#
New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra ${\mathit{A}}_{\mathbf{2}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

- The Kaup–Newell Equation [6], also known as dNLS-I:$${\partial}_{t}q+{\partial}_{x}^{2}q\pm i{\partial}_{x}\left({q\left|q\right|}^{2}\right)=0.$$
- The Chen–Lee–Liu Equation [7], also known as dNLS-II:$$i{\partial}_{t}q+{\partial}_{x}^{2}q-{\left|q\right|}^{2}{\partial}_{x}q=0.$$
- The Gerdjikov–Ivanov Equation [8], also known as dNLS-III:$$i{\partial}_{t}q+{\partial}_{x}^{2}q-{\left|q\right|}^{2}{\partial}_{x}{q}^{\ast}+\frac{1}{2}{\left|q\right|}^{4}q=0,$$

## 2. Preliminaries

## 3. Lax Pair and Recursion Relations

- The potentials are elements of the corresponding eigenspaces of C$$\begin{array}{cc}\hfill {U}_{0}(x,t)\in {\mathfrak{g}}^{\left(0\right)}& ,\phantom{\rule{1.em}{0ex}}{U}_{1}(x,t)\in {\mathfrak{g}}^{\left(1\right)},\phantom{\rule{1.em}{0ex}}{V}_{k}(x,t)\in {\mathfrak{g}}^{(kmod3)},\hfill \\ & K\in {\mathfrak{g}}^{\left(1\right)},\phantom{\rule{1.em}{0ex}}J\in {\mathfrak{g}}^{\left(2\right)}.\hfill \end{array}$$We will also assume that the potentials vanish at spatial infinity, i.e.,$$\underset{x\to \pm \infty}{lim}{U}_{k}(x,t)=0,\phantom{\rule{1.em}{0ex}}\underset{x\to \pm \infty}{lim}{V}_{k}(x,t)=0.$$Usually, an ever more restrictive condition on the asymptotic behavior of the potentials is required. For the purposes of this paper, we will assume them to be Schwartz functions, but this might be too restrictive. This, of course, needs to be studied more rigorously and this will be accomplished in future works.
- The explicit form of the elements of L is:$$J={\mathcal{H}}_{1}^{\left(1\right)}=\left(\begin{array}{ccc}\omega & 0& 0\\ 0& {\omega}^{2}& 0\\ 0& 0& 1\end{array}\right),\phantom{\rule{1.em}{0ex}}{U}_{0}=\left(\begin{array}{ccc}0& {q}_{1}& {q}_{2}\\ {q}_{2}& 0& {q}_{1}\\ {q}_{1}& {q}_{2}& 0\end{array}\right),\phantom{\rule{1.em}{0ex}}{U}_{1}=\left(\begin{array}{ccc}0& {\omega}^{2}{q}_{3}& {\omega}^{2}{q}_{4}\\ \omega {q}_{4}& 0& {q}_{3}\omega \\ {q}_{3}& {q}_{4}& 0\end{array}\right)$$
- The choice of J determines the inverse of ${ad}_{J}$ to be (see Appendix C)$${\mathrm{ad}}_{J}^{-1}=-\frac{1}{27}{\mathrm{ad}}_{J}^{5}.$$
- The elements of M are:$$K={\mathcal{H}}_{1}^{\left(2\right)}=\left(\begin{array}{ccc}{\omega}^{2}& 0& 0\\ 0& \omega & 0\\ 0& 0& 1\end{array}\right),\phantom{\rule{1.em}{0ex}}{V}_{k}={v}_{1}^{\left(k\right)}{\mathcal{E}}_{{\alpha}_{1}}^{\left(k\right)}+{v}_{1}^{\left(k\right)}{\mathcal{E}}_{{\alpha}_{1}+{\alpha}_{2}}^{\left(k\right)}+{v}_{0}^{\left(k\right)}{\mathcal{H}}_{1}^{\left(k\right)},$$
