Korteweg–De Vries equation

In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM).[2] In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation.

Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9).

Numerical solution of the KdV equation

u t + u u x + δ 2 u x x x = 0

(

δ = 0.022

) with an initial condition

u (x, 0) = cos(π x)

. Time evolution was done by the Zabusky–Kruskal scheme.[1] The initial cosine wave evolves into a train of solitary-type waves.

Two-soliton solution to the KdV equation

The KdV equation was first introduced by Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895),[3][4] who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967.

Definition

The KdV equation is a nonlinear, dispersive partial differential equation for a function $\phi$ of two dimensionless real variables, $x$ and $t$ which are proportional to space and time respectively:[5]

\partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,

with $\partial _{x}$ and $\partial _{t}$ denoting partial derivatives with respect to $x$ and $t$ . For modelling shallow water waves, $\phi$ is the height displacement of the water surface from its equilibrium height.

The constant $6$ in front of the last term is conventional but of no great significance: multiplying $t$ , $x$ , and $\phi$ by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.

The $\partial _{x}^{3}\phi$ introduces dispersion while $\phi \partial _{x}\phi$ is an advection term.

Soliton solutions

One-soliton solution

Consider solutions in which a fixed wave form (given by $f(X)$ ) maintains its shape as it travels to the right at phase speed $c$ . Such a solution is given by $\varphi (x,t)=f(x-ct-a)=f(X)$ . Substituting it into the KdV equation gives the ordinary differential equation

-c{\frac {df}{dX}}+{\frac {d^{3}f}{dX^{3}}}-6f{\frac {df}{dX}}=0,

or, integrating with respect to $X$ ,

-cf+{\frac {d^{2}f}{dX^{2}}}-3f^{2}=A

where $A$ is a constant of integration. Interpreting the independent variable $X$ above as a virtual time variable, this means $f$ satisfies Newton's equation of motion of a particle of unit mass in a cubic potential

V(f)=-\left(f^{3}+{\frac {1}{2}}cf^{2}+Af\right)

If

A=0,\,c>0

then the potential function $V(f)$ has local maximum at $f=0$ , there is a solution in which $f(X)$ starts at this point at 'virtual time' $-\infty$ , eventually slides down to the local minimum, then back up the other side, reaching an equal height, then reverses direction, ending up at the local maximum again at time $\infty$ . In other words, $f(X)$ approaches $0$ as $X\to -\infty$ . This is the characteristic shape of the solitary wave solution.

More precisely, the solution is

\phi (x,t)=-{\frac {1}{2}}\,c\,\operatorname {sech} ^{2}\left[{{\sqrt {c}} \over 2}(x-c\,t-a)\right]

where $\operatorname {sech}$ stands for the hyperbolic secant and $a$ is an arbitrary constant.[6] This describes a right-moving soliton with velocity $c$ .

N-soliton solution

There is a known expression for a solution which is an $N$ -soliton solution, which at late times resolves into $N$ separate single solitons.[7] The solution depends on an decreasing positive set of parameters $\chi _{1},\cdots ,\chi _{N}>0$ and a non-zero set of parameters $\beta _{1},\cdots ,\beta _{N}$ . The solution is given in the form

\phi (x,t)=-2{\frac {\partial ^{2}}{\partial x^{2}}}\mathrm {log} [\mathrm {det} A(x,t)]

where the components of the matrix $A(x,t)$ are given by $A_{nm}(x,t)=\delta _{nm}+{\frac {\beta _{n}e^{8\chi _{n}^{3}t}e^{-(\chi _{n}+\chi _{m})x}}{\chi _{n}+\chi _{m}}}.$

This is derived using the inverse scattering method.

Integrals of motion

The KdV equation has infinitely many integrals of motion (Miura, Gardner & Kruskal 1968), which do not change with time. They can be given explicitly as

\int _{-\infty }^{+\infty }P_{2n-1}(\phi ,\,\partial _{x}\phi ,\,\partial _{x}^{2}\phi ,\,\ldots )\,{\text{d}}x\,

where the polynomials $P_{n}$ are defined recursively by

{\begin{aligned}P_{1}&=\phi ,\\P_{n}&=-{\frac {dP_{n-1}}{dx}}+\sum _{i=1}^{n-2}\,P_{i}\,P_{n-1-i}\quad {\text{ for }}n\geq 2.\end{aligned}}

The first few integrals of motion are:

the mass $\int \phi \,\mathrm {d} x,$
the momentum $\int \phi ^{2}\,\mathrm {d} x,$
the energy $\int \left[2\phi ^{3}-\left(\partial _{x}\phi \right)^{2}\right]\,\mathrm {d} x$ .

