On a Model of Propagation of Two Component Nonlinear OpticalWaves

1. Structure of the Lax Operator for a Quadratic Spectral Bundle

Let the Lax pair be given by the equations

$$\begin{array}{cc}\hfill & i{\Psi }_t+U\Psi =\mathbf{0},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & i{\Psi }_z+V\Psi =\mathbf{0},\hfill \end{array}$$

(3)

where

$$U={\lambda }^{2}J+\lambda Q_1+Q_2,V={\lambda }^{2}K+\lambda S_1+S_2,$$

(4)

$J,K$ are constant matrices, $\lambda$ is a spectral parameter.

The condition of compatibility of the equations (2) and (3) has the form

$$i{V}_t-i{U}_z+UV-VU=\mathbf{0}$$

or

$$\begin{array}{c}{\lambda }^4(JK-KJ)+{\lambda }^3(J{S}_1-S_{1}J+Q_{1}K-K{Q}_1)\hfill \\ \hfill +{\lambda }^2(J{S}_2-S_{2}J+Q_1{S}_1-S_1{Q}_1+Q_{2}K-K{Q}_2)+\cdots =\mathbf{0}.\end{array}$$

(5)

The condition (5), along with integrable nonlinear equations, gives algebraic constraints on the elements of the matrices U and V. These restrictions have the following form

$$\begin{array}{cc}\hfill & JK=KJ,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & [J,S_1]=[K,Q_1],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & [J,S_2]+Q_1{S}_1=[K,Q_2]+S_1{Q}_1.\hfill \end{array}$$

(8)

Usually solutions of the equation (7) is written using the relations

$$S_1=[K,R],Q_1=[J,R],$$

(9)

where R is some matrix of the same size.

Substituting (9) into (8) and simplifying, we get

$$\begin{array}{c}J(S_2+RKR-R^{2}K)-(S_2+RKR-R^{2}K)J\\ =K(Q_2+RJR-R^{2}J)-(Q_2+RJR-R^{2}J)K.\end{array}$$

or

$$[J,S_2+[RK,R]]=[K,Q_2+[RJ,R]].$$

Therefore, the following representation of the matrices $S_2$ and $Q_2$ can be used

$$\begin{array}{cc}\hfill & S_2+[RK,R]=[K,R_2],\hfill \\ \hfill & Q_2+[RJ,R]=[J,R_2]\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill & S_2=[K,R_2]+[R,RK],\hfill \\ \hfill & Q_2=[J,R_2]+[R,RJ],\hfill \end{array}$$

(10)

where $R_2$ is a new matrix of the same size. Note that in the right-hand sides of the equalities (9) and (10) it is possible to add linear combinations of matrices commuting with J and K.

Taking into account the remaining constraints from the condition (5) leads to the following matrix of operators (2) and (3)

$$\begin{array}{cc}\hfill & S_1=[K,R],Q_1=[J,R],\hfill \\ \hfill & S_2=[K,R_2]+[R,RK]+s_{20}K,\hfill \\ \hfill & Q_2=[J,R_2]+[R,RJ]+q_{20}J,\hfill \end{array}$$

(11)

where $s_{20}$ and $q_{20}$ are some functions. It is easy to see that the matrix J completely defines the structure of the operator (2). For example, if $J=\mathrm{diag}(1,-1)$, then

$$\begin{array}{cc}\hfill & Q_1=\left(\begin{array}{cc}0& 2r_{12}\\ -2r_{21}& 0\end{array}\right),\hfill \\ \hfill & Q_2=\left(\begin{array}{cc}0& 2{\tilde{r}}_{12}\\ -2{\tilde{r}}_{21}& 0\end{array}\right)+\left(\begin{array}{cc}2r_{12}{r}_{21}& -2r_{11}{r}_{12}\\ 2r_{21}{r}_{22}& -2r_{12}{r}_{21}\end{array}\right)+\left(\begin{array}{cc}{q}_{20}& 0\\ 0& -q_{20}\end{array}\right).\hfill \end{array}$$