- The zero-curvature condition $[L,M]=0$ leads to the following set of recursion relations with $N=4$:$$\begin{array}{cccc}\hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{N+2}:\hfill & \hfill \left[J,K\right]& =0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{N+1}:\hfill & \hfill \left[J,{V}_{N-1}\right]-\left[K,{U}_{1}\right]& =0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{N}:\hfill & \hfill \left[J,{V}_{N-2}\right]-\left[K,{U}_{0}\right]-\left[{U}_{1},{V}_{N-1}\right]& =0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{s}:\hfill & \hfill i{\partial}_{x}{V}_{s}+\left[{U}_{0},{V}_{s}\right]+\left[{U}_{1},{V}_{s-1}\right]-\left[J,{V}_{s-2}\right]& =0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{1}:\hfill & \hfill -i{\partial}_{t}{U}_{1}+i{\partial}_{x}{V}_{1}+\left[{U}_{0},{V}_{1}\right]+\left[{U}_{1},{V}_{0}\right]& =0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\lambda}^{0}:\hfill & \hfill -i{\partial}_{t}{U}_{0}+i{\partial}_{x}{V}_{0}+\left[{U}_{0},{V}_{0}\right]& =0.\hfill \end{array}$$
- The recursion relations can be solved by noting that each $X\in \mathfrak{g}$ can be decomposed as$$X={X}^{\perp}+{X}^{\parallel},\phantom{\rule{1.em}{0ex}}{\mathrm{ad}}_{J}{X}^{\parallel}=[J,{X}^{\parallel}]=0,$$
- This leads to the following solutions:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {V}_{N-1}={\mathrm{ad}}_{J}^{-1}\left[K,{U}_{1}\right],\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {V}_{N-2}={\mathrm{ad}}_{J}^{-1}\left(\left[K,{U}_{0}\right]+\left[{U}_{1},{V}_{N-1}\right]\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {V}_{s-2}^{\perp}={\mathrm{ad}}_{J}^{-1}\left(i{\partial}_{x}{V}_{s}+\left[{U}_{0},{V}_{s}^{\perp}\right]+\left[{U}_{0},{V}_{s}^{\parallel}\right]+\left[{U}_{1},{V}_{s-1}^{\perp}\right]+\left[{U}_{1},{V}_{s-1}^{\parallel}\right]\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {V}_{s}^{\parallel}=i{\partial}_{x}^{-1}\left({\left[{U}_{0},{V}_{s}^{\perp}\right]}^{\parallel}+{\left[{U}_{1},{V}_{s-1}^{\perp}\right]}^{\parallel}\right),\hfill \end{array}$$$${\partial}_{x}^{-1}f\left(x\right)={\int}_{-\infty}^{x}f\left(y\right)dy.$$Note that for any function vanishing at $-\infty $, this is equivalent to integrating and setting any constant of integration to zero. The above solutions to the recursion relations can be formalized with the help of recursion operators $\mathrm{\Lambda}$; see, for example [14,17]. However, calculating their explicit form in the case of polynomial Lax operators is more involved and writing their explicit form presents considerable difficulties.

