Nonlinear Dynamics, ISSN 0924-090X, 04/2018, Volume 92, Issue 2, pp. 221 - 234

Based on the Lax pair, the nonlocal symmetries to -dimensional Korteweg-de Vries equation are investigated, which are also constructed by the truncated...

Nonlocal symmetry | Similarity reduction | Interaction solutions | (2 + 1)-dimensional Korteweg–de Vries equation | INTEGRABLE MODELS | MECHANICS | SOLITONS | CRE SOLVABILITY | (2+1)-DIMENSIONAL KDV EQUATION | ENGINEERING, MECHANICAL | (2+1)-dimensional Korteweg-de Vries equation | BACKLUND TRANSFORMATION | Numerical analysis | Similarity | Computer simulation | Solitary waves | Symmetry

Nonlocal symmetry | Similarity reduction | Interaction solutions | (2 + 1)-dimensional Korteweg–de Vries equation | INTEGRABLE MODELS | MECHANICS | SOLITONS | CRE SOLVABILITY | (2+1)-DIMENSIONAL KDV EQUATION | ENGINEERING, MECHANICAL | (2+1)-dimensional Korteweg-de Vries equation | BACKLUND TRANSFORMATION | Numerical analysis | Similarity | Computer simulation | Solitary waves | Symmetry

Journal Article

Journal of King Saud University - Science, ISSN 1018-3647, 01/2019, Volume 31, Issue 1, pp. 8 - 13

In this article, the analytical solution of (3+1)-dimensional Korteweg-de Vries Benjamin–Bona–Mahony equation, Kadomtsev–Petviashvili Benjamin–Bona–Mahony...

Modified Korteweg-de Vries–Zakharov–Kuznetsov equation | Korteweg-de Vries Benjamin–Bona–Mahony equation | Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation | MULTIDISCIPLINARY SCIENCES | Korteweg-de Vries Benjamin-Bona-Mahony equation | ION-ACOUSTIC-WAVES | Modified Korteweg-de Vries-Zakharov-Kuznetsov equation | NONLINEAR SCHRODINGER-EQUATION | Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation | KDV-MKDV EQUATION

Modified Korteweg-de Vries–Zakharov–Kuznetsov equation | Korteweg-de Vries Benjamin–Bona–Mahony equation | Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation | MULTIDISCIPLINARY SCIENCES | Korteweg-de Vries Benjamin-Bona-Mahony equation | ION-ACOUSTIC-WAVES | Modified Korteweg-de Vries-Zakharov-Kuznetsov equation | NONLINEAR SCHRODINGER-EQUATION | Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation | KDV-MKDV EQUATION

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 08/2016, Volume 455, pp. 44 - 51

The nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation governs the behavior of weakly nonlinear ion-acoustic waves...

Magnetized electron–positron plasma | Modified Korteweg–de Vries–Zakharov–Kuznetsov equation | Ion-acoustic waves | Fractional extended direct algebraic method | Magnetized electron-positron plasma | Modified Korteweg-de Vries-Zakharov-Kuznetsov equation | PHYSICS, MULTIDISCIPLINARY | ACOUSTIC SOLITARY WAVES | BURGERS EQUATION | DOUBLE-LAYERS | INSTABILITIES | Electric fields | Electric potential | Perturbation methods | Mathematical analysis | Traveling waves | Nonlinearity | Electrostatic fields | Electron-positron plasmas | Three dimensional

Magnetized electron–positron plasma | Modified Korteweg–de Vries–Zakharov–Kuznetsov equation | Ion-acoustic waves | Fractional extended direct algebraic method | Magnetized electron-positron plasma | Modified Korteweg-de Vries-Zakharov-Kuznetsov equation | PHYSICS, MULTIDISCIPLINARY | ACOUSTIC SOLITARY WAVES | BURGERS EQUATION | DOUBLE-LAYERS | INSTABILITIES | Electric fields | Electric potential | Perturbation methods | Mathematical analysis | Traveling waves | Nonlinearity | Electrostatic fields | Electron-positron plasmas | Three dimensional

Journal Article

Pramana, ISSN 0304-4289, 10/2015, Volume 85, Issue 4, pp. 583 - 592

This paper presents the exact solutions for the fractional Korteweg–de Vries equations and the coupled Korteweg–de Vries equations with time-fractional...