In the case of the main variants of the derivative nonlinear Schrödinger equations, we have ($q=-p^{*}$)

• in the case of the Kaup-Newell equation [15,16,17,20,25,26,29,30,32,33]

  $$r_{12}=ip/2,r_{21}=-iq/2,{\tilde{r}}_{12}=r_{11}{r}_{12},{\tilde{r}}_{21}=r_{21}{r}_{22},q_{20}=-2r_{12}{r}_{21};$$
• in the case of the Chen-Lee-Liu equation [22,25,26,29,30,32,33]

  $$r_{12}=ip/2,r_{21}=-iq/2,{\tilde{r}}_{12}=r_{11}{r}_{12},{\tilde{r}}_{21}=r_{21}{r}_{22},q_{20}=-pq/4;$$
• in the case of the Gerdjikov-Ivanov equation [25,26,27,28,29,30,32,33]

  $$r_{12}=ip/2,r_{21}=-iq/2,{\tilde{r}}_{12}=r_{11}{r}_{12},{\tilde{r}}_{21}=r_{21}{r}_{22},q_{20}=0.$$

By choosing other values of $q_{20}$, other special cases of the generalized derivative nonlinear Schrödinger equation can be obtained [31,33,67,68,69]. Note that the solutions of these equations are connected by a gauge transformation preserving the amplitude (see for ex. [26,32,33,70,71,72]).

It is easy to notice that in these models, the matrices $Q_1$ and $Q_2$ depend only on two functions p and q, and the matrix $Q_2$ is diagonal. Therefore, the addition of new functions u and v to the non-diagonal terms of the matrix $Q_2$ allows us to explore a new nonlinear integrable model:

$$r_{12}=p/2,r_{21}=q/2,{\tilde{r}}_{12}=r_{11}{r}_{12}+u/2,{\tilde{r}}_{21}=r_{21}{r}_{22}+v/2,q_{20}=0.$$

This model at $u=v=0$ is transformed into one of the special cases of the generalized derivative nonlinear Schrödinger equation [33]. The matrices J, $Q_1$ and $Q_2$ in the case of the new model are equal

$$J=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),Q_1=\left(\begin{array}{cc}0& p\\ -q& 0\end{array}\right),Q_2={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}pq& 2u\\ -2v& -pq\end{array}\right).$$

(12)

2. The Monodromy Matrix

The monodromy matrix is a key object of spectral analysis of periodic solutions of the integrable nonlinear models. Spectral data in the case of periodic nonlinear signals is the spectral curve, its genus and its parameters. The equation of the spectral curve is the characteristic equation of the monodromy matrix M, which is a polynomial of the spectral parameter $\lambda$

$$M=\sum _{j=0}^N{m}_j\left(t\right){\lambda }^j,$$

and which satisfies the equation [73]

$$i{M}_x+UM-MU=\mathbf{0}.$$

(13)

The monodromy matrix also exists in the limiting cases when the solution periods become infinite. In particular, solitary waves are the limiting cases of periodic waves when the period of a nonlinear wave becomes infinitely large.

From equation (13), where the matrix U is determined by the equalities (4), (12), the following structure of the matrix M follows

$$M=W_n+\sum _{k=1}^{n-1}{c}_k{W}_{n-k}+c_{n}U+c_{n+1}{W}_{-1}+c_{n+2}J,$$

(14)

where $W_{-1}=\lambda J+Q_1$, $U=\lambda W_{-1}+Q_2$, $W_1=\lambda U+W_1^0$,

$$W_{k+1}=\lambda V_k+W_{k+1}^0,W_k^0=\left(\begin{array}{cc}{F}_k& H_k\\ -G_k& -F_k\end{array}\right),k\ge 1,$$

