An integrable shallow water equation with peaked solitons

Abstract

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.

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Cited by in corpus (625)

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• Generalized indefinite strings with purely discrete spectrum
• The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus
• Geodesic motion on the groups of diffeomorphisms with $H^1$ metric as geometric generalised Lagrangian mean theory
• Shape analysis via gradient flows on diffeomorphism groups
• Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces
• Spectral stability of shock-fronted travelling waves under viscous relaxation
• Singular solutions for geodesic flows of Vlasov moments
• On the Cauchy problem of a periodic 2-component $\mu$-Hunter-Saxton equation
• Integrable systems with quadratic nonlinearity in Fourier space
• Patched peakon weak solutions of the modified Camassa-Holm equation
• Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions
• Topological Bifurcations and Reconstruction of Travelling Waves
• A new integrable structure associated to the Camassa-Holm peakons
• An integrable (2+1)-dimensional Camassa-Holm hierarchy with peakon solutions
• A linearly implicit structure-preserving scheme for the Camassa-Holm equation based on multiple scalar auxiliary variables approach
• A generalised multicomponent system of Camassa-Holm-Novikov equations
• The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
• Asymptotics in shallow water waves
• Reciprocal link for a three-component Camassa-Holm type equation
• Vortices with source, FQHE and nontrivial statistics in 2+1 dimensions
• Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy
• Stability of smooth periodic traveling waves in the Camassa-Holm equation
• Exact solutions for the Generalized Modified Degasperis-Procesi equation
• Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in $2+1$ dimensions
• Exactly solvable model for nonlinear pulse propagation in optical fibers
• Deep Learning: Hydrodynamics, and Lie-Poisson Hamilton-Jacobi Theory
• Symmetry classification and conservation laws for higher order Camassa-Holm equation
• Existence of peakons for a cubic generalization of the Camassa-Holm equation
• A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B_{p,1}^{1+\frac{1}{p}},1\le p<+\infty$
• On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
• Multipeakons and the Classical Moment Problem
• On the Cauchy problem of a weakly dissipative $\mu$HS equation
• The Cauchy problem for the shallow water typ equations in low regularity spaces on the circle
• Orientation Waves in a Director Field With Rotational Inertia
• Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian
• A geometric Hamilton--Jacobi theory on a Nambu-Jacobi manifold
• Unique Continuation Properties for solutions to the Camassa-Holm equation and other non-local equations
• Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics
• Camassa-Holm type equations for axisymmetric Poiseuille pipe flows
• Non-uniform continuity of periodic Holm-Staley b-family of equations
• On Negative Order KdV Equations
• The 2-component dispersionless Burgers equation arising in the modelling of blood flow
• Analysis of singular solutions for two nonlinear wave equations
• Integrable U(1)-invariant peakon equations from the NLS hierarchy
• Persistence Properties and Unique Continuation for a Dispersionless Two-Component Camassa-Holm System with Peakon and Weak Kink Solutions
• Well-posedness and peakons for a higher-order $\mu$-Camassa-Holm equation
• Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude
• The Cauchy problem for fractional Camassa-Holm equation in Besov space
• Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry
• Solitons and peaked solitons for the general Degasperis-Procesi model
• Existence and Uniqueness of Global Weak solutions of the Camassa-Holm Equation with a Forcing
• The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws
• Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system
• Blow-up phenomena for the rotation-two-component Camassa-Holm system
• Stability of peakons for the generalized modified Camassa-Holm equation
• Uniqueness of Dissipative Solution for Camassa-Holm Equation with Peakon-Antipeakon Initial Data
• Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric
• Pseudolocalized Three-Dimensional Solitary Waves as Quasi-Particles
• Unique continuation results for abstract quasi-linear evolution equations in Banach spaces
• Global solutions of the Hunter-Saxton equation
• Orbital and asymptotic stability of a train of peakons for the Novikov equation
• Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity
• Geometric aspects of two- and threepeakons
• Shock creation and Painlevé property of colliding peakons in the Degasperis-Procesi Equation
• Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy
• On Questions of Decay and Existence for the Viscous Camassa-Holm Equations
• A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations
• Stability of periodic peakons for the Novikov equation
• Canonically conjugate variables for the periodic Camassa-Holm equation
• Discrete Variational Derivative Methods for the EPDiff equation
• Stability of peakons for a modified Camassa-Holm equation
• A reciprocal transformation for a 3-component Camassa-Holm type system
• On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
• Symmetries and multipeakon solutions for the modified two-component Camassa-Holm system
• Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities
• An inverse problem for the modified Camassa-Holm equation and multi-point Padé approximants
• Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function
• A four component cubic peakon (4CH) equations
• Existence and Stability of Traveling Waves for an Integro-differential Equation for Slow Erosion
• On the well-posedness of a weakly dispersive Boussinesq system
• Multi-soliton solution to the two-component Hunter-Saxton equation
• Miura Reciprocal transformation for two integrable hierarchies in 1+1 dimensions
• Convergence of the dispersion Camassa-Holm N-Soliton
• Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation
• Geometric Discretization of the EPDiff Equations
• Liouville correspondences between multi-component integrable hierarchies
• Large Time Behavior and Convergence for the Camassa-Holm Equations with Fractional Laplacian Viscosity
• Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data
• Dynamical behavior near self-similar blowup waves for the generalized b-equation
• Dynamics of conservative peakons in a system of Popowicz
• On the group of A-P diffeomorphisms and its exponential map
• An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation
• The Modified Camassa-Holm Equation: Wave breaking, Classification of Traveling Waves and Explicit Elliptic Peakons
