Sine-Gordon Equation

Author

Eric W. Weisstein
March 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/Sine-GordonEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Sine-GordonEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Equation

(((∂^2)v)/(∂t ∂t))-(((∂^2)v)/(∂x ∂x))+sin(v)==0

DSolve[(∂_(t,t))v[x,t]-(∂_(x,x))v[x,t]+Sin[v[x,t]]==0,v[x,t],{x,t}]

ODE Obtained by Substitution

DSolve[z f''[z]+f'[z]==Sin[f[z]],f[z],z]

FullSimplify[z f''[z]+f'[z]-Sin[f[z]] /.f:>Function[z,Log[g[z]]/I]]

Collect[Expand[(-((g[z])^3)-2 z ((g'[z])^2)+g[z] (1+2 g'[z]+2 z g''[z]))/2/z/g[z]],z]

-(((g'[z])^2)/(g[z]))+((1-((g[z])^2)+2g'[z])/(2z))+g''[z]

DSolve[g''[z]-g'[z]^2/g[z]+(2g'[z]-g[z]^2+1)/2/z==0,g[z],z]

Solution of the Form 4 (tan^(-1))(((ϕ(x))/(ψ(t))))

FullSimplify[Subtract@@{ArcTan[θ],π/2-ArcTan[1/θ]},θ>0]

<<Graphics`Colors`

Plot[{ArcTan[θ],π/2-ArcTan[1/θ]},{θ,-10,10},PlotStyle->{{Thickness[.015],Red},Blue}]

eqn=FullSimplify[D[v[x,t],{t,2}]-D[v[x,t],{x,2}]+Sin[v[x,t]]/.v->Function[{x,t},4ArcTan[((ϕ[x])/(ψ[t]))]]]

FactorTerms[Numerator[eqn]]/4

Expand[Apart/@((FactorTerms[Numerator[eqn]]/4)/(ϕ[x]ψ[t]))]

(Function[f,Plus@@Cases[-((ϕ[x])^2)+((ψ[t])^2)+2 ((ϕ'[x])^2)+2 ((ψ'[t])^2)-ϕ[x] ϕ''[x]-ψ[t] ψ''[t],_?(!FreeQ[#,f]&)]])/@{ψ,ϕ}

((((ψ[t])^2) ϕ''[x])/(ϕ[x]))+((((ϕ[x])^2) ψ''[t])/(ψ[t]))==(((ψ[t])^2)+2 ((ψ'[t])^2)-ψ[t] ψ''[t])+(-((ϕ[x])^2)+2 ((ϕ'[x])^2)-ϕ[x] ϕ''[x])

Expand[-(1/(2ψ[t]ψ'[t]ϕ[x]ϕ'[x]))D[-((ϕ[x])^2)+((ψ[t])^2)+2 ((ϕ'[x])^2)+2 ((ψ'[t])^2)-ϕ[x] ϕ''[x]-((((ψ[t])^2) ϕ''[x])/(ϕ[x]))-((((ϕ[x])^2) ψ''[t])/(ψ[t]))-ψ[t] ψ''[t],x,t]]

Expand[((D[ϕ''[x]/ϕ[x],x])/(ϕ[x]ϕ'[x]))]

Expand[((D[((ψ''[t])/(ψ[t])),t])/(ψ[t]ψ'[t]))]

Integrate[ϕ[x]ϕ'[x],x]

Integrate[ ϕ[x]^2ϕ'[x],x]

FullSimplify[Integrate[(1/(Sqrt[-k^2 ϕ^3+m ϕ+n^2])),ϕ],k>0&&m>0&&n>0]

FullSimplify[Integrate[(1/(Sqrt[k^2 ψ^3+(Sqrt[m^2-1]) ψ-n^2])),ψ],k>0&&m>0&&n>0]

FullSimplify[Integrate[(1/(Sqrt[k^2 ψ^3+m β ψ-n^2])),ψ],k>0&&m>0&&n>0]

Single Soliton Solution (k=n=0, m>1)

DSolve[{ϕ'[x]==m ϕ[x],ϕ[0]==1},ϕ[x],x]

DSolve[{ψ'[t]==Sqrt[m^2-1]ψ[t],ψ[0]==1},ψ[t],t]

FullSimplify[4ArcTan[(((E^(m x)) )/(E^((Sqrt[-1+(m^2)]) t)))]]

Solve[β==((Sqrt[(m^2)-1])/m),m]

<<Graphics`Colors`

Plot[Evaluate[4ArcTan[Exp[# z]]&/@{1,-1}],{z,-5,5},AxesLabel->TraditionalForm/@{z,4ArcTan[± Exp[z]]},
PlotStyle->{Black,Red}]

Two-Soliton Solution (k=0, m>1)

I can't figure out what Infeld and Rowlands (1990, p. 200) are doing here...

Subtract@@{ϕ'[x],ϕ[x]^3+m ϕ[x]+(n^2)}/.ϕ:>Function[x,Sinh[β m x]]/.β->((Sqrt[(m^2)-1])/m)//FullSimplify

ExpToTrig[DSolve[{ϕ'[x]==m ϕ[x]+n^2,ϕ[0]==0},ϕ[x],x]]//FullSimplify

DSolve[{ψ'[t]==(Sqrt[m^2-1])ψ[t]+(n^2),ψ[0]==1},ψ[t],t]//FullSimplify

Soliton-Antisoliton Solution (k≠0, n=0, m>1)

DSolve[{ϕ'[x]==-(k^2)ϕ[x]^3+m ϕ[x]},ϕ[x],x]

DSolve[{ψ'[t]==k^2 ψ[t]^3+(Sqrt[m^2-1])ψ[t]},ψ[t],t]