Mathematical Optics

Example 3.

It is well known that (6.96) has N-soliton solution. The single soliton solution at rest for real η is given by [20]

$q=\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)e^{i{\eta }^{2}t}.$

(6.113)

(The solution can be extended to a moving soliton by the Galilean transformation.)

Let us consider the leading term

$q_0=\mathcal{A}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x).$

(6.114)

If we choose $\mathcal{A}=\eta /\sqrt{\beta }$ then Equations Equation 6.109, Equation 6.114, and Equation 6.116 with successive iterations yield

$\begin{array}{l}{q}_1=(i{\eta }^{2}t)\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_1^{\ast }=(-i{\eta }^{2}t)\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_2=\frac{{(i{\eta }^{2}t)}^2}{2}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_2^{\ast }=\frac{{(i{\eta }^{2}t)}^2}{2}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_3=\frac{{(i{\eta }^{2}t)}^3}{6}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_3^{\ast }=-\frac{{(i{\eta }^{2}t)}^3}{6}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ ⋮\\ q_n=\frac{{(i{\eta }^{2}t)}^n}{n!}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)\\ q_n^{\ast }=(-1{)}^n\frac{{(i{\eta }^{2}t)}^n}{n!}\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x).\end{array}$

Thus (6.102) yields the one-soliton solution of the NLS equation (6.96) and is given by

$q=\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)e^{(i{\eta }^{2}t)}.$

(6.115)

Similarly,

$q^{\ast }=\frac{\eta }{\sqrt{\beta }}\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}(\eta x)e^{-(i}$