## 4. Derivative NLS Equations

## 5. Fundamental Analytic Solutions of L and Scattering Data

- 1.
- The continuous spectrum of L fills up the set of rays ${l}_{\nu}$, $\nu =0,\cdots 11$ in the complex $\lambda $-plane for which (see Figure 1)$$\begin{array}{c}\hfill \mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}\alpha \left(J\right)=0,\end{array}$$$$\begin{array}{c}\hfill {l}_{\nu}=arg\lambda =\frac{\pi (\nu +\frac{1}{2})}{6},\phantom{\rule{2.em}{0ex}}\nu =0,\cdots 11.\end{array}$$Each ray is related to a subalgebra ${\mathfrak{g}}_{\nu}$ with root systems ${\delta}_{\nu}$ whose roots satisfy$$\begin{array}{c}\hfill {\delta}_{\nu}\equiv \{\alpha \in {\delta}_{\nu}\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}\mathrm{I}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}\alpha \left(J\right)=0,\phantom{\rule{1.em}{0ex}}\forall \lambda \in {l}_{\nu}\}.\end{array}$$More specifically, for this particular case we have$$\begin{array}{cc}\hfill {\delta}_{0}& \equiv \{\pm ({e}_{1}-{e}_{3})\},\hfill \\ \hfill {\delta}_{1}& \equiv \{\pm ({e}_{1}-{e}_{2})\},\hfill \\ \hfill {\delta}_{2}& \equiv \{\pm ({e}_{2}-{e}_{3})\},\hfill \end{array}$$
- 2.
- The regions of analyticity of the FAS ${\xi}_{\nu}(x,t,\lambda )$ are the sectors$$\begin{array}{c}\hfill {\mathrm{\Omega}}_{\nu}\equiv \left\{\frac{\pi (\nu +\frac{1}{2})}{6}\le arg\lambda \le \frac{\pi (\nu +1+\frac{1}{2})}{6}\right\}.\end{array}$$The FAS are introduced as the solutions of the following set of integral equations (written component-wise)$$\begin{array}{c}\hfill {\xi}_{\nu ,jk}(x,t,\lambda )={\delta}_{jk}+i{\int}_{{s}_{jk}\infty}^{x}dy\phantom{\rule{0.277778em}{0ex}}{e}^{-i{\lambda}^{2}({J}_{jj}-{J}_{kk})(x-y)}{\left(({U}_{0}(y,t)+\lambda {U}_{1}(y,t)){\xi}_{\nu}(y,t,\lambda )\right)}_{jk},\end{array}$$
- 3.
- In each sector ${\mathrm{\Omega}}_{\nu}$, the roots are ordered as follows: the root $\alpha $ is called $\nu $-positive (resp. $\nu $-negative) if $\mathrm{I}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}\alpha \left(J\right)>0$ (resp. $\mathrm{I}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}\alpha \left(J\right)<0$) for $\lambda \in {\mathrm{\Omega}}_{\nu}$. For example, the sets of positive roots of the subalgebras ${\mathfrak{g}}_{\nu}$ are$$\begin{array}{cc}\hfill {\delta}_{0}^{+}& \equiv \left\{({e}_{1}-{e}_{3})\right\},\hfill \\ \hfill {\delta}_{1}^{+}& \equiv \left\{({e}_{1}-{e}_{2})\right\},\hfill \\ \hfill {\delta}_{2}^{+}& \equiv \{-({e}_{2}-{e}_{3})\}.\hfill \end{array}$$Note that the root systems ${\delta}_{\nu}$ are isomorphic to the root system of $sl\left(2\right)$.
- 4.
- The scattering data is obtained by the limits of the FAS along both sides of the rays ${l}_{\nu}{e}^{\pm i0}$:$$\begin{array}{cc}\hfill \underset{x\to -\infty}{lim}{e}^{-i{\lambda}^{2}Jx}{\xi}_{\nu}(x,t,\lambda ){e}^{i{\lambda}^{2}Jx}& ={S}_{\nu}^{+}(t,\lambda ),\hfill \\ \hfill \underset{x\to \infty}{lim}{e}^{-i{\lambda}^{2}Jx}{\xi}_{\nu}(x,t,\lambda ){e}^{i{\lambda}^{2}Jx}& ={T}_{\nu}^{-}(t,\lambda ){D}_{\nu}^{+}\left(\lambda \right),\hfill \end{array}\phantom{\rule{1.em}{0ex}}\forall \lambda \in {l}_{\nu}{e}^{+i0},$$$$\begin{array}{cc}\hfill \underset{x\to -\infty}{lim}{e}^{-i{\lambda}^{2}Jx}{\xi}_{\nu}(x,t,\lambda ){e}^{i{\lambda}^{2}Jx}& ={S}_{\nu +1}^{-}(t,\lambda ),\hfill \\ \hfill \underset{x\to \infty}{lim}{e}^{-i{\lambda}^{2}Jx}{\xi}_{\nu}(x,t,\lambda ){e}^{i{\lambda}^{2}Jx}& ={T}_{\nu +1}^{+}(t,\lambda ){D}_{\nu +1}^{-}\left(\lambda \right),\hfill \end{array}\phantom{\rule{1.em}{0ex}}\forall \lambda \in {l}_{\nu +1}{e}^{-i0},$$$$\begin{array}{cc}\hfill {S}_{\nu}^{\pm}(t,\lambda )& =exp\left(\sum _{\alpha \in {\delta}_{\nu}^{+}}^{}{s}_{\alpha ,\nu}^{\pm}(t,\lambda ){E}_{\pm \alpha}\right),\hfill \\ \hfill {T}_{\nu}^{\pm}(t,\lambda )& =exp\left(\sum _{\alpha \in {\delta}_{\nu}^{+}}^{}{\tau}_{\alpha ,\nu}^{\pm}(t,\lambda ){E}_{\pm \alpha}\right),\hfill \\ \hfill {D}_{\nu}^{\pm}\left(\lambda \right)& =exp\left(\sum _{\alpha \in {\delta}_{\nu}^{+}}^{}{d}_{\alpha ,\nu}^{\pm}\left(\lambda \right){H}_{\alpha}\right),\hfill \end{array}$$$${T}_{\nu}^{-}{D}_{\nu}^{+}\widehat{{S}_{\nu}^{+}}={T}_{\nu}^{+}{D}_{\nu}^{-}\widehat{{S}_{\nu}^{-}}={T}_{\nu}(t,\lambda ),\phantom{\rule{2.em}{0ex}}\lambda \in {l}_{\nu}.$$Equation (37) is the Gauss decomposition of the scattering matrix ${T}_{\nu}(t,\lambda )$. Note that the functions ${D}_{\nu}^{+}$ and ${D}_{\nu +1}^{-}$ are analytic in the sector ${\mathrm{\Omega}}_{\nu}$.
- 5.
- It can be shown, that the fundamental analytic solutions ${\xi}_{\nu}(x,t,\lambda )$ satisfy a (multiplicative) Riemann–Hilbert problem (RHP):$$\begin{array}{cc}\hfill {\xi}_{\nu +1}(x,t,\lambda )& ={\xi}_{\nu}(x,t,\lambda ){G}_{\nu}(x,t,\lambda ),\hfill \\ \hfill \phantom{\rule{2.em}{0ex}}{G}_{\nu}(x,t,\lambda )& ={e}^{i{\lambda}^{2}Jx}{\widehat{S}}_{\nu +1}^{-}(t,\lambda ){S}_{\nu +1}^{+}(t,\lambda ){e}^{-i{\lambda}^{2}Jx},\hfill \end{array}$$$$\underset{\lambda \to \infty}{lim}{\xi}_{\nu}(x,t,\lambda )=1\phantom{\rule{-3.3pt}{0ex}}1.$$It follows from the generalization of Zakharov–Shabat theorem for an L operator, quadratic in the spectral parameter [18], that the solution of the RHP (38) with canonical normalization is an FAS of the system$$i\frac{\partial {\xi}_{\nu}}{\partial x}+({U}_{0}+\lambda {U}_{1}){\xi}_{\nu}(x,t,\lambda )-{\lambda}^{2}[J,{\xi}_{\nu}(x,t,\lambda )]=0,$$To make this more precise, since ${\xi}_{\nu}(x,t,\lambda )$ is canonically normalized, it has an asymptotic form given by$$\begin{array}{c}\hfill {\xi}_{\nu}(x,t,\lambda )=exp\left(\mathcal{Q}(x,t,\lambda )\right),\end{array}$$$$\begin{array}{c}\hfill \mathcal{Q}(x,t,\lambda )=\sum _{s=1}^{\infty}{\lambda}^{-s}{Q}_{s}(x,t).\end{array}$$Following the idea of Gel’fand and Dikii [23], it can be shown [18] that for quadratic Lax operators only the first two terms are needed$$\mathcal{Q}(x,t,\lambda )=\frac{{Q}_{1}(x,t)}{\lambda}+\frac{{Q}_{2}(x,t)}{{\lambda}^{2}}+....$$$$\begin{array}{cc}\hfill {U}_{1}& ={\mathrm{ad}\phantom{\rule{0.166667em}{0ex}}}_{J}{Q}_{1},\hfill \\ \hfill {U}_{0}& ={\mathrm{ad}\phantom{\rule{0.166667em}{0ex}}}_{J}{Q}_{2}-\frac{1}{2}{\mathrm{ad}\phantom{\rule{0.166667em}{0ex}}}_{{Q}_{1}}^{2}J.\hfill \end{array}$$The above can be inverted, allowing us to express ${Q}_{1}(x,t)$ and ${Q}_{2}(x,t)$ in terms of the potentials$$\begin{array}{cc}\hfill {Q}_{1}& ={\mathrm{a}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}}_{J}^{-1}{U}_{1},\hfill \\ \hfill {Q}_{2}& ={\mathrm{a}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}}_{J}^{-1}\left({U}_{0}-\frac{1}{2}\left[{U}_{1},{\mathrm{a}\mathrm{d}\phantom{\rule{0.166667em}{0ex}}}_{J}^{-1}{U}_{1}\right]\right),\hfill \end{array}$$