Only the odd-numbered terms $P_{2n+1}$ result in non-trivial (meaning non-zero) integrals of motion (Dingemans 1997, p. 733).

Lax pairs

The KdV equation

\partial _{t}\phi =6\,\phi \,\partial _{x}\phi -\partial _{x}^{3}\phi

can be reformulated as the Lax equation

L_{t}=[L,A]\equiv LA-AL\,

with $L$ a Sturm–Liouville operator:

{\begin{aligned}L&=-\partial _{x}^{2}+\phi ,\\A&=4\partial _{x}^{3}-3\left[2\phi \,\partial _{x}+\partial _{x}\phi \right]\end{aligned}}

and this accounts for the infinite number of first integrals of the KdV equation (Lax 1968).

In fact, $L$ is the time-independent Schrödinger operator (disregarding constants) with potential $\phi (x,t)$ . It can be shown that due to this Lax formulation that in fact the eigenvalues do not depend on $t$ .

Zero-curvature representation

Setting the components of the Lax connection to be

L_{x}={\begin{pmatrix}0&1\\\phi -\lambda &0\end{pmatrix}},L_{t}={\begin{pmatrix}-\phi _{x}&2\phi +4\lambda \\2\phi ^{2}-\phi _{xx}+2\phi \lambda -4\lambda ^{2}&\phi _{x}\end{pmatrix}},

the KdV equation is equivalent to the zero-curvature equation for the Lax connection,

\partial _{t}L_{x}-\partial _{x}L_{t}+[L_{x},L_{t}]=0.

Least action principle

The Korteweg–De Vries equation

\partial _{t}\phi +6\phi \,\partial _{x}\phi +\partial _{x}^{3}\phi =0,

is the Euler–Lagrange equation of motion derived from the Lagrangian density, ${\mathcal {L}}\,$

{\mathcal {L}}:={\frac {1}{2}}\partial _{x}\psi \,\partial _{t}\psi +\left(\partial _{x}\psi \right)^{3}-{\frac {1}{2}}\left(\partial _{x}^{2}\psi \right)^{2}

(1)

with $\phi$ defined by

{\displaystyle \phi

Derivation of Euler–Lagrange equations

Since the Lagrangian (eq (1)) contains second derivatives, the Euler–Lagrange equation of motion for this field is

\partial _{\mu \mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu \mu }\psi )}}\right)-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.

(2)

where $\partial$ is a derivative with respect to the $\mu$ component.

A sum over $\mu$ is implied so eq (2) really reads,

\partial _{tt}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{tt}\psi )}}\right)+\partial _{xx}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{xx}\psi )}}\right)-\partial _{t}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\psi )}}\right)-\partial _{x}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x}\psi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.

(3)

Evaluate the five terms of eq (3) by plugging in eq (1),

\partial _{tt}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{tt}\psi )}}\right)=0

\partial _{xx}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{xx}\psi )}}\right)=\partial _{xx}\left(-\partial _{xx}\psi \right)

\partial _{t}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\psi )}}\right)=\partial _{t}\left({\frac {1}{2}}\partial _{x}\psi \right)

\partial _{x}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x}\psi )}}\right)=\partial _{x}\left({\frac {1}{2}}\partial _{t}\psi +3(\partial _{x}\psi )^{2}\right)\,

{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0

Remember the definition $\phi =\partial _{x}\psi$ , so use that to simplify the above terms,

\partial _{xx}\left(-\partial _{xx}\psi \right)=-\partial _{xxx}\phi

\partial _{t}\left({\frac {1}{2}}\partial _{x}\psi \right)={\frac {1}{2}}\partial _{t}\phi

\partial _{x}\left({\frac {1}{2}}\partial _{t}\psi +3(\partial _{x}\psi )^{2}\right)={\frac {1}{2}}\partial _{t}\phi +3\partial _{x}(\phi )^{2}={\frac {1}{2}}\partial _{t}\phi +6\phi \partial _{x}\phi

Finally, plug these three non-zero terms back into eq (3) to see

\left(-\partial _{xxx}\phi \right)-\left({\frac {1}{2}}\partial _{t}\phi \right)-\left({\frac {1}{2}}\partial _{t}\phi +6\phi \partial _{x}\phi \right)=0,

which is exactly the KdV equation

\partial _{t}\phi +6\phi \,\partial _{x}\phi +\partial _{x}^{3}\phi =0.

