Whistler precursor and intrinsic variability of quasi-perpendicular shocks

The structure of a whistler precursor in a quasi-perpendicular shock is studied within two-fluid approach in one-dimensional case. The complete set of equations is reduced to the KdV equation, if no dissipation is included. With a phenomenological resistive dissipation the structure is described with the KdV–Burgers equation. The shock profile is intrinsically time dependent. For sufficiently strong dissipation, temporal evolution of a steepening profile results in generation of a stationary decaying whistler ahead of the shock front. With the decrease of the dissipation parameter, whistler wave trains begin to detach and propagate toward the upstream and the ramp is weakly time dependent. In the weakly dissipative regime the shock front experiences reformation.

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Complete set of infinitesimal transformations about n-soliton solutions of the KdV equation

Journal of Physics A General Physics ◽

10.1088/0305-4470/15/2/012 ◽

1982 ◽

Vol 15 (2) ◽

pp. 397-403 ◽

Cited By ~ 2

Author(s):

R N Aiyer

Keyword(s):

Kdv Equation ◽

Soliton Solutions ◽

The Kdv Equation ◽

Complete Set ◽

Infinitesimal Transformations

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Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients

Abstract and Applied Analysis ◽

10.1155/2014/853578 ◽

2014 ◽

Vol 2014 ◽

pp. 1-13

Author(s):

Li-hua Zhang

Keyword(s):

Conservation Laws ◽

Exact Solutions ◽

Kdv Equation ◽

Nonlinear Partial Differential Equations ◽

Time Dependent ◽

Group Method ◽

New Exact Solutions ◽

Petviashvili Equation ◽

The Kdv Equation ◽

Special Case

The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions oft. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.

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Integrable Models

10.1093/oso/9780198805021.003.0009 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Fluid Mechanics ◽

Soliton Solution ◽

Water Waves ◽

Heisenberg Model ◽

Kdv Equation ◽

Short Review ◽

Lax Pair ◽

Shallow Water Waves ◽

Initial Value ◽

The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽

10.1007/s42286-020-00046-6 ◽

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.

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Wave breaking in the KdV equation on a flow with constant vorticity

European Journal of Mechanics - B/Fluids ◽

10.1016/j.euromechflu.2017.12.006 ◽

2019 ◽

Vol 73 ◽

pp. 48-54 ◽

Cited By ~ 3

Author(s):

Amutha Senthilkumar ◽

Henrik Kalisch

Keyword(s):

Wave Breaking ◽

Kdv Equation ◽

Constant Vorticity ◽

The Kdv Equation

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0147 ◽

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.

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