**Theorem**

**1.**

**Proof.**

- Looking at the first equation from (36), the set of scattering data $\mathcal{T}$ uniquely determines the matrices ${S}_{1}^{\pm}(t,\lambda )$ for $\lambda \in {l}_{1}{e}^{\pm i0}$, ${S}_{2}^{\pm}(t,\lambda )$ for $\lambda \in {l}_{2}{e}^{\pm i0}$ and ${S}_{3}^{\pm}(t,\lambda )$ for $\lambda \in {l}_{3}{e}^{\pm i0}$. The Coxeter reduction implies that$$\begin{array}{cc}\hfill {S}_{3\nu}^{\pm}(t,\lambda )& ={c}^{\nu}{S}_{0}^{\pm}(t,{\omega}^{-\nu}\lambda ){c}^{-\nu},\phantom{\rule{2.em}{0ex}}\lambda \in {l}_{\nu}{e}^{\pm i0},\hfill \\ \hfill {S}_{3\nu +1}^{\pm}(t,\lambda )& ={c}^{\nu}{S}_{1}^{\pm}(t,{\omega}^{-\nu}\lambda ){c}^{-\nu},\phantom{\rule{2.em}{0ex}}\lambda \in {l}_{\nu +1}{e}^{\pm i0},\hfill \\ \hfill {S}_{3\nu +2}^{\pm}(t,\lambda )& ={c}^{\nu}{S}_{2}^{\pm}(t,{\omega}^{-\nu}\lambda ){c}^{-\nu},\phantom{\rule{2.em}{0ex}}\lambda \in {l}_{\nu +2}{e}^{\pm i0}.\hfill \end{array}$$This determines ${S}_{\nu}^{\pm}\left(\lambda \right)$ on the rest of the rays.
- ${D}_{\nu}^{\pm}\left(\lambda \right)$ are determined uniquely by ${S}_{\nu}^{\pm}(t,\lambda )$. The regularity of ${\xi}_{\nu}(x,t,\lambda )$ implies that the functions ${D}_{\nu}^{\pm}\left(\lambda \right)$ are also regular, i.e., have no zeros or singularities. This also means that the functions ${d}_{\alpha ,\nu}^{\pm}\left(\lambda \right)$ from the last equation in (36) are analytical. We will use the regularity of ${D}_{\nu}^{\pm}\left(\lambda \right)$ along with (37). In what follows, the reader is assumed to have some familiarity with the representation theory of simple Lie algebras.Let ${\omega}_{\nu ,j}^{+}$ be the j-th fundamental weight of the subalgebra ${\mathfrak{g}}_{\nu}$ evaluated (with respect to the index j) with the ordering in ${\mathrm{\Omega}}_{\nu}$. Let $|{\omega}_{\nu ,j}^{\pm}\rangle $ (here, we are using standard Bra–ket notation) be the corresponding weight vector in the fundamental representation of ${\mathfrak{g}}_{\nu}$ that has highest weight ${\omega}_{\nu ,j}^{+}$ (respectively, ${\omega}_{\nu ,j}^{-}$ is the lowest weight). Then, considering that for all $\alpha \in {\delta}_{\nu}^{+}$$${E}_{\alpha}|{\omega}_{\nu ,j}^{+}\rangle =0,\phantom{\rule{1.em}{0ex}}{E}_{-\alpha}|{\omega}_{\nu ,j}^{-}\rangle =0,\phantom{\rule{1.em}{0ex}}\langle {\omega}_{\nu ,j}^{+}|{E}_{-\alpha}=0,\phantom{\rule{1.em}{0ex}}\langle {\omega}_{\nu ,j}^{-}|{E}_{\alpha}=0,$$$$\begin{array}{cccc}\hfill \langle {\omega}_{\nu ,j}^{+}|{S}_{\nu}^{-}& =\langle {\omega}_{\nu ,j}^{+}|{T}_{\nu}^{-}=\langle {\omega}_{\nu ,j}^{+}|,\hfill & \hfill \phantom{\rule{1.em}{0ex}}\langle {\omega}_{\nu ,j}^{-}|{S}_{\nu}^{+}& =\langle {\omega}_{\nu ,j}^{-}|{T}_{\nu}^{+}=\langle {\omega}_{\nu ,j}^{-}|,\hfill \\ \hfill {S}_{\nu}^{+}|{\omega}_{\nu ,j}^{+}\rangle & ={T}_{\nu}^{+}|{\omega}_{\nu ,j}^{+}\rangle =|{\omega}_{\nu ,j}^{+}\rangle ,\hfill & \hfill \phantom{\rule{1.em}{0ex}}{S}_{\nu}^{-}|{\omega}_{\nu ,j}^{-}\rangle & ={T}_{\nu}^{-}|{\omega}_{\nu ,j}^{-}\rangle =|{\omega}_{\nu ,j}^{-}\rangle \hfill \end{array}$$$$\begin{array}{cc}\hfill {D}_{\nu}^{+}|{\omega}_{\nu ,j}^{+}\rangle & ={e}^{{d}_{{\alpha}_{j},\nu}^{+}}|{\omega}_{\nu ,j}^{+}\rangle ,\phantom{\rule{2.em}{0ex}}{D}_{\nu}^{-}|{\omega}_{\nu ,j}^{-}\rangle ={e}^{-{d}_{{\alpha}_{j},\nu}^{-}}|{\omega}_{\nu ,j}^{-}\rangle .\hfill \end{array}$$Analogous relations can also be derived for the inverses. Note that we can rewrite (37) in the form of$$\begin{array}{c}\hfill {D}_{\nu}^{-}\widehat{{S}_{\nu}^{-}}{S}_{\nu}^{+}{\widehat{D}}_{\nu}^{+}={\widehat{T}}_{\nu}^{+}{T}_{\nu}^{-},\phantom{\rule{2.em}{0ex}}\lambda \in {l}_{\nu}.\end{array}$$Then, by squeezing (51) between $\langle {\omega}_{\nu ,j}^{\pm}|\dots |{\omega}_{\nu ,j}^{\pm}\rangle $ we obtain that$$\begin{array}{cc}\hfill {d}_{{\alpha}_{j},\nu}^{+}-{d}_{{\alpha}_{j},\nu}^{-}& =ln\langle {\omega}_{\nu ,j}^{-}\left|{\widehat{S}}_{\nu}^{-}{S}_{\nu}^{+}\right|{\omega}_{\nu ,j}^{-}\rangle \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-ln\langle {\omega}_{\nu ,j}^{+}\left|{\widehat{T}}_{\nu}^{+}{T}_{\nu}^{-}\right|{\omega}_{\nu ,j}^{+}\rangle .\hfill \end{array}$$The functions ${d}_{{\alpha}_{j},\nu}^{\pm}$ can be recovered uniquely from their analyticity properties and from the jumps (52) along the rays ${l}_{\nu}$. This in turn determines ${D}_{\nu}^{\pm}\left(\lambda \right)$. The exact details are essentially the same as for linear Lax operators and can be found in [16]).
- The matrices ${T}_{\nu}^{\pm}(t,\lambda )$ are recovered as the Gauss factors of the right hand side of (51).
- Finally, the corresponding potentials are reconstructed from the regular solutions of the RHP (38) by taking the limit $\lambda \to \infty $ in$$\begin{array}{c}\hfill \left(\frac{\partial {\xi}_{\nu}}{\partial x}(x,t,\lambda )\right){\widehat{\xi}}_{\nu}(x,t,\lambda )+{U}_{0}(x,t)+\lambda {U}_{1}(x,t)={\lambda}^{2}\left(J-{\xi}_{\nu}(x,t,\lambda )J{\widehat{\xi}}_{\nu}(x,t,\lambda )\right).\end{array}$$

