Nonlinear Dynamics, ISSN 0924-090X, 9/2017, Volume 89, Issue 4, pp. 2979 - 2994

By using the multiplier approach, we construct the conservation laws and the corresponding conserved quantities for the modified Camassa–Holm equation and...

Engineering | Vibration, Dynamical Systems, Control | Classical Mechanics | Multiplier | Automotive Engineering | Camassa–Holm equation | Mechanical Engineering | Soliton solutions | Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation | Semi-inverse variational principle | TRAVELING-WAVE SOLUTIONS | MECHANICS | PARTIAL-DIFFERENTIAL-EQUATIONS | Zakharov-Kuznetsov-Benjamin-Bona-Mahoney equation | BOUSSINESQ EQUATIONS | SYSTEMS | EVOLUTION-EQUATIONS | Camassa-Holm equation | ENGINEERING, MECHANICAL | Environmental law | Conservation laws | Solitary waves

Engineering | Vibration, Dynamical Systems, Control | Classical Mechanics | Multiplier | Automotive Engineering | Camassa–Holm equation | Mechanical Engineering | Soliton solutions | Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation | Semi-inverse variational principle | TRAVELING-WAVE SOLUTIONS | MECHANICS | PARTIAL-DIFFERENTIAL-EQUATIONS | Zakharov-Kuznetsov-Benjamin-Bona-Mahoney equation | BOUSSINESQ EQUATIONS | SYSTEMS | EVOLUTION-EQUATIONS | Camassa-Holm equation | ENGINEERING, MECHANICAL | Environmental law | Conservation laws | Solitary waves

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 12/2011, Volume 269, Issue 3, pp. 1137 - 1153

In this paper we present a geometric interpretation of the Degasperis–Procesi (DP) equation as the geodesic flow of a right-invariant symmetric linear...

Euler equation | 58D05 | Mathematics, general | Diffeomorphisms group of the circle | Mathematics | Degasperis–Procesi equation | 35Q53 | Degasperis-Procesi equation | INTEGRABILITY | DIFFEOMORPHISM GROUP | CAMASSA-HOLM EQUATION | WELL-POSEDNESS | SHALLOW-WATER EQUATION | SHOCK-WAVES | MATHEMATICS | MOTION | GEODESIC-FLOW | BLOW-UP PHENOMENA | WAVE BREAKING | Mathematical Physics | Physics

Euler equation | 58D05 | Mathematics, general | Diffeomorphisms group of the circle | Mathematics | Degasperis–Procesi equation | 35Q53 | Degasperis-Procesi equation | INTEGRABILITY | DIFFEOMORPHISM GROUP | CAMASSA-HOLM EQUATION | WELL-POSEDNESS | SHALLOW-WATER EQUATION | SHOCK-WAVES | MATHEMATICS | MOTION | GEODESIC-FLOW | BLOW-UP PHENOMENA | WAVE BREAKING | Mathematical Physics | Physics

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 03/2013, Volume 219, Issue 14, pp. 7480 - 7492

A version of the method of the simplest equation called modified method of simplest equation is applied to the extended Korteweg–de Vries equation and to...

Water waves | Method of simplest equation | Exact traveling-wave solutions | Generalized Camassa–Holm equation | Extended Korteweg–de Vries equation | Generalized Camassa-Holm equation | Extended Korteweg-de Vries equation | SYSTEM | MATHEMATICS, APPLIED | INFINITE PRANDTL NUMBER | SINE-GORDON EQUATION | PDES | MODEL | FLUID LAYER | UPPER-BOUNDS | DYNAMICS | ECOLOGY | CONVECTIVE HEAT-TRANSPORT | Methods

Water waves | Method of simplest equation | Exact traveling-wave solutions | Generalized Camassa–Holm equation | Extended Korteweg–de Vries equation | Generalized Camassa-Holm equation | Extended Korteweg-de Vries equation | SYSTEM | MATHEMATICS, APPLIED | INFINITE PRANDTL NUMBER | SINE-GORDON EQUATION | PDES | MODEL | FLUID LAYER | UPPER-BOUNDS | DYNAMICS | ECOLOGY | CONVECTIVE HEAT-TRANSPORT | Methods

Journal Article

Computer Physics Communications, ISSN 0010-4655, 2011, Volume 182, Issue 3, pp. 616 - 627

In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary...

