32 Painlevé Transcendents Applications 32.12 Asymptotic Approximations for Complex Variables 32.14 Combinatorics

§32.13 Reductions of Partial Differential Equations

ⓘ

Keywords:: Painlevé transcendents, applications, partial differential equations, reductions of partial differential equations
Permalink:: http://dlmf.nist.gov/32.13
See also:: Annotations for Ch.32

§32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations

ⓘ

Keywords:: KdV equation, Korteweg–de Vries equation, Painlevé transcendents, applications, modified Korteweg–de Vries equation, modified Korteweg–de Vries equation
Notes:: See Ablowitz and Segur (1977).
Permalink:: http://dlmf.nist.gov/32.13.i
See also:: Annotations for §32.13 and Ch.32

The modified Korteweg–de Vries (mKdV) equation

32.13.1		$v_{t}-6v^{2}v_{x}+v_{xxx}=0,$
ⓘ Symbols: $v(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E1 Encodings: TeX, pMML, png See also: Annotations for §32.13(i), §32.13 and Ch.32

has the scaling reduction

32.13.2	$\displaystyle z$	$\displaystyle=x(3t)^{-1/3},$
32.13.2	$\displaystyle v(x,t)$	$\displaystyle=(3t)^{-1/3}w(z),$
ⓘ Symbols: $z$ : real and $v(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.13(i), §32.13 and Ch.32

where $w(z)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha$ a constant of integration.

The Korteweg–de Vries (KdV) equation

32.13.3		$u_{t}+6uu_{x}+u_{xxx}=0,$
ⓘ Symbols: $u(x,t)$ : solution Referenced by: §32.13(i) Permalink: http://dlmf.nist.gov/32.13.E3 Encodings: TeX, pMML, png See also: Annotations for §32.13(i), §32.13 and Ch.32

has the scaling reduction

32.13.4	$\displaystyle z$	$\displaystyle=x(3t)^{-1/3},$
32.13.4	$\displaystyle u(x,t)$	$\displaystyle=-(3t)^{-2/3}(w^{\prime}+w^{2}),$
ⓘ Symbols: $z$ : real and $u(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.13(i), §32.13 and Ch.32

where $w(z)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ .

Equation (32.13.3) also has the similarity reduction

32.13.5	$\displaystyle z$	$\displaystyle=x+3\lambda t^{2},$
32.13.5	$\displaystyle u(x,t)$	$\displaystyle=W(z)-\lambda t,$
ⓘ Symbols: $z$ : real and $u(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.13(i), §32.13 and Ch.32

where $\lambda$ is an arbitrary constant and $W(z)$ is expressible in terms of solutions of $\mbox{P}_{\mbox{\scriptsize I}}$ . See Fokas and Ablowitz (1982) and P. J. Olver (1993b, p. 194).

§32.13(ii) Sine-Gordon Equation

ⓘ

Keywords:: Painlevé transcendents, applications, sine-Gordon equation
Notes:: See Ablowitz and Segur (1977).
Referenced by:: §32.11(v), Erratum (V1.0.1) for References, Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/32.13.ii
Correction (effective with 1.0.7):: The reference Wong and Zhang (2009a), formerly incorrectly placed in this subsection, has been moved to the end of §32.11(v).
Amendment (effective with 1.0.1):: Two references were added at the end of this subsection.
See also:: Annotations for §32.13 and Ch.32

The sine-Gordon equation

32.13.6		$u_{xt}=\sin u,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $u(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E6 Encodings: TeX, pMML, png See also: Annotations for §32.13(ii), §32.13 and Ch.32

has the scaling reduction

32.13.7	$\displaystyle z$	$\displaystyle=xt,$
32.13.7	$\displaystyle u(x,t)$	$\displaystyle=v(z),$
ⓘ Symbols: $z$ : real, $u(x,t)$ : solution and $v(z)$ : solution Permalink: http://dlmf.nist.gov/32.13.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.13(ii), §32.13 and Ch.32

where $v(z)$ satisfies (32.2.10) with $\alpha=\tfrac{1}{2}$ and $\gamma=0$ . In consequence if $w=\exp\left(-iv\right)$ , then $w(z)$ satisfies $\mbox{P}_{\mbox{\scriptsize III}}$ with $\alpha=-\beta=\tfrac{1}{2}$ and $\gamma=\delta=0$ .

§32.13(iii) Boussinesq Equation

ⓘ

Keywords:: Boussinesq equation, Painlevé transcendents, applications, partial differential equations
Notes:: See Ablowitz and Segur (1977).
Permalink:: http://dlmf.nist.gov/32.13.iii
See also:: Annotations for §32.13 and Ch.32

The Boussinesq equation

32.13.8		$u_{tt}=u_{xx}-6(u^{2})_{xx}+u_{xxxx},$
ⓘ Symbols: $u(x,t)$ : solution Permalink: http://dlmf.nist.gov/32.13.E8 Encodings: TeX, pMML, png See also: Annotations for §32.13(iii), §32.13 and Ch.32

has the traveling wave solution

32.13.9	$\displaystyle z$	$\displaystyle=x-ct,$
32.13.9	$\displaystyle u(x,t)$	$\displaystyle=v(z),$
ⓘ Symbols: $z$ : real, $u(x,t)$ : solution, $c$ : constant and $v(z)$ : solution Permalink: http://dlmf.nist.gov/32.13.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.13(iii), §32.13 and Ch.32

where $c$ is an arbitrary constant and $v(z)$ satisfies

32.13.10		$v^{\prime\prime}=6v^{2}+(c^{2}-1)v+Az+B,$
ⓘ Symbols: $z$ : real, $c$ : constant, $v(z)$ : solution, $A$ : integration constant and $B$ : integration constant Permalink: http://dlmf.nist.gov/32.13.E10 Encodings: TeX, pMML, png See also: Annotations for §32.13(iii), §32.13 and Ch.32

with $A$ and $B$ constants of integration. Depending whether $A=0$ or $A\neq 0$ , $v(z)$ is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of $\mbox{P}_{\mbox{\scriptsize I}}$ , respectively.

32.12 Asymptotic Approximations for Complex Variables 32.14 Combinatorics

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§32.13 Reductions of Partial Differential Equations

Contents

§32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations

§32.13(ii) Sine-Gordon Equation

§32.13(iii) Boussinesq Equation

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.