An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

Modular Korteweg - de Vries equation: Riemann, cnoidal and solitary waves

2020 ◽  
Author(s):  
Efim Pelinovsky ◽  
Anna Kokorina ◽  
Alexey Slunyaev ◽  
Elena Tobisch
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Scattering Transform ◽  
De Vries ◽  
Dispersionless Limit ◽  
Pseudo Spectral Method ◽  
Appl Math

<p>The review paper by Oleg Rudenko [1] suggests several examples of elastic systems with so-called modular nonlinearities. In this study we consider the modular Korteweg - de Vries (KdV) equation in the form u_t + 6 u u_x + u_{xxx} = 0. This equation is not integrable by means of the Inverse Scattering Transform in the general case, but sign-defined functions which never change the sign satisfy the integrable KdV equation, and hence possess an exact solution. Firstly, we consider the dispersionless limit of the modular KdV equation and analyze the evolution of a simple nonlinear wave (Riemann wave) and its Fourier transform including the asymptotics when the wave tends to break [2]. Then, we study the structure of travelling waves. If the waves propagate on a pedestal and do not cross the zero level u = 0, they coincide with the well-known travelling wave solutions of the classic KdV equation in the form of cnoidal and solitary waves. If the pedestal is zero, the structure of sign-varying travelling waves is expressed through Jacobi elliptic functions. The interaction of solitary waves of different polarities is studied numerically using an implicit pseudo-spectral method. The simulation has revealed the inelastic character of the collision; in the course of the interaction the solitons can alter their amplitudes (the small soliton decreases and the large one grows) and emit small-amplitude waves. The inelastic effects are most pronounced when the solitons’ amplitudes are close. When their amplitudes differ significantly, the maximum wave height which is attained during the absorb-emit interaction tends to the sum of the heights of the solitons with the polarity inherited from the large soliton, as predicted in the frameworks of different long-wave integrable models in [3, 4]. As a result of the collision the solitons may experience non-classic phase shifts as they both jump back.</p><p>[1] O.V. Rudenko. Physics – Uspekhi, Vol. 56(7), 683-690 (2013).</p><p>[2] E. Tobisch, and E. Pelinovsky. Appl. Math. Lett., Vol. 97, 1-5 (2019).</p><p>[3] A.V. Slunyaev, and E.N. Pelinovsky. Phys. Rev. Lett., Vol. 117, 214501 (2016).</p><p>[4] A. Slunyaev. Stud. Appl. Math., Vol. 142, 385-413 (2019).</p>

On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

scholarly journals Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
A. R. Seadawy ◽  
K. El-Rashidy
Keyword(s):  
Differential Equations ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Transformation Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Equations Of Mathematical Physics

Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

SPHERICAL KORTEWEG – DE VRIES SOLITONS FROM RELATIVISTIC HYDRODYNAMICS

2007 ◽  
Vol 16 (09) ◽  
pp. 3019-3023 ◽  
Author(s):  
D. A. FOGAÇA ◽  
F. S. NAVARRA
Keyword(s):  
Fluid Dynamics ◽  
Equation Of State ◽  
Kdv Equation ◽  
Relativistic Hydrodynamics ◽  
Spherical Coordinates ◽  
Relativistic Fluid Dynamics ◽  
Continuity Equations ◽  
Relativistic Fluid ◽  
The Kdv Equation ◽  
De Vries

We study the conditions for the formation and propagation of Korteweg – de Vries (KdV) solitons in relativistic fluid dynamics using an appropriate equation of state. It is known that in cartesian coordinates the combination of the Euler and continuity equations leads to the KdV equation. Recently it has been shown that this is true also in relativistic hydrodynamics. Here we show that a KdV-like equation can be obtained in relativistic hydrodynamics in spherical coordinates.

Approximate travelling waves for generalized KPP equations and classical mechanics

1994 ◽  
Vol 446 (1928) ◽  
pp. 529-554 ◽  
Keyword(s):  
Travelling Waves ◽  
Classical Mechanics ◽  
Initial Distribution ◽  
Travelling Wave ◽  
Step Function ◽  
Initial Position ◽  
Classical Path ◽  
Huygens Principle ◽  
Kpp Equations ◽  
Complete Riemannian Manifolds

We consider the existence of approximate travelling waves of generalized KPP equations in which the initial distribution can depend on a small parameter μ which in the limit μ → 0 is the sum of some δ -functions or a step function. Using the method of Elworthy & Truman (1982) we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears. By the Feynman–Kac formula and the Maruyama–Girsanov–Cameron–Martin transformation we obtain an identity from which, with a late caustic assumption, we see the propagation of the global wave front and the shape of the trough. Our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave. Finally, we prove a Huygens principle for KPP equations on complete riemannian manifolds without cut locus, with some bounds on their volume element, in particular Cartan–Hadamard manifolds.

scholarly journals A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1254
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Sobia Sultana ◽  
Zakia Hammouch ◽  
Rasool Shah ◽  
...  
Keyword(s):  
Nonlinear Waves ◽  
Analytical Procedure ◽  
Integral Transform ◽  
Kdv Equation ◽  
Analytical Techniques ◽  
Transform Method ◽  
Fractional Pdes ◽  
Approximate Analytical Solutions ◽  
The Kdv Equation ◽  
De Vries

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

2021 ◽  
Author(s):  
Murat Polat ◽  
Ömer Oruç
Keyword(s):  
Conservation Laws ◽  
Lie Group ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
High Order ◽  
Basis Functions ◽  
Numerical Integrator ◽  
The Kdv Equation ◽  
De Vries ◽  
Novel Method

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

scholarly journals On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control

2018 ◽  
Vol 24 (1) ◽  
pp. 237-263 ◽  
Author(s):  
Behzad Azmi ◽  
Anne-Céline Boulanger ◽  
Karl Kunisch
Keyword(s):  
Model Predictive Control ◽  
Numerical Experiment ◽  
Exponential Stability ◽  
Predictive Control ◽  
Vries Equation ◽  
Kdv Equation ◽  
Bounded Interval ◽  
De Vries ◽  
The Stability ◽  
Theoretical Results

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

scholarly journals Second Order Scheme For Korteweg-De Vries (KDV) Equation

2019 ◽  
Vol 43 (1) ◽  
pp. 85-93
Author(s):  
Khandaker Md Eusha Bin Hafiz ◽  
Laek Sazzad Andallah
Keyword(s):  
Stability Condition ◽  
Kdv Equation ◽  
Convex Combination ◽  
Second Order ◽  
Combination Method ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
The Kdv Equation ◽  
De Vries ◽  
Order Scheme

The kinematics of the solitary waves is formed by Korteweg-de Vries (KdV) equation. In this paper, a third order general form of the KdV equation with convection and dispersion terms is considered. Explicit finite difference schemes for the numerical solution of the KdV equation is investigated and stability condition for a first-order scheme using convex combination method is determined. Von Neumann stability analysis is performed to determine the stability condition for a second order scheme. The well-known qualitative behavior of the KdV equation is verified and error estimation for comparisons is performed. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 85-93, 2019

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