Long-time asymptotics

It can be shown that any sufficiently fast decaying smooth solution will eventually split into a finite superposition of solitons travelling to the right plus a decaying dispersive part travelling to the left. This was first observed by Zabusky & Kruskal (1965) and can be rigorously proven using the nonlinear steepest descent analysis for oscillatory Riemann–Hilbert problems.[8]

History

The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870 and, finally, Korteweg and De Vries in 1895.

The KdV equation was not studied much after this until Zabusky & Kruskal (1965) discovered numerically that its solutions seemed to decompose at large times into a collection of "solitons": well separated solitary waves. Moreover, the solitons seems to be almost unaffected in shape by passing through each other (though this could cause a change in their position). They also made the connection to earlier numerical experiments by Fermi, Pasta, Ulam, and Tsingou by showing that the KdV equation was the continuum limit of the FPUT system. Development of the analytic solution by means of the inverse scattering transform was done in 1967 by Gardner, Greene, Kruskal and Miura.[9][10]

The KdV equation is now seen to be closely connected to Huygens' principle.[11][12]

Applications and connections

The KdV equation has several connections to physical problems. In addition to being the governing equation of the string in the Fermi–Pasta–Ulam–Tsingou problem in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including:

shallow-water waves with weakly non-linear restoring forces,
long internal waves in a density-stratified ocean,
ion acoustic waves in a plasma,
acoustic waves on a crystal lattice.

The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.

KdV equation and the Gross–Pitaevskii equation

Considering the simplified solutions of the form

\phi (x,t)=\phi (x\pm t)

we obtain the KdV equation as

\pm \partial _{x}\phi +\partial _{x}^{3}\phi +6\,\phi \,\partial _{x}\phi =0\,

or

\pm \partial _{x}\phi +\partial _{x}(\partial _{x}^{2}\phi +3\phi ^{2})=0\,

Integrating and taking the special case in which the integration constant is zero, we have:

-\partial _{x}^{2}\phi -3\phi ^{2}=\pm \phi \,

which is the $\lambda =1$ special case of the generalized stationary Gross–Pitaevskii equation (GPE)

-\partial _{x}^{2}\phi -3\phi ^{\lambda }\phi =\pm \phi \,

Therefore, for the certain class of solutions of generalized GPE ( $\lambda =4$ for the true one-dimensional condensate and $\lambda =2$ while using the three dimensional equation in one dimension), two equations are one. Furthermore, taking the $\lambda =3$ case with the minus sign and the $\phi$ real, one obtains an attractive self-interaction that should yield a bright soliton.

Variations

Many different variations of the KdV equations have been studied. Some are listed in the following table.

Name	Equation
Korteweg–De Vries (KdV)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u+6u\partial _{x}u=0$
KdV (cylindrical)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u-6u\partial _{x}u+{\tfrac {1}{2t}}u=0$
KdV (deformed)	$\displaystyle \partial _{t}u+\partial _{x}\left({\frac {\partial _{x}^{2}u-2\eta u^{3}-3u(\partial _{x}u)^{2}}{2(\eta +u^{2})}}\right)=0$
KdV (generalized)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u=\partial _{x}^{5}u$
KdV (generalized)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0$
KdV (modified)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u\pm 6u^{2}\partial _{x}u=0$
KdV (modified modified)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u-{\tfrac {1}{8}}(\partial _{x}u)^{3}+(\partial _{x}u)(Ae^{au}+B+Ce^{-au})=0$
KdV (spherical)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u-6u\partial _{x}u+{\tfrac {1}{t}}u=0$
KdV (super)	$\displaystyle {\begin{cases}\partial _{t}u=6u\partial _{x}u-\partial _{x}^{3}u+3w\partial _{x}^{2}w\\\partial _{t}w=3(\partial _{x}u)w+6u\partial _{x}w-4\partial _{x}^{3}w\end{cases}}$
KdV (transitional)	$\displaystyle \partial _{t}u+\partial _{x}^{3}u-6f(t)u\partial _{x}u=0$
KdV (variable coefficients)	$\displaystyle \partial _{t}u+\beta t^{n}\partial _{x}^{3}u+\alpha t^{n}u\partial _{x}u=0$
Korteweg–De Vries–Burgers equation[13]	$\displaystyle \partial _{t}u+\mu \partial _{x}^{3}u+2u\partial _{x}u-\nu \partial _{x}^{2}u=0$
non-homogeneous KdV	$\partial _{t}u+\alpha u+\beta \partial _{x}u+\gamma \partial _{x}^{2}u=Ai(x),\quad u(x,0)=f(x)$

