Benjamin-Ono equation

The Benjamin-Ono equation (BO equation) is a nonlinear partial differential equation (evolution equation) with soliton solution. It is exactly integrable and an integro-differential equation of the form:

{\ displaystyle u_ {t} + H (u_ {xx}) + 2uu_ {x} = 0}

where the Hilbert transform is: ${\ displaystyle H}$

{\ displaystyle Hf (x) = {\ frac {1} {\ pi}} \ operatorname {CH} \ int _ {- \ infty} ^ {\ infty} {\ frac {f (y)} {yx}} dy}

and stands for Cauchy's principal value . Subscripts mean partial derivatives . ${\ displaystyle \ operatorname {CH}}$

The Benjamin-Ono equation can be exactly solved by Inverse Scattering Transformation (IST) and was introduced in 1967 to describe internal water waves at great depths. Inner waves arise z. B. at interfaces of liquid layers of different densities .

It is named after H. Ono and Brooke Benjamin .

Intermediate long wave equation

The ILW (Intermediate Long Wave) equation is sometimes used to describe internal waves. It is also an integro-differential equation with a singular integral operator , but compared to the Benjamin-Ono equation it has an additional term in which the constant indicates the depth: ${\ displaystyle {\ frac {u_ {x}} {\ delta}}}$ ${\ displaystyle \ delta}$

{\ displaystyle u_ {t} + {\ frac {u_ {x}} {\ delta}} + T (u_ {xx}) + 2uu_ {x} = 0}

with the integral operator ( convolution of with hyperbolic cotangent ) ${\ displaystyle u_ {xx}}$

{\ displaystyle T (f (x)) = {\ frac {1} {2 \ delta}} \ operatorname {CH} \ int _ {- \ infty} ^ {\ infty} \ coth ({\ frac {\ pi } {2 \ delta}} (yx)) f (y) dy}

For (deep water) the intermediate long wave equation changes into the Benjamin-Ono equation and for (shallow water) into the Korteweg-de-Vries equation . The IST for the ILW mediates between the IST schemes of these two borderline cases. ${\ displaystyle \ delta \ rightarrow \ infty}$ ${\ displaystyle \ delta \ rightarrow 0}$

The IST scheme of the Benjamin Ono equation has more the form of an IST scheme for multi-dimensional problems (two dimensions). The investigation of the ILW and BO equations is therefore also of mathematical interest for the transition from one to multi-dimensional IST schemes.

There are variants of the equation that can also be exactly integrated.

literature

Ablowitz, Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991, p. 163ff (chapter 4)

Web links

Benjamin Ono equation, Dispersive wiki, University of Toronto

Individual evidence

↑ According to Clarkson, Ablowitz, see literature, sometimes in a slightly modified form, e.g. without coefficient 2
^ MJ Ablowitz , AS Fokas, The inverse scattering transform for the Benjamin-Ono equation --- a pivot to multidimensional problems. Stud. Appl. Math., 68: 1-10 (1983)
^ T. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech, 29: 559-562 (1967)
^ H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan, 39: 1082-1091 (1975)
↑ Ablowitz, Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991, p. 3

[1] According to Clarkson, Ablowitz, see literature, sometimes in a slightly modified form, e.g. without coefficient 2

[2] MJ Ablowitz , AS Fokas, The inverse scattering transform for the Benjamin-Ono equation --- a pivot to multidimensional problems. Stud. Appl. Math., 68: 1-10 (1983)

[3] T. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech, 29: 559-562 (1967)

[4] H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan, 39: 1082-1091 (1975)

[5] Ablowitz, Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991, p. 3