Multi-symplectic | Wavelet collocation method | Camassa–Holm equation | Nonlinear Schrödinger equation | Camassa-Holm equation | Nonlinear Schrodinger equation | HIGH-ORDER | PHYSICS, MATHEMATICAL | TRIGONOMETRICALLY-FITTED FORMULAS | SCHEME | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | PARTIAL-DIFFERENTIAL EQUATIONS | CONVERGENCE | Energy conservation | Analysis | Methods

Multi-symplectic | Wavelet collocation method | Camassa–Holm equation | Nonlinear Schrödinger equation | Camassa-Holm equation | Nonlinear Schrodinger equation | HIGH-ORDER | PHYSICS, MATHEMATICAL | TRIGONOMETRICALLY-FITTED FORMULAS | SCHEME | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | PARTIAL-DIFFERENTIAL EQUATIONS | CONVERGENCE | Energy conservation | Analysis | Methods

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2017, Volume 262, Issue 1, pp. 506 - 558

Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order...

Integrable system | Initial–boundary value problem | Dirichlet-to-Neumann map | Riemann–Hilbert problem | Coupled nonlinear Schrödinger equation | MKDV EQUATION | INTEGRABILITY | CAMASSA-HOLM EQUATION | LONG-TIME ASYMPTOTICS | SINE-GORDON EQUATION | EVOLUTION-EQUATIONS | MATHEMATICS | Riemann-Hilbert problem | Coupled nonlinear Schrodinger equation | SOLITONS | HALF-LINE | Initial-boundary value problem | X-3 LAX PAIRS | Analysis | Methods | Differential equations

Integrable system | Initial–boundary value problem | Dirichlet-to-Neumann map | Riemann–Hilbert problem | Coupled nonlinear Schrödinger equation | MKDV EQUATION | INTEGRABILITY | CAMASSA-HOLM EQUATION | LONG-TIME ASYMPTOTICS | SINE-GORDON EQUATION | EVOLUTION-EQUATIONS | MATHEMATICS | Riemann-Hilbert problem | Coupled nonlinear Schrodinger equation | SOLITONS | HALF-LINE | Initial-boundary value problem | X-3 LAX PAIRS | Analysis | Methods | Differential equations

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 10/2016, Volume 31, pp. 388 - 408

Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using...

The (2+1)-dimensional Boussinesq equation | Bell’s polynomials | Soliton solution | Periodic wave solution | Infinite conservation laws | Bäcklund transformation | Bell's polynomials | POLYNOMIALS | MATHEMATICS, APPLIED | CAMASSA-HOLM EQUATION | Backlund transformation | RATIONAL CHARACTERISTICS | Environmental law

The (2+1)-dimensional Boussinesq equation | Bell’s polynomials | Soliton solution | Periodic wave solution | Infinite conservation laws | Bäcklund transformation | Bell's polynomials | POLYNOMIALS | MATHEMATICS, APPLIED | CAMASSA-HOLM EQUATION | Backlund transformation | RATIONAL CHARACTERISTICS | Environmental law

Journal Article

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Bilinear equations and new multi-soliton solution for the modified Camassa–Holm equation

Applied Mathematics Letters, ISSN 0893-9659, 09/2016, Volume 59, pp. 18 - 23

The bilinear equations, which directly derive the modified Camassa–Holm equation, are given from the reduction of the extended KP hierarchy with negative flow....

Soliton solution | Modified Camassa–Holm equation | Bilinear equation | Modified Camassa-Holm equation | MATHEMATICS, APPLIED | SOLITONS | BACKLUND-TRANSFORMATIONS | HEREDITARY SYMMETRIES | Functions (mathematics) | Reduction | Hierarchies | Fluid dynamics | Partial differential equations | Mathematical analysis | Byproducts | Determinants

Soliton solution | Modified Camassa–Holm equation | Bilinear equation | Modified Camassa-Holm equation | MATHEMATICS, APPLIED | SOLITONS | BACKLUND-TRANSFORMATIONS | HEREDITARY SYMMETRIES | Functions (mathematics) | Reduction | Hierarchies | Fluid dynamics | Partial differential equations | Mathematical analysis | Byproducts | Determinants

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2017, Volume 451, Issue 2, pp. 990 - 1025

In this study, we consider the Cauchy problem for the second-order Camassa–Holm equation with periodic initial data . Using the vanishing viscosity method, the...