## 6. Time Dependence of the Scattering Data

## 7. Concluding Remarks

- The first is a rigorous study of the mapping between potential $({U}_{0},{U}_{1})$ and scattering matrix ${T}_{\nu}$. In defining the FAS, we assumed that Equation (32) has a solution which is obviously not true for all classes of potentials (it is true for potentials on compact support and for Schwartz functions). The first step is a mathematically rigorous definition of the class of admissible potentials, such that the mapping $({U}_{0},{U}_{1})\mapsto {T}_{\nu}$ and its inverse are correctly defined. This problem in the case of linear L operators [16,17,23] is rather involved and the same is expected to be true for the quadratic case.
- There is a hierarchy of integrable systems of equations related to a single L operator. This can be derived with the help of the recursion operators. Finding their explicit form is somewhat difficult. One can infer from the solution of the recursion relations, i.e., Equation (36), that for an L operator of order m in $\lambda $, the recursion operator will have m arguments (i.e., a tensor of rank $m+1$).
- The hierarchy of integrable equations admits a Hamiltonian formulation. Since the factors ${D}_{\nu}^{\pm}$ generate the integrals of motion for the system, they can be used to find the Hamiltonian. In the general case, the Hamiltonian can also be found by using the recursion operators [14].
- Finding the multi-soliton solutions of the corresponding equations. This can be done, for example, by using the dressing method, with the procedure being more involved for polynomial L operators [18].
- If the soliton solutions are found by the dressing method, then the soliton dynamics and interactions can be studied by considering the asymptotic behavior of the dressing factor.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Simple Lie Algebra **A**_{2}

