubic nonlinearity ⋮ EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS ⋮ Global dissipative solutions of a modified two-component Camassa–Holm shallow water system ⋮ Global existence and blow-up phenomena for a periodic modified Camassa–Holm equation (MOCH) ⋮ Global conservative and dissipative solutions of a coupled Camassa-Holm equations ⋮ Lower order regularity for the generalized Camassa–Holm equation ⋮ Quadratic operator pencils associated with the conservative Camassa-Holm flow ⋮ Conserved quantities, continuation and compactly supported solutions of some shallow water models
Cites Work
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