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.

Lump-type solutions, interaction solutions and periodic wave solutions of a (3+1)-dimensional Korteweg–de Vries equation

2019 ◽  
Vol 33 (27) ◽  
pp. 1950319 ◽  
Author(s):  
Hongfei Tian ◽  
Jinting Ha ◽  
Huiqun Zhang
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
Hirota Bilinear

Based on the Hirota bilinear form, lump-type solutions, interaction solutions and periodic wave solutions of a (3[Formula: see text]+[Formula: see text]1)-dimensional Korteweg–de Vries (KdV) equation are obtained. The interaction between a lump-type soliton and a stripe soliton including two phenomena: fission and fusion, are illustrated. The dynamical behaviors are shown more intuitively by graphics.

GENERATING SYMMETRY OPERATORS FOR THE KdV EQUATION AND SEEKING THEIR SUPERSYMMETRIC GENERALIZATIONS

1998 ◽  
Vol 13 (18) ◽  
pp. 3203-3213 ◽  
Author(s):  
B. BAGCHI ◽  
J. BECKERS ◽  
N. DEBERGH
Keyword(s):  
Nonlinear Equation ◽  
Vries Equation ◽  
Kdv Equation ◽  
Lie Superalgebra ◽  
Conserved Quantities ◽  
Supersymmetric Version ◽  
The Kdv Equation ◽  
De Vries ◽  
Symmetry Operators

Exploiting the Lax form of the Korteweg-de Vries equation, we determine the whole set of its symmetry operators and relate them to conserved quantities. We also study a KdV supersymmetric version and put in evidence its even (bosonic) and odd (fermionic) symmetries. We then get the Lie superalgebra characterizing this supersymmetric nonlinear equation.

Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg—de Vries equation

2011 ◽  
Vol 141 (2) ◽  
pp. 287-317 ◽  
Author(s):  
Yvan Martel ◽  
Frank Merle
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Time Asymptotics ◽  
The Kdv Equation ◽  
De Vries ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Different Sources ◽  
Integrability Theory

We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

scholarly journals A universal form for the emergence of the Korteweg–de Vries equation

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120707 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Dispersion Relation ◽  
Solitary Waves ◽  
Vries Equation ◽  
Kdv Equation ◽  
Universal Form ◽  
Conventional View ◽  
The Kdv Equation ◽  
De Vries ◽  
New Perspective ◽  
Background State

A new perspective on the emergence of the Korteweg–de Vries (KdV) equation is presented. The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of partial differential equations has the appropriate form, and the nonlinearity is quadratic. The assumptions of this paper imply that the usual spectral and nonlinearity assumptions for the derivation of the KdV equation are met. In addition to a new mechanism, the theory shows that the emergence of the KdV equation always takes a universal form, where the coefficients in the KdV equation are completely determined from the properties of the background state—even an apparently trivial background state. Moreover, the mechanism for the emergence of the KdV equation is simplified, reducing it to a single condition. Well-known examples, such as the KdV equation in shallow-water hydrodynamics and in the emergence of dark solitary waves, are predicted by the new theory for emergence of KdV.

scholarly journals The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments

1974 ◽  
Vol 65 (2) ◽  
pp. 289-314 ◽  
Author(s):  
Joseph L. Hammack ◽  
Harvey Segur
Keyword(s):  
Initial Data ◽  
Water Waves ◽  
Entire Range ◽  
Vries Equation ◽  
Kdv Equation ◽  
Asymptotic Shape ◽  
Wide Range ◽  
The Kdv Equation ◽  
De Vries ◽  
Wave Forms

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

scholarly journals On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source

2019 ◽  
Vol 50 (3) ◽  
pp. 281-291 ◽  
Author(s):  
G. U. Urazboev ◽  
A. K. Babadjanova
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Scattering Data ◽  
Self Consistent ◽  
De Vries ◽  
The Matrix

In this work we deduce laws of the evolution of the scattering  data for the matrix Zakharov Shabat system with the potential that is the solution of the matrix modied KdV equation with a self consistent source.

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