_{2}

## Appendix B. Basis in ${\mathit{A}}_{\mathbf{2}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$

## Appendix C. The Inverse of ${\mathbf{ad}}_{\mathit{J}}$

## References

- Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett.
**1967**, 19, 1095–1097. [Google Scholar] [CrossRef] - Lax, P.D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.
**1968**, 21, 467–490. [Google Scholar] [CrossRef] - Zakharov, V.E.; Shabat, A.B. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I Funct. Anal. Appl.
**1974**, 8, 226–235. [Google Scholar] [CrossRef] - Zakharov, V.E.; Shabat, A.B. Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl.
**1979**, 13, 166–174. [Google Scholar] [CrossRef] - Manakov, S.V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP
**1974**, 38, 248–253. [Google Scholar] - Kaup, D.J.; Newell, A.C. An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys.
**1978**, 19, 798–801. [Google Scholar] [CrossRef] - Chen, H.H.; Lee, Y.C.; Liu, C.S. Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method. Phys. Scr.
**1979**, 20, 490. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Ivanov, M.I. The quadratic bundle of general form and the nonlinear evolution equations: Expansions over the “squared” solutions-generalized Fourier transform. Bulg. J. Phys.
**1983**, 10, 130. [Google Scholar] - Mikhailov, A.V. The reduction problem and the inverse scattering method. Phys. Nonlinear Phenom.
**1981**, 3, 73–117. [Google Scholar] [CrossRef] - Gerdjikov, V.S. Derivative Nonlinear Schrödinger Equations with Z
_{N}and D_{N}–Reductions. Rom. J. Phys.**2013**, 58, 573–582. [Google Scholar] - Drinfeld, V.G.; Sokolov, V.V. Equations of KdV type and simple Lie algebras. Sov. Math. Dokl.
**1981**, 23, 457–462. [Google Scholar] - Drinfel’d, V.G.; Sokolov, V.V. Lie algebras and equations of Korteweg-de Vries type. Sov. J. Math.
**1985**, 30, 1975–2036. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Mladenov, D.M.; Stefanov, A.A.; Varbev, S.K. Integrable equations and recursion operators related to the affine Lie algebras ${A}_{r}^{\left(1\right)}$. J. Math. Phys.
**2015**, 56, 052702. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Stefanov, A.A.; Iliev, I.D.; Boyadjiev, G.P.; Smirnov, A.O.; Matveev, V.B.; Pavlov, M.V. Recursion operators and hierarchies of mKdV equations related to the Kac–Moody algebras ${D}_{\left(1\right)}^{4}$, ${D}_{\left(2\right)}^{4}$, and ${D}_{\left(3\right)}^{4}$. Theor. Math. Phys.
**2020**, 204, 1110–1129. [Google Scholar] [CrossRef] - Beals, R.; Coifman, R.R. Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math.
**1984**, 37, 39–90. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Yanovski, A.B. Completeness of the eigenfunctions for the Caudrey-Beals-Coifman system. J. Math. Phys.
**1994**, 35, 3687–3725. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Yanovski, A.B. CBC systems with Mikhailov reductions by Coxeter Automorphism: I. Spectral Theory of the Recursion Operators. Stud. Appl. Math.
**2015**, 134, 145–180. [Google Scholar] [CrossRef] - Gerdjikov, V.S.; Stefanov, A.A. Riemann-Hilbert problems, polynomial Lax pairs, integrable equations and their soliton solutions. Symmetry
**2023**, 15, 1933. [Google Scholar] [CrossRef] - Krishnaswami, G.S.; Vishnu, T.R. An introduction to Lax pairs and the zero curvature representation. arXiv
**2020**, arXiv:2004.05791v1. [Google Scholar] - Carter, R. Lie Algebras of Finite and Affine Type; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Kac, V. Infinite Dimensional Lie Algebras; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Helgasson, S. Differential Geometry, Lie Groups and Symmetric Spaces; Graduate Studies in Mathematics 34; AMS: Providence, RI, USA, 2012. [Google Scholar]
- Gel’fand, I.M.; Dikii, L.A. The calculus of jets and nonlinear Hamiltonian systems. Funct. Anal. Appl.
**1978**, 12, 81–94. [Google Scholar] [CrossRef]

**Figure 1.**The continuous spectrum of L fills up the rays ${l}_{\nu}$, $\nu =0,\cdots ,11$. Here, ${\mathrm{\Omega}}_{\nu}$ denotes the sectors of analyticity of the FAS ${\xi}_{\nu}(x,t,\lambda )$.

**Table 1.**The signs ${s}_{jk}$ in (32). The table contains the values of ${s}_{jk}$ for $j<k$ because ${s}_{kj}=-{s}_{jk}$. Also, the signs for ${\mathrm{\Omega}}_{\nu +3}$ and ${\mathrm{\Omega}}_{\nu +9}$ are opposite to the signs for ${\mathrm{\Omega}}_{\nu}$, $\nu =0,\cdots ,2$, while the signs for ${\mathrm{\Omega}}_{\nu +6}$ are the same. The signs are chosen in a way that ensures that the integral Equation (32) is correctly defined.

$(1,2)$ | $(1,3)$ | $(2,3)$ | |
---|---|---|---|

${\mathrm{\Omega}}_{0}$ | − | + | + |

${\mathrm{\Omega}}_{1}$ | + | + | + |

${\mathrm{\Omega}}_{2}$ | + | + | − |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stefanov, A.A.
New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra *Dynamics* **2024**, *4*, 81-96.
https://doi.org/10.3390/dynamics4010005

**AMA Style**

Stefanov AA.
New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra *Dynamics*. 2024; 4(1):81-96.
https://doi.org/10.3390/dynamics4010005

**Chicago/Turabian Style**

Stefanov, Aleksander Aleksiev.
2024. "New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra *Dynamics* 4, no. 1: 81-96.
https://doi.org/10.3390/dynamics4010005