Astrophysics and Astroparticles | Korteweg–de Vries equation | functional variable method | Physics, general | coupled Korteweg–de Vries equation | Physics | Astronomy, Observations and Techniques | Korteweg-de Vries equation | Coupled Korteweg-de Vries equation | Functional variable method | WAVE | SOLITON-SOLUTIONS | 1ST INTEGRAL METHOD | PHYSICS, MULTIDISCIPLINARY | coupled Korteweg-de Vries equation | PARTIAL-DIFFERENTIAL-EQUATIONS | KDV EQUATION

Astrophysics and Astroparticles | Korteweg–de Vries equation | functional variable method | Physics, general | coupled Korteweg–de Vries equation | Physics | Astronomy, Observations and Techniques | Korteweg-de Vries equation | Coupled Korteweg-de Vries equation | Functional variable method | WAVE | SOLITON-SOLUTIONS | 1ST INTEGRAL METHOD | PHYSICS, MULTIDISCIPLINARY | coupled Korteweg-de Vries equation | PARTIAL-DIFFERENTIAL-EQUATIONS | KDV EQUATION

Journal Article

Results in Physics, ISSN 2211-3797, 2017, Volume 7, pp. 1143 - 1149

The Boussinesq equation with dual dispersion and modified Korteweg–de Vries–Kadomtsev–Petviashvili equations describe weakly dispersive and small amplitude...

Modified Korteweg–de Vries–Kadomtsev–Petviashvili equation | Breaking soliton equation | Boussinesq equation

Modified Korteweg–de Vries–Kadomtsev–Petviashvili equation | Breaking soliton equation | Boussinesq equation

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 07/2013, Volume 82, Issue 283, pp. 1401 - 1432

-norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial...

Mathematical problems | Error rates | Simulations | Approximation | Mathematics | Polynomials | Numerical schemes | Legendre polynomials | Waves | Galerkin methods | Conservation laws | Korteweg-de Vries equation | Error estimates | Discontinuous Galerkin methods | MATHEMATICS, APPLIED | NONLINEAR DISPERSIVE WAVES | DEVRIES EQUATION | HIGH-ORDER | EVOLUTION-EQUATIONS | MODEL EQUATIONS | conservation laws | SOLITARY WAVES | PARTIAL-DIFFERENTIAL-EQUATIONS | error estimates | FINITE-ELEMENT-METHOD | BBM-EQUATION | NUMERICAL ASPECTS

Mathematical problems | Error rates | Simulations | Approximation | Mathematics | Polynomials | Numerical schemes | Legendre polynomials | Waves | Galerkin methods | Conservation laws | Korteweg-de Vries equation | Error estimates | Discontinuous Galerkin methods | MATHEMATICS, APPLIED | NONLINEAR DISPERSIVE WAVES | DEVRIES EQUATION | HIGH-ORDER | EVOLUTION-EQUATIONS | MODEL EQUATIONS | conservation laws | SOLITARY WAVES | PARTIAL-DIFFERENTIAL-EQUATIONS | error estimates | FINITE-ELEMENT-METHOD | BBM-EQUATION | NUMERICAL ASPECTS

Journal Article

Automatica, ISSN 0005-1098, 01/2018, Volume 87, pp. 210 - 217

This paper presents an output feedback control law for the Korteweg–de Vries equation. The control design is based on the backstepping method and the...

Backstepping | Nonlinear system | Output feedback | Korteweg–de Vries equation | Korteweg-de Vries equation | LINEAR KDV EQUATION | NULL CONTROLLABILITY | GLOBAL STABILIZATION | Backstepping Nonlinear system | SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Numerical analysis | Mathematics | Engineering Sciences | Analysis of PDEs | Automatic

Backstepping | Nonlinear system | Output feedback | Korteweg–de Vries equation | Korteweg-de Vries equation | LINEAR KDV EQUATION | NULL CONTROLLABILITY | GLOBAL STABILIZATION | Backstepping Nonlinear system | SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Numerical analysis | Mathematics | Engineering Sciences | Analysis of PDEs | Automatic

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2007, Volume 187, Issue 2, pp. 1368 - 1372

With the help of the symbolic computation system Maple, we present Korteweg–de Vries equation-based sub-equation method. Being concise and straightforward, it...