Continuous semigroup | Camassa–Holm equation | Periodic second-order | Lipschitz metric | Local weak solution | Relabeling | MATHEMATICS | MATHEMATICS, APPLIED | WELL-POSEDNESS | Camassa-Holm equation | SHALLOW-WATER EQUATION | GLOBAL WEAK SOLUTIONS | CONSERVATIVE SOLUTIONS

Continuous semigroup | Camassa–Holm equation | Periodic second-order | Lipschitz metric | Local weak solution | Relabeling | MATHEMATICS | MATHEMATICS, APPLIED | WELL-POSEDNESS | Camassa-Holm equation | SHALLOW-WATER EQUATION | GLOBAL WEAK SOLUTIONS | CONSERVATIVE SOLUTIONS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 02/2019, Volume 470, Issue 1, pp. 647 - 658

For the Fokas–Olver–Rosenau–Qiao equation (FORQ), on both the line and the circle, it is proved that there exist initial data in the Sobolev spaces , with ,...

Well-posedness in Sobolev spaces | Peakon solutions | Non-uniqueness and ill-posedness | Fokas–Olver–Rosenau–Qiao equation | Integrable Camassa–Holm type equations | Cauchy problem | DEGASPERIS-PROCESI EQUATION | NOVIKOV EQUATION | MATHEMATICS, APPLIED | CAMASSA-HOLM EQUATION | CAUCHY-PROBLEM | ILL-POSEDNESS | SHALLOW-WATER EQUATION | Fokas-Olver-Rosenau-Qiao equation | FAMILY | MATHEMATICS | TRAVELING-WAVE SOLUTIONS | SOLITONS | Integrable Camassa Holm type equations

Well-posedness in Sobolev spaces | Peakon solutions | Non-uniqueness and ill-posedness | Fokas–Olver–Rosenau–Qiao equation | Integrable Camassa–Holm type equations | Cauchy problem | DEGASPERIS-PROCESI EQUATION | NOVIKOV EQUATION | MATHEMATICS, APPLIED | CAMASSA-HOLM EQUATION | CAUCHY-PROBLEM | ILL-POSEDNESS | SHALLOW-WATER EQUATION | Fokas-Olver-Rosenau-Qiao equation | FAMILY | MATHEMATICS | TRAVELING-WAVE SOLUTIONS | SOLITONS | Integrable Camassa Holm type equations

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 07/2018, Volume 336, pp. 317 - 337

In the present paper, a three-step iterative algorithm for solving a two-component Camassa–Holm (2CH) equation is presented. In the first step, the...

Combined compact difference scheme | Inhomogeneous Helmholtz equation | Two-component Camassa–Holm equation | peakon–antipeakon | Hamiltonians | Casimir function | Two-component Camassa-Holm equation | SYSTEM | MATHEMATICS, APPLIED | FINITE-DIFFERENCE SCHEMES | SOLITONS | peakon-antipeakon | Algorithms

Combined compact difference scheme | Inhomogeneous Helmholtz equation | Two-component Camassa–Holm equation | peakon–antipeakon | Hamiltonians | Casimir function | Two-component Camassa-Holm equation | SYSTEM | MATHEMATICS, APPLIED | FINITE-DIFFERENCE SCHEMES | SOLITONS | peakon-antipeakon | Algorithms

Journal Article

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Asymptotic expansions and solitons of the Camassa–Holm – nonlinear Schrödinger equation

Physics Letters A, ISSN 0375-9601, 12/2017, Volume 381, Issue 48, pp. 3965 - 3971

We study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa–Holm NLS, hereafter referred to as CH–NLS equation. We...

Camassa–Holm NLS | Multiscale expansion | Dark solitons | Antidark solitons | PHYSICS, MULTIDISCIPLINARY | INVERSE SCATTERING TRANSFORM | AMPLITUDE | INSTABILITIES | SOLITARY WAVES | SYSTEMS | MEDIA | DARK OPTICAL SOLITONS | KORTEWEG-DE-VRIES | Camassa Holm NLS | Mechanical engineering | Analysis | Numerical analysis

Camassa–Holm NLS | Multiscale expansion | Dark solitons | Antidark solitons | PHYSICS, MULTIDISCIPLINARY | INVERSE SCATTERING TRANSFORM | AMPLITUDE | INSTABILITIES | SOLITARY WAVES | SYSTEMS | MEDIA | DARK OPTICAL SOLITONS | KORTEWEG-DE-VRIES | Camassa Holm NLS | Mechanical engineering | Analysis | Numerical analysis

Journal Article

ANALYSIS AND APPLICATIONS, ISSN 0219-5305, 01/2007, Volume 5, Issue 1, pp. 1 - 27

This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent...