• The Calogero-Françoise integrable system: algebraic geometry, Higgs fields, and the inverse problem
• A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon
• Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations
• Traveling surface waves of moderate amplitude in shallow water
• On cusped solitary waves in finite water depth
• Accelerating dynamical peakons and their behaviour
• A Lipschitz metric for the Camassa--Holm equation
• Global estimates and blow-up criteria for the generalized Hunter-Saxton system
• Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation
• Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation
• Bi-Hamiltonian Systems in (2+1) and Higher Dimensions Defined by Novikov Algebras
• The Replacement Rule for Nonlinear Shallow Water Waves
• Time-dependent defects in integrable soliton equations
• Application of the modified method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg - de Vries equation and generalized Camassa-Holm equation
• Local and global analyticity for a generalized Camassa-Holm system
• An alternative approach to the quasi-Periodic solutions of the Hunter-Saxton hierarchy
• Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space
• The Canada Day Theorem
• Self-Organized Networks, Darwinian Evolution of Self-Organized Networks, Darwinian Evolution of Dynein Rings, Stalks and Stalk Heads
• Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces
• On the Uniform Controllability of the Inviscid and Viscous Burgers-$\alpha$ Systems
• Local well-posedness and blow-up for a family of $U(1)$-invariant peakon equations
• Characterising arbitrary dark solitons in trapped one-dimensional Bose-Einstein condensates
• Global existence and blow-up phenomena for a periodic 2-component Camassa-Holm equation with vorticity
• Self-adjointness of a generalized Camassa-Holm equation
• Hölder Continuity of the Data to Solution Map for HR in the Weak Topology
• Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation
• On the Cauchy problem for a two-component Degasperis-Procesi system
• Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type
• Orbital stability of two-component peakons
• Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $\lambda$-family equations
• Similarity reductions of peakon equations: the $b$-family
• Stability of peakons of the Camassa-Holm equation beyond wave breaking
• The algebro-geometric solutions for Hunter-Saxton hierarchy
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
• Persistence Properties and Unique Continuation of solutions of the Camassa-Holm equation
• Classical Solutions of the Fornberg-Whitham Equation
• On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces
• The conservative Camassa-Holm flow with step-like irregular initial data
• On the Integrability of a Class of Nonlinear Dispersive Wave Equations
• Superconformal Algebras and Supersymmetric Integrable Flows
• An integrable pseudospherical equation with pseudo-peakon solutions
• Stochastic multisymplectic PDEs and their structure-preserving numerical methods
• Time-optimal control of a two-peakon collision
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• Integrable invariant Sobolev metrics on the Abelian extension of the diffeomorphism group of the circle and two-component generalizations of the Camassa-Holm equation
• The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces
• Analyticity of Riemannian exponential maps on ${\rm Diff}(\T)$
• The continuous dependence and non-uniform dependence of the rotation Camassa-Holm equation in Besov spaces
• The Sato Grassmannian and the CH hierarchy
• An explicit finite difference scheme for the Camassa-Holm equation
• On the Cauchy problem and asymptotic behavior for a three-component Novikov system
• Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation
• Global weak solutions for a two-component Novikov system
• Painleve II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit
• Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space
• Exterior Differential Systems, Prolongations and the Integrability of Two Nonlinear Partial Differential Equations
• Factorization Problem on the Hilbert-Schmidt Group and the Camassa-Holm Equation
• Local well-posedness and global existence for a multi-component Novikov equation
• Cauchy Biorthogonal Polynomials
• Short-Time Existence for Scale-Invariant Hamiltonian Waves
• On Calogero-Francoise-type Lax matrices and their dynamical r-matrices
• Conformal and geometric properties of the Camassa-Holm hierarchy
• Non-uniform dependence for the Novikov equation in Besov spaces
• Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations
• Generalized Heisenberg Ferromagnet type Equation and Modified Camassa-Holm Equation: Geometric Formulation, Soliton Solutions and Equivalence
• On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations
• Attractors for the viscous Camassa-Holm equation
• Some Remarks on the KP System of the Camassa-Holm Hierarchy
• Local and global analyticity for $\mu$-Camassa-Holm equations
• The modified Camassa-Holm equation in Lagrangian coordinates
• Global classification of two-component approximately integrable evolution equations
• The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid
• Stability of peakons for the Degasperis-Procesi equation
• Peakompactons: Peaked compact nonlinear waves
• Lipschitz metric for the Novikov equation
• A study of two new generalized negative KdV type equations
• Alpha-modeling strategy for LES of turbulent mixing
• R-matrix for a geodesic flow associated with a new integrable peakon equation
• Algebro-Geometric Solutions of the Camassa--Holm hierarchy
• Real-Valued Algebro-Geometric Solutions of the Camassa--Holm hierarchy
• Third order differential equations and local isometric immersions of pseudospherical surfaces
• Integrable hierarchy, $3\times 3$ constrained systems, and parametric and peaked stationary solutions
• The blow-up phenomena and exponential decay of solutions for a three-component Camassa-Holm equations
• The Cauchy problem for a higher order shallow water type equation on the circle
• Riemannian geometry of the contactomorphism group
• Least action principle for an integrable shallow water equation
• Integrable vs Nonintegrable Geodesic Soliton Behavior
• Acoustic Scattering and the Extended Korteweg deVries hierarchy
• The scattering approach for the Camassa-Holm equation
• On the Pseudolocalized Solutions in Multi-dimension of Boussinesq Equation
• On Asymptotic Variational Wave Equations
• Spatial asymptotic expansions in the incompressible Euler equation
• Well-posedness and persistence properties for two-component higher order Camassa-Holm systems with fractional inertia operator
• Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
• The Cauchy problem for higher-order modified Camassa-Holm equations on the circle