Notes

Zabusky, N. J.; Kruskal, M. D. (9 August 1965). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6): 240–243. doi:10.1103/PhysRevLett.15.240.
Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (6 November 1967). "Method for Solving the Korteweg-deVries Equation". Physical Review Letters. 19 (19): 1095–1097. doi:10.1103/PhysRevLett.19.1095. Retrieved 14 July 2023.
Korteweg, D. J.; de Vries, G. (May 1895). "XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 39 (240): 422–443. doi:10.1080/14786449508620739.
Darrigol, O. (2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, p. 84, ISBN 9780198568438
See e.g. Newell, Alan C. (1985), Solitons in mathematics and physics, SIAM, ISBN 0-89871-196-7, p. 6. Or Lax (1968), without the factor 6.
Alexander F. Vakakis (31 January 2002). Normal Modes and Localization in Nonlinear Systems. Springer. pp. 105–108. ISBN 978-0-7923-7010-9. Retrieved 27 October 2012.
Dunajski, Maciej (2015). Solitons, instantons, and twistors (1. publ., corrected 2015 ed.). Oxford: Oxford University Press. ISBN 978-0198570639.
See e.g. Grunert & Teschl (2009)
Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M (1967), "Method for solving the Korteweg–De Vries equation", Physical Review Letters, 19 (19): 1095–1097, Bibcode:1967PhRvL..19.1095G, doi:10.1103/PhysRevLett.19.1095.
Dauxois, Thierry; Peyrard, Michel (2006), Physics of Solitons, Cambridge University Press, ISBN 0-521-85421-0
Fabio A. C. C. Chalub and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"
Berest, Yuri Y.; Loutsenko, Igor M. (1997). "Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–De Vries Equation". Communications in Mathematical Physics. 190 (1): 113–132. arXiv:solv-int/9704012. Bibcode:1997CMaPh.190..113B. doi:10.1007/s002200050235. S2CID 14271642.
Shu, Jian-Jun (1987). "The proper analytical solution of the Korteweg–De Vries–Burgers equation". Journal of Physics A: Mathematical and General. 20 (2): 49–56. arXiv:1403.3636. Bibcode:1987JPhA...20L..49J. doi:10.1088/0305-4470/20/2/002.

References

Boussinesq, J. (1877), Essai sur la theorie des eaux courantes, Memoires presentes par divers savants ` l’Acad. des Sci. Inst. Nat. France, XXIII, pp. 1–680
de Jager, E.M. (2006). "On the origin of the Korteweg–De Vries equation". arXiv:math/0602661v1.
Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, vol. 13, World Scientific, Singapore, ISBN 981-02-0427-2, 2 Parts, 967 pages
Drazin, P. G. (1983), Solitons, London Mathematical Society Lecture Note Series, vol. 85, Cambridge: Cambridge University Press, pp. viii+136, doi:10.1017/CBO9780511662843, ISBN 0-521-27422-2, MR 0716135
Grunert, Katrin; Teschl, Gerald (2009), "Long-Time Asymptotics for the Korteweg–De Vries Equation via Nonlinear Steepest Descent", Math. Phys. Anal. Geom., vol. 12, no. 3, pp. 287–324, arXiv:0807.5041, Bibcode:2009MPAG...12..287G, doi:10.1007/s11040-009-9062-2, S2CID 8740754
Kappeler, Thomas; Pöschel, Jürgen (2003), KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 45, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-08054-2, ISBN 978-3-540-02234-3, MR 1997070
Korteweg, D. J.; De Vries, G. (1895), "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves", Philosophical Magazine, 39 (240): 422–443, doi:10.1080/14786449508620739
Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves", Communications on Pure and Applied Mathematics, 21 (5): 467–490, doi:10.1002/cpa.3160210503
Miles, John W. (1981), "The Korteweg–De Vries equation: A historical essay", Journal of Fluid Mechanics, 106: 131–147, Bibcode:1981JFM...106..131M, doi:10.1017/S0022112081001559, S2CID 122811526.
Miura, Robert M.; Gardner, Clifford S.; Kruskal, Martin D. (1968), "Korteweg–De Vries equation and generalizations. II. Existence of conservation laws and constants of motion", J. Math. Phys., 9 (8): 1204–1209, Bibcode:1968JMP.....9.1204M, doi:10.1063/1.1664701, MR 0252826
Takhtadzhyan, L.A. (2001) [1994], "Korteweg-de Vries equation", Encyclopedia of Mathematics, EMS Press

