Korteweg–De Vries equation

E387064

evolution equation integrable system nonlinear partial differential equation soliton equation

The Korteweg–De Vries equation is a fundamental nonlinear partial differential equation that models shallow water waves and solitons, playing a central role in the theory of integrable systems.

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All labels observed (4)

Label	Occurrences
Korteweg–de Vries equation	2
KdV equation	1
Korteweg–De Vries equation canonical	1
modified Korteweg–de Vries equation	1

How this entity was disambiguated

This entity first appeared as the object of triple T3771947 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Korteweg–De Vries equation
Context triple: [Martin David Kruskal, notableWork, Korteweg–De Vries equation]

A. Gross–Pitaevskii equation
The Gross–Pitaevskii equation is a nonlinear Schrödinger-type equation that describes the macroscopic wavefunction and dynamics of weakly interacting Bose gases at ultra-cold temperatures.
B. Yang–Yang equation
The Yang–Yang equation is a fundamental integral equation in statistical mechanics that describes the thermodynamic properties of one-dimensional interacting Bose gases within the Bethe ansatz framework.
C. Euler equations
The Euler equations are fundamental partial differential equations in fluid dynamics that describe the motion of an ideal (inviscid) fluid without viscosity.
D. Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
E. d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Korteweg–De Vries equation
Target entity description: The Korteweg–De Vries equation is a fundamental nonlinear partial differential equation that models shallow water waves and solitons, playing a central role in the theory of integrable systems.

A. Gross–Pitaevskii equation
The Gross–Pitaevskii equation is a nonlinear Schrödinger-type equation that describes the macroscopic wavefunction and dynamics of weakly interacting Bose gases at ultra-cold temperatures.
B. Yang–Yang equation
The Yang–Yang equation is a fundamental integral equation in statistical mechanics that describes the thermodynamic properties of one-dimensional interacting Bose gases within the Bethe ansatz framework.
C. Euler equations
The Euler equations are fundamental partial differential equations in fluid dynamics that describe the motion of an ideal (inviscid) fluid without viscosity.
D. Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
E. d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
F. None of above. chosen

Statements (54)

Predicate	Object
instanceOf	evolution equation ⓘ integrable system ⓘ nonlinear partial differential equation ⓘ soliton equation ⓘ
appearsIn	internal waves in stratified fluids ⓘ lattice dynamics ⓘ plasma physics ⓘ theory of shallow water waves ⓘ
dependentVariable	u(x,t) ⓘ
describes	balance between nonlinearity and dispersion ⓘ unidirectional propagation of waves ⓘ
field	applied mathematics ⓘ fluid dynamics ⓘ mathematical physics ⓘ
hasAbbreviation	Korteweg–De Vries equation self-linksurface differs ⓘ surface form: KdV equation
hasCanonicalForm	u_t + 6 u u_x + u_{xxx} = 0 ⓘ
hasConservedQuantity	energy ⓘ mass ⓘ momentum ⓘ
hasLaxPair	L_t = [P,L] with L = -\partial_x^2 + u(x,t) ⓘ P = -4\partial_x^3 + 3(u\partial_x + \partial_x u) ⓘ
hasNonlinearityType	quadratic nonlinearity ⓘ
hasOneSolitonSolutionForm	u(x,t) = 2 k^2 \operatorname{sech}^2(k(x - 4k^2 t - x_0)) ⓘ
hasOrder	first order in time ⓘ third order in space ⓘ
hasProperty	Lax pair representation ⓘ bi-Hamiltonian structure ⓘ complete integrability ⓘ infinite number of conservation laws ⓘ soliton collisions are elastic ⓘ
hasSolutionType	breather-like solution under perturbations ⓘ cnoidal wave ⓘ multi-soliton solution ⓘ periodic solution ⓘ solitary wave ⓘ
hasTerm	6 u u_x ⓘ u_t ⓘ u_{xxx} ⓘ
independentVariable	t ⓘ x ⓘ
isIntegrableBy	inverse scattering transform ⓘ
isPrototypeFor	integrable nonlinear wave equations ⓘ soliton theory ⓘ
isRelatedTo	Kadomtsev–Petviashvili equation ⓘ modified Korteweg–De Vries equation ⓘ nonlinear Schrödinger equation ⓘ
isSpecialCaseOf	general KdV-type equations ⓘ
models	long waves in shallow channels ⓘ shallow water waves ⓘ solitons ⓘ
namedAfter	Diederik Johannes Korteweg ⓘ Gustav de Vries ⓘ
publishedIn	Philosophical Magazine ⓘ
yearProposed	1895 ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Martin David Kruskal → notableWork → Korteweg–De Vries equation ⓘ

Korteweg–De Vries equation → hasAbbreviation → Korteweg–De Vries equation self-linksurface differs ⓘ

this entity surface form: KdV equation

Robert Miura → studies → Korteweg–De Vries equation ⓘ

this entity surface form: Korteweg–de Vries equation

Robert Miura → studies → Korteweg–De Vries equation ⓘ

this entity surface form: modified Korteweg–de Vries equation

Clifford S. Gardner → studied → Korteweg–De Vries equation ⓘ