The Regular Benjamin-Ono Equation is a Soliton Equation.  It has 1-Soliton, 2-Soliton , and in general N-Soliton Solutions.

One-Soliton Solution

The One Soliton Solution of the B-O equation is given by a Lorentzian Profile.

u (x, t) = (4 a)/((x - v t)^2 + a^2) , with a>0

This corresponds to the following:

u^+(x) = (2  )/(x - v t +  a), and u^- (x) = (-2  )/(x - v t -  a)   with u (x) = u^+(x) + u^-(x)

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An animation of the above follows below (each time frame is plotted in a separate frame, but shown in rapid succession)

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Multi-Soliton Solution

The N-Soliton Solutions of the Benjamin-Ono Equation are given by a series of assignments and formulas.

Im (x_i) >0, i = 1, 2. .. N

v (t) = Overscript[x, .] _l (t) = Underoverscript[∑, j = 1, j≠i, arg3] (2 )/(x_j - x_i) + Underoverscript[∑, j = 1, arg3] (-2 )/((x) _j - x_i)

L_lm = δ_lmOverscript[x, .] _l + (1 - δ_lm) (-2 )/(x_l - x_m)

Z_lm = δ_lmx_l (t_0) + (t - t_0) L_lm (t_0), notice that matrix Z depends only on initial conditions .

u = /2∂_x ln (det (x I - Z^*))/(det (x I - Z))

The following is an animation for a 2 - Soliton Plot of the Benjamin - Ono Solitons, with the initial values x_i ' s set as {4 + , -7 + 12 } :

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The following is an animation for a 3 - Soliton Plot of the Benjamin - Ono Solitons, with the initial values x_i ' s set as {3 + 7, -7 + 11 , -4 + 9 } :

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One-Periodic Soliton Solution

The Single Soliton Solution for the Periodic Conditions is given by:

u (x, t) = (2 Sinh (a))/(Cosh (a) - Cos (x))

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See the animation of the evolution of the above soliton here:

Multi-Periodic Soliton Solution

The generalized Multi - Periodic Soliton Solution of the Benjamin - Ono Equation is described by a series of assignments and formulas .

Let p_i ' s and q_i ' s for i = 1. .. N be parameters that satisfy the following relation :

0<q_1<p_1<q_2<p_2< ... <q_N<p_N

Then we define several new quantities based on these given parameters as follows:

φ_j = lnp_j/q_j^(1/2),      exp (Overscript[A_ij, _]) = ((p_i - p_j) (q_i - q_j))/((p_i - q_j) (q_i - p_j)) , for i≠j

F (x, t) = (f_jk (x, t)),        F ' (x, t) = (f ' _jk (x, t))

Finally, we have that :

u (x, t) = 2 ∂_x ln (| F ' (x, t) |)/(| F (x, t) |)

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The animation of the evolution of this soliton is as follows:

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