(2 + 1)-dimensional Korteweg–de Vries equation | Korteweg–de Vries equation-based sub-equation method | Nonlinear partial differential equations | N-soliton solution | (2 + 1)-dimensional Korteweg-de Vries equation | Korteweg-de Vries equation-based sub-equation method | MATHEMATICS, APPLIED | BREAKING SOLITON EQUATION | EXPLICIT EXACT-SOLUTIONS | DIFFERENTIAL-EQUATIONS | RATIONAL EXPANSION METHOD | PERIODIC-WAVE SOLUTIONS | HOMOGENEOUS BALANCE METHOD | nonlinear partial differential equations | SYMBOLIC COMPUTATION | BURGERS-EQUATION | NONLINEAR EVOLUTION-EQUATIONS | TANH-FUNCTION METHOD | (2+1)-dimensional Korteweg-de Vries equation

(2 + 1)-dimensional Korteweg–de Vries equation | Korteweg–de Vries equation-based sub-equation method | Nonlinear partial differential equations | N-soliton solution | (2 + 1)-dimensional Korteweg-de Vries equation | Korteweg-de Vries equation-based sub-equation method | MATHEMATICS, APPLIED | BREAKING SOLITON EQUATION | EXPLICIT EXACT-SOLUTIONS | DIFFERENTIAL-EQUATIONS | RATIONAL EXPANSION METHOD | PERIODIC-WAVE SOLUTIONS | HOMOGENEOUS BALANCE METHOD | nonlinear partial differential equations | SYMBOLIC COMPUTATION | BURGERS-EQUATION | NONLINEAR EVOLUTION-EQUATIONS | TANH-FUNCTION METHOD | (2+1)-dimensional Korteweg-de Vries equation

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 09/2017, Volume 50, Issue 39, p. 395204

In this paper, we implement the Fokas method in order to study initial boundary value problems of the coupled modified Korteweg-de Vries equation formulated on...

Riemann-Hilbert problem | integrable system | initial-boundary value problem | Dirichlet to Neumann map | coupled modified Korteweg-de Vries equation | INTERVAL | PHYSICS, MULTIDISCIPLINARY | DEVRIES EQUATION | MODIFIED KDV EQUATIONS | SINE-GORDON EQUATION | PDES | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | UNIFIED TRANSFORM METHOD | X-3 LAX PAIRS

Riemann-Hilbert problem | integrable system | initial-boundary value problem | Dirichlet to Neumann map | coupled modified Korteweg-de Vries equation | INTERVAL | PHYSICS, MULTIDISCIPLINARY | DEVRIES EQUATION | MODIFIED KDV EQUATIONS | SINE-GORDON EQUATION | PDES | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | UNIFIED TRANSFORM METHOD | X-3 LAX PAIRS

Journal Article

Modern Physics Letters A, ISSN 0217-7323, 12/2018, Volume 33, Issue 37, p. 1850217

In this work, we consider the propagation of one-dimensional nonlinear unmagnetized dusty plasma, by using the reductive perturbation technique to formulate...

Unmagnetized dusty plasma | Dust and ion solitary wave solutions | Electrostatic potential and pressure | Further modified Korteweg-de Vries equation | Mathematical methods | BURGERS EQUATION | SCHRODINGER-EQUATION | PHYSICS, NUCLEAR | PHYSICS, MATHEMATICAL | electrostatic potential and pressure | STABILITY ANALYSIS | HIGHER-ORDER | mathematical methods | ASTRONOMY & ASTROPHYSICS | unmagnetized dusty plasma | dust and ion solitary wave solutions | BRIGHT | PHYSICS, PARTICLES & FIELDS | Plasma physics

Unmagnetized dusty plasma | Dust and ion solitary wave solutions | Electrostatic potential and pressure | Further modified Korteweg-de Vries equation | Mathematical methods | BURGERS EQUATION | SCHRODINGER-EQUATION | PHYSICS, NUCLEAR | PHYSICS, MATHEMATICAL | electrostatic potential and pressure | STABILITY ANALYSIS | HIGHER-ORDER | mathematical methods | ASTRONOMY & ASTROPHYSICS | unmagnetized dusty plasma | dust and ion solitary wave solutions | BRIGHT | PHYSICS, PARTICLES & FIELDS | Plasma physics

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 10/2016, Volume 333, pp. 99 - 106

Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation....