$c_k$ are some real constants. From Equation (6) it also follows the recurrence relations on the elements of the matrix $W_k^0$:

$$\begin{array}{cc}\hfill & H_1=-{\displaystyle \frac{i}{2}}{p}_t,G_1={\displaystyle \frac{i}{2}}{q}_t,\hfill \\ \hfill & H_2=p{F}_1-{\displaystyle \frac{i}{2}}{u}_t,G_2=q{F}_1+{\displaystyle \frac{i}{2}}{v}_t,\hfill \\ \hfill & H_{k+2}=p{F}_{k+1}+u{F}_k-{\displaystyle \frac{1}{2}}pq{H}_k-{\displaystyle \frac{i}{2}}{\partial }_t{H}_k,\hfill \\ \hfill & G_{k+2}=q{F}_{k+1}+v{F}_k-{\displaystyle \frac{1}{2}}pq{G}_k+{\displaystyle \frac{i}{2}}{\partial }_t{G}_k,\hfill \\ \hfill & {\partial }_t{F}_k=i\left(v{H}_k-u{G}_k+q{H}_{k+1}-p{G}_{k+1}\right).\hfill \end{array}$$

(15)

In particular,

$$\begin{array}{cc}\hfill & F_1={\displaystyle \frac{1}{2}}(pv+qu),\hfill \\ \hfill & H_2={\displaystyle \frac{1}{2}}(pqu+p^{2}v-i{u}_t),\hfill \\ \hfill & G_2={\displaystyle \frac{1}{2}}(pqv+q^{2}u+i{v}_x),\hfill \\ \hfill & F_2={\displaystyle \frac{1}{2}}uv-{\displaystyle \frac{1}{8}}{p}^2{q}^2+{\displaystyle \frac{i}{4}}(p{q}_t-q{p}_t),\hfill \\ \hfill & H_3=uvp+{\displaystyle \frac{1}{2}}{u}^{2}q-{\displaystyle \frac{1}{8}}{p}^3{q}^2+{\displaystyle \frac{i}{4}}{p}^2{q}_t-{\displaystyle \frac{1}{4}}{p}_{tt},\hfill \\ \hfill & G_3=uvq+{\displaystyle \frac{1}{2}}{v}^{2}p-{\displaystyle \frac{1}{8}}{p}^2{q}^3-{\displaystyle \frac{i}{4}}{q}^2{p}_t-{\displaystyle \frac{1}{4}}{q}_{tt},\hfill \\ \hfill & F_3={\displaystyle \frac{1}{4}}(uq+pv)pq+{\displaystyle \frac{i}{4}}(p{v}_t-v{p}_t+u{q}_t-q{u}_t),\hfill \\ \hfill & H_4={\displaystyle \frac{1}{2}}{u}^{2}v-{\displaystyle \frac{1}{8}}{p}^2{q}^{2}u-{\displaystyle \frac{i}{4}}(2qu+3pv)p_t+{\displaystyle \frac{i}{4}}(u{q}_t-q{u}_t)p-{\displaystyle \frac{1}{4}}{u}_{tt},\hfill \\ \hfill & G_4={\displaystyle \frac{1}{2}}u{v}^2-{\displaystyle \frac{1}{8}}{p}^2{q}^{2}v+{\displaystyle \frac{i}{4}}(3qu+2pv)q_t+{\displaystyle \frac{i}{4}}(p{v}_t-v{p}_t)q-{\displaystyle \frac{1}{4}}{v}_{tt},\hfill \\ \hfill & F_4=pquv-{\displaystyle \frac{1}{16}}{p}^3{q}^3+{\displaystyle \frac{3}{8}}(p^2{v}^2+q^2{u}^2)+{\displaystyle \frac{1}{8}}{p}_t{q}_t\hfill \\ \hfill & +{\displaystyle \frac{i}{4}}(u{v}_t-v{u}_t)-{\displaystyle \frac{1}{8}}(p{q}_{tt}+q{p}_{tt}).\hfill \end{array}$$

It follows from recurrent relations (15) that when the reality conditions

$$q=\sigma p^{*},v=\sigma u^{*},\sigma =\pm 1,$$

(