External links

Korteweg–De Vries equation at EqWorld: The World of Mathematical Equations.
Korteweg–De Vries equation at NEQwiki, the nonlinear equations encyclopedia.
Cylindrical Korteweg–De Vries equation at EqWorld: The World of Mathematical Equations.
Modified Korteweg–De Vries equation at EqWorld: The World of Mathematical Equations.
Modified Korteweg–De Vries equation at NEQwiki, the nonlinear equations encyclopedia.
Weisstein, Eric W. "Korteweg–deVries Equation". MathWorld.
Derivation of the Korteweg–De Vries equation for a narrow canal.
Three Solitons Solution of KdV Equation –
Three Solitons (unstable) Solution of KdV Equation –
Mathematical aspects of equations of Korteweg–De Vries type are discussed on the Dispersive PDE Wiki.
Solitons from the Korteweg–De Vries Equation by S. M. Blinder, The Wolfram Demonstrations Project.
Solitons & Nonlinear Wave Equations

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[ZK-1] Zabusky, N. J.; Kruskal, M. D. (9 August 1965). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6): 240–243. doi:10.1103/PhysRevLett.15.240.

[GGKM-2] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (6 November 1967). "Method for Solving the Korteweg-deVries Equation". Physical Review Letters. 19 (19): 1095–1097. doi:10.1103/PhysRevLett.19.1095. Retrieved 14 July 2023.

[KdV-3] Korteweg, D. J.; de Vries, G. (May 1895). "XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 39 (240): 422–443. doi:10.1080/14786449508620739.

[4] Darrigol, O. (2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, p. 84, ISBN 9780198568438

[5] See e.g. Newell, Alan C. (1985), Solitons in mathematics and physics, SIAM, ISBN 0-89871-196-7, p. 6. Or Lax (1968), without the factor 6.

[Vakakis2002-6] Alexander F. Vakakis (31 January 2002). Normal Modes and Localization in Nonlinear Systems. Springer. pp. 105–108. ISBN 978-0-7923-7010-9. Retrieved 27 October 2012.

[dunajski-7] Dunajski, Maciej (2015). Solitons, instantons, and twistors (1. publ., corrected 2015 ed.). Oxford: Oxford University Press. ISBN 978-0198570639.

[8] See e.g. Grunert & Teschl (2009)

[9] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M (1967), "Method for solving the Korteweg–De Vries equation", Physical Review Letters, 19 (19): 1095–1097, Bibcode:1967PhRvL..19.1095G, doi:10.1103/PhysRevLett.19.1095.

[10] Dauxois, Thierry; Peyrard, Michel (2006), Physics of Solitons, Cambridge University Press, ISBN 0-521-85421-0

[11] Fabio A. C. C. Chalub and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"

[12] Berest, Yuri Y.; Loutsenko, Igor M. (1997). "Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–De Vries Equation". Communications in Mathematical Physics. 190 (1): 113–132. arXiv:solv-int/9704012. Bibcode:1997CMaPh.190..113B. doi:10.1007/s002200050235. S2CID 14271642.

[13] Shu, Jian-Jun (1987). "The proper analytical solution of the Korteweg–De Vries–Burgers equation". Journal of Physics A: Mathematical and General. 20 (2): 49–56. arXiv:1403.3636. Bibcode:1987JPhA...20L..49J. doi:10.1088/0305-4470/20/2/002.