Perturbation theory | Korteweg–de Vries equation | Whitham modulation theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | INSTABILITY | PHYSICS, MULTIDISCIPLINARY | DEVRIES EQUATION | WAVETRAINS | INTEGRABLE EQUATIONS | PHYSICS, MATHEMATICAL | LINEAR DISPERSIVE WAVES | Korteweg-de Vries equation | TRAINS | SOLITARY WAVES | MODULATION | WATER | Physics - Pattern Formation and Solitons

Perturbation theory | Korteweg–de Vries equation | Whitham modulation theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | INSTABILITY | PHYSICS, MULTIDISCIPLINARY | DEVRIES EQUATION | WAVETRAINS | INTEGRABLE EQUATIONS | PHYSICS, MATHEMATICAL | LINEAR DISPERSIVE WAVES | Korteweg-de Vries equation | TRAINS | SOLITARY WAVES | MODULATION | WATER | Physics - Pattern Formation and Solitons

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 09/2016, Volume 321, pp. 776 - 796

The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude...

Discontinuous Galerkin method | Stability | Korteweg–de Vries equation | Conservation | Korteweg-de Vries equation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RECURRENCE | CONSERVATION-LAWS | PHYSICS, MATHEMATICAL | FINITE-ELEMENT-METHOD | ENERGY-CONSERVATION | SCHEMES | Analysis | Methods | Algorithms

Discontinuous Galerkin method | Stability | Korteweg–de Vries equation | Conservation | Korteweg-de Vries equation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RECURRENCE | CONSERVATION-LAWS | PHYSICS, MATHEMATICAL | FINITE-ELEMENT-METHOD | ENERGY-CONSERVATION | SCHEMES | Analysis | Methods | Algorithms

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 04/2019, Volume 77, Issue 8, pp. 2255 - 2262

In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation...

(2+1)-dimensional Korteweg–de Vries equation | Improved fractional sub-equation method | Modified Riemann–Liouville derivative | (2+1)-dimensional Burgers equation | TRANSFORMATION | ORDER | MATHEMATICS, APPLIED | CALCULUS | Modified Riemann-Liouville derivative | DIFFERENTIAL-EQUATIONS | (2+1)-dimensional Korteweg-de Vries equation | Graphical representations | Linear equations | Burgers equation | Solitary waves | Exact solutions

(2+1)-dimensional Korteweg–de Vries equation | Improved fractional sub-equation method | Modified Riemann–Liouville derivative | (2+1)-dimensional Burgers equation | TRANSFORMATION | ORDER | MATHEMATICS, APPLIED | CALCULUS | Modified Riemann-Liouville derivative | DIFFERENTIAL-EQUATIONS | (2+1)-dimensional Korteweg-de Vries equation | Graphical representations | Linear equations | Burgers equation | Solitary waves | Exact solutions

Journal Article

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ISSN 0363-0129, 2019, Volume 57, Issue 4, pp. 2467 - 2486

We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a...

Korteweg-de Vries equation | MATHEMATICS, APPLIED | flatness approach | NULL CONTROLLABILITY | BOUNDARY CONTROLLABILITY | STABILIZATION | 1D SCHRODINGER-EQUATION | exact controllability | controllability to the trajectories | Gevrey class | AUTOMATION & CONTROL SYSTEMS | smoothing effect | Analysis of PDEs | Mathematics

Korteweg-de Vries equation | MATHEMATICS, APPLIED | flatness approach | NULL CONTROLLABILITY | BOUNDARY CONTROLLABILITY | STABILIZATION | 1D SCHRODINGER-EQUATION | exact controllability | controllability to the trajectories | Gevrey class | AUTOMATION & CONTROL SYSTEMS | smoothing effect | Analysis of PDEs | Mathematics

Journal Article

Automatica, ISSN 0005-1098, 02/2019, Volume 100, pp. 260 - 273

We consider distributed stabilization of 1-D Korteweg–de Vries–Burgers (KdVB) equation in the presence of constant input delay. The delay may be uncertain, but...