MATHEMATICS | BREAKING | MATHEMATICS, APPLIED | conservation law | BANACH-SPACES | WAVE-EQUATION | dissipative solutions | non-local source | WEAK SOLUTIONS | Camassa-Holm equation | SHALLOW-WATER EQUATION | CONSERVATIVE SOLUTIONS

MATHEMATICS | BREAKING | MATHEMATICS, APPLIED | conservation law | BANACH-SPACES | WAVE-EQUATION | dissipative solutions | non-local source | WEAK SOLUTIONS | Camassa-Holm equation | SHALLOW-WATER EQUATION | CONSERVATIVE SOLUTIONS

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 09/2018, Volume 62, pp. 378 - 385

In this paper, a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCHKP) equation is investigated, which describes the role of dispersion in...

Solitary wave | Breather wave | A generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation | Homoclinic breather limit approach | Rogue wave | SCHRODINGER-EQUATIONS | MATHEMATICS, APPLIED | SYMMETRIES | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Camassa-Holm-Kadomtsev-Petviashvili equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SOLITONS | DYNAMICS | A generalized (2+1)-dimensional | RATIONAL CHARACTERISTICS | EXPLICIT | Water waves

Solitary wave | Breather wave | A generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation | Homoclinic breather limit approach | Rogue wave | SCHRODINGER-EQUATIONS | MATHEMATICS, APPLIED | SYMMETRIES | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Camassa-Holm-Kadomtsev-Petviashvili equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SOLITONS | DYNAMICS | A generalized (2+1)-dimensional | RATIONAL CHARACTERISTICS | EXPLICIT | Water waves

Journal Article

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Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Journal of Differential Equations, ISSN 0022-0396, 10/2017, Volume 263, Issue 7, pp. 4355 - 4381

In this paper, we prove well-posedness of the Fornberg–Whitham equation in Besov spaces in both the periodic and non-periodic cases. This will imply the...

DEGASPERIS-PROCESI EQUATION | SOLUTION MAP | MATHEMATICS | NONUNIFORM DEPENDENCE | CAMASSA-HOLM EQUATION | CH EQUATION | STOKES WAVES | TRAJECTORIES | CAUCHY-PROBLEM | INITIAL DATA | SHALLOW-WATER EQUATION

DEGASPERIS-PROCESI EQUATION | SOLUTION MAP | MATHEMATICS | NONUNIFORM DEPENDENCE | CAMASSA-HOLM EQUATION | CH EQUATION | STOKES WAVES | TRAJECTORIES | CAUCHY-PROBLEM | INITIAL DATA | SHALLOW-WATER EQUATION

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2019, Volume 346, pp. 86 - 95

In this paper, we develop two linear conservative Fourier pseudo-spectral schemes for the Camassa–Holm equation. We first apply the Fourier pseudo-spectral...

Camassa–Holm equation | Fourier pseudo-spectral method | Momentum-preserving | Linear conservative scheme | MATHEMATICS, APPLIED | PRESERVING ALGORITHMS | INTEGRATION | CONVERGENCE | Camassa-Holm equation | FINITE-DIFFERENCE SCHEME | SHALLOW-WATER EQUATION | MULTI-SYMPLECTIC FORMULATIONS

Camassa–Holm equation | Fourier pseudo-spectral method | Momentum-preserving | Linear conservative scheme | MATHEMATICS, APPLIED | PRESERVING ALGORITHMS | INTEGRATION | CONVERGENCE | Camassa-Holm equation | FINITE-DIFFERENCE SCHEME | SHALLOW-WATER EQUATION | MULTI-SYMPLECTIC FORMULATIONS

Journal Article

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, 05/2018, Volume 474, Issue 2213, p. 20180052

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincare equations defined on...

