Add Benjamin-Ono equation

[MikaelSlevinsky]

If the Hilbert transform of the second partial derivative is diagonal in some basis, then time-evolution could be taken care of using a fourth-order time-stepper for stiff PDEs, like this.

[dlfivefifty]

Would have to be the rational basis since soliton solutions have 1/x decay, probably LaurentDirichlet:

S=LaurentDirichlet([-Inf,Inf]) isdaig(Hilbert(S)) # returns true

On 18 Dec 2015, at 3:57 AM, Richard Mikael Slevinsky notifications@github.com wrote:

If the Hilbert transform of the second partial derivative is diagonal in some basis, then time-evolution could be taken care of using a fourth-order time-stepper for stiff PDEs, like this https://peoplemathsoxacuk/trefethen/publication/PDF/2005_111pdf — Reply to this email directly or view it on GitHub https://github.com/ApproxFun/SingularIntegralEquations.jl/issues/64.

[MikaelSlevinsky]

We could start with a large periodic domain to compare with this:

http://www.diva-portal.org/smash/get/diva2:617038/FULLTEXT01.pdf

I think the solitons decay as 1/x^2 on the real line right?

Cheers,

Mikael

On Dec 18, 2015, at 1:12 AM, Sheehan Olver notifications@github.com wrote:

Would have to be the rational basis since soliton solutions have 1/x decay, probably LaurentDirichlet:

S=LaurentDirichlet([-Inf,Inf]) isdaig(Hilbert(S)) # returns true

On 18 Dec 2015, at 3:57 AM, Richard Mikael Slevinsky notifications@github.com wrote:

If the Hilbert transform of the second partial derivative is diagonal in some basis, then time-evolution could be taken care of using a fourth-order time-stepper for stiff PDEs, like this https://peoplemathsoxacuk/trefethen/publication/PDF/2005_111pdf — Reply to this email directly or view it on GitHub https://github.com/ApproxFun/SingularIntegralEquations.jl/issues/64.

— Reply to this email directly or view it on GitHub.

[dlfivefifty]

Why do we need diagonal operators? Can't we just do a rational approximation of the exponential and use \?

Sent from my iPhone

On 18 Dec 2015, at 12:47, Richard Mikael Slevinsky notifications@github.com wrote:

We could start with a large periodic domain to compare with this:

http://www.diva-portal.org/smash/get/diva2:617038/FULLTEXT01.pdf

I think the solitons decay as 1/x^2 on the real line right?

Cheers,

Mikael

On Dec 18, 2015, at 1:12 AM, Sheehan Olver notifications@github.com wrote:

Would have to be the rational basis since soliton solutions have 1/x decay, probably LaurentDirichlet:

S=LaurentDirichlet([-Inf,Inf]) isdaig(Hilbert(S)) # returns true

On 18 Dec 2015, at 3:57 AM, Richard Mikael Slevinsky notifications@github.com wrote:

If the Hilbert transform of the second partial derivative is diagonal in some basis, then time-evolution could be taken care of using a fourth-order time-stepper for stiff PDEs, like this https://peoplemathsoxacuk/trefethen/publication/PDF/2005_111pdf — Reply to this email directly or view it on GitHub https://github.com/ApproxFun/SingularIntegralEquations.jl/issues/64.

— Reply to this email directly or view it on GitHub.

— Reply to this email directly or view it on GitHub.