Distributed control | LMI | Lyapunov–Krasovskii method | Korteweg–de Vries–Burgers equation | Time delay | FINITE DETERMINING PARAMETERS | NONLINEAR DISSIPATIVE SYSTEMS | FEEDBACK-CONTROL | Lyapunov-Krasovskii method | STABILITY | Korteweg-de Vries-Burgers equation | BOUNDARY CONTROL | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC

Distributed control | LMI | Lyapunov–Krasovskii method | Korteweg–de Vries–Burgers equation | Time delay | FINITE DETERMINING PARAMETERS | NONLINEAR DISSIPATIVE SYSTEMS | FEEDBACK-CONTROL | Lyapunov-Krasovskii method | STABILITY | Korteweg-de Vries-Burgers equation | BOUNDARY CONTROL | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC

Journal Article

Physics Reports, ISSN 0370-1573, 02/2013, Volume 523, Issue 2, pp. 61 - 126

In the past years there was a huge interest in experimental and theoretical studies in the area of few-optical-cycle pulses and in the broader fast growing...

Few-cycle dissipative solitons | Generalized Kadomtsev–Petviashvili equation | Two-level atoms | Circular polarization | Maxwell–Bloch equations | Few-optical-cycle solitons | Modified Korteweg–de Vries equation | Density matrix | Long-wave approximation | Few-cycle pulses | Reductive perturbation method | Half-cycle optical solitons | Short-wave approximation | Unipolar pulses | Few-cycle light bullets | Linear polarization | Sine–Gordon equation | Complex modified Korteweg–de Vries equation | Modified Korteweg-de Vries equation | Generalized Kadomtsev-Petviashvili equation | Complex modified Korteweg-de Vries equation | Sine-Gordon equation | Maxwell-Bloch equations | ULTRA-SHORT PULSES | SELF-INDUCED TRANSPARENCY | SOLITARY-WAVE SOLUTIONS | QUADRATIC NONLINEAR MEDIA | PHYSICS, MULTIDISCIPLINARY | DE-VRIES EQUATION | KADOMTSEV-PETVIASHVILI EQUATION | SINE-GORDON EQUATIONS | TI-SAPPHIRE LASER | SHORT ELECTROMAGNETIC PULSES | SHORT-PULSE EQUATION | Analysis | Models | Wave propagation

Few-cycle dissipative solitons | Generalized Kadomtsev–Petviashvili equation | Two-level atoms | Circular polarization | Maxwell–Bloch equations | Few-optical-cycle solitons | Modified Korteweg–de Vries equation | Density matrix | Long-wave approximation | Few-cycle pulses | Reductive perturbation method | Half-cycle optical solitons | Short-wave approximation | Unipolar pulses | Few-cycle light bullets | Linear polarization | Sine–Gordon equation | Complex modified Korteweg–de Vries equation | Modified Korteweg-de Vries equation | Generalized Kadomtsev-Petviashvili equation | Complex modified Korteweg-de Vries equation | Sine-Gordon equation | Maxwell-Bloch equations | ULTRA-SHORT PULSES | SELF-INDUCED TRANSPARENCY | SOLITARY-WAVE SOLUTIONS | QUADRATIC NONLINEAR MEDIA | PHYSICS, MULTIDISCIPLINARY | DE-VRIES EQUATION | KADOMTSEV-PETVIASHVILI EQUATION | SINE-GORDON EQUATIONS | TI-SAPPHIRE LASER | SHORT ELECTROMAGNETIC PULSES | SHORT-PULSE EQUATION | Analysis | Models | Wave propagation

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2017, Volume 310, pp. 97 - 111

This paper considers the stochastic Korteweg–de Vries equation on a bounded domain. The existence of a weak martingale solution is discussed by the Galerkin’s...

Stochastic Korteweg–de Vries equation | Unique Continuation Property | Weak martingale solution | MATHEMATICS, APPLIED | SOLITONS | DECAY | KDV EQUATION | DRIVEN | Stochastic Korteweg-de Vries equation | CONTROLLABILITY

Stochastic Korteweg–de Vries equation | Unique Continuation Property | Weak martingale solution | MATHEMATICS, APPLIED | SOLITONS | DECAY | KDV EQUATION | DRIVEN | Stochastic Korteweg-de Vries equation | CONTROLLABILITY

Journal Article