Hunter-saxton equation | Multisymplectic partial differential equations | Camassa-holm equation | Korteweg-de vries equation | Variational principles | Virasoro-bott group | multisymplectic partial differential equations | Virasoro-Bott group | MULTIDISCIPLINARY SCIENCES | POISSON BRACKETS | SHALLOW-WATER EQUATION | PRINCIPLES | Korteweg-de Vries equation | WAVES | MECHANICS | Hunter-Saxton equation | variational principles | DYNAMICS | KDV EQUATION | Camassa-Holm equation | MULTI-SYMPLECTIC INTEGRATION | SCHEMES | 1008 | Camassa–Holm equation | Hunter–Saxton equation | 120 | Virasoro–Bott group

Hunter-saxton equation | Multisymplectic partial differential equations | Camassa-holm equation | Korteweg-de vries equation | Variational principles | Virasoro-bott group | multisymplectic partial differential equations | Virasoro-Bott group | MULTIDISCIPLINARY SCIENCES | POISSON BRACKETS | SHALLOW-WATER EQUATION | PRINCIPLES | Korteweg-de Vries equation | WAVES | MECHANICS | Hunter-Saxton equation | variational principles | DYNAMICS | KDV EQUATION | Camassa-Holm equation | MULTI-SYMPLECTIC INTEGRATION | SCHEMES | 1008 | Camassa–Holm equation | Hunter–Saxton equation | 120 | Virasoro–Bott group

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 03/2016, Volume 133, pp. 161 - 199

The initial value problem for a novel 4-parameter family of evolution equations, which are nonlinear and nonlocal and possess peakon traveling wave solutions,...

Well-posedness in Sobolev spaces | Novikov equation | Integrable equations | Camassa–Holm equation | Commutator estimate | Approximate solutions | Cauchy problem | Conserved quantities | Degasperis–Procesi equation | Fokas–Olver–Rosenau–Qiao equation | Non-uniform dependence on initial data | Hölder continuity | Peakon traveling waves | Fokas-Olver-Rosenau-Qiao equation | Degasperis-Procesi equation | Camassa-Holm equation | MATHEMATICS, APPLIED | CH EQUATION | STABILITY | WELL-POSEDNESS | SHALLOW-WATER EQUATION | MATHEMATICS | Holder continuity | SOLITONS | ROSENAU-QIAO EQUATION | NONUNIFORM DEPENDENCE | BLOW-UP PHENOMENA | Sobolev space | Mathematical analysis | Norms | Traveling waves | Nonlinearity | Evolution | Initial value problems

Well-posedness in Sobolev spaces | Novikov equation | Integrable equations | Camassa–Holm equation | Commutator estimate | Approximate solutions | Cauchy problem | Conserved quantities | Degasperis–Procesi equation | Fokas–Olver–Rosenau–Qiao equation | Non-uniform dependence on initial data | Hölder continuity | Peakon traveling waves | Fokas-Olver-Rosenau-Qiao equation | Degasperis-Procesi equation | Camassa-Holm equation | MATHEMATICS, APPLIED | CH EQUATION | STABILITY | WELL-POSEDNESS | SHALLOW-WATER EQUATION | MATHEMATICS | Holder continuity | SOLITONS | ROSENAU-QIAO EQUATION | NONUNIFORM DEPENDENCE | BLOW-UP PHENOMENA | Sobolev space | Mathematical analysis | Norms | Traveling waves | Nonlinearity | Evolution | Initial value problems

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 11/2018, Volume 51, Issue 49, p. 495202

In this work, we study solitary waves in a (2 + 1)-dimensional variant of the defocusing nonlinear Schrodinger (NLS) equation, the so-called Camassa-Holm-NLS...

antidark solitons | dark solitons | Camassa-Holm-NLS | multiscale expansion | PHYSICS, MULTIDISCIPLINARY | RING DARK SOLITONS | MULTISCALE EXPANSIONS | SYSTEMS | MEDIA | AMPLITUDE | PHYSICS, MATHEMATICAL | Camassa-Hohn-NLS

antidark solitons | dark solitons | Camassa-Holm-NLS | multiscale expansion | PHYSICS, MULTIDISCIPLINARY | RING DARK SOLITONS | MULTISCALE EXPANSIONS | SYSTEMS | MEDIA | AMPLITUDE | PHYSICS, MATHEMATICAL | Camassa-Hohn-NLS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 10/2018, Volume 265, Issue 7, pp. 2886 - 2896

In this paper, the wave breaking of the Fornberg–Whitham equation is studied. We first investigate the local well-posedness and show a blow-up scenario by...

Blow-up scenario | Fornberg–Whitham equation | Wave breaking | MATHEMATICS | CAMASSA-HOLM | Fornberg-Whitham equation | CAUCHY-PROBLEM | WELL-POSEDNESS | FAMILY

Blow-up scenario | Fornberg–Whitham equation | Wave breaking | MATHEMATICS | CAMASSA-HOLM | Fornberg-Whitham equation | CAUCHY-PROBLEM | WELL-POSEDNESS | FAMILY

Journal Article

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