STRINGS, WAVES, DRUMS: SPECTRA AND INVERSE PROBLEMS

This survey treats a number of interconnected topics related in one way or another to the famous paper of Mark Kac, "Can one hear the shape of a drum?": wave motion, classical and quantum inverse problems, integrable systems, and the relations between spectra and geometry. We sketch the history and some of the principal developments from the vibrating string to quantum inverse problems, the KdV equation and integrable systems, spectral geometry and the index problem.

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Diffeomorphisms on S1, projective structures and integrable systems

The ANZIAM Journal ◽

10.1017/s1446181100008026 ◽

2002 ◽

Vol 44 (1) ◽

pp. 169-180 ◽

Cited By ~ 3

Author(s):

Partha Guha

Keyword(s):

Evolution Equation ◽

Integrable Systems ◽

Kdv Equation ◽

Lax Pair ◽

Projective Connection ◽

Solution Curve ◽

Projective Structures ◽

Lax Equation ◽

Homogeneous Coordinates ◽

The Kdv Equation

AbstractIn this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

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The motion integrals for the KdV equation and related inverse problems

Inverse Problems ◽

10.1088/0266-5611/12/1/005 ◽

1996 ◽

Vol 12 (1) ◽

pp. 55-75

Author(s):

Mikhail Novitskii

Keyword(s):

Inverse Problems ◽

Kdv Equation ◽

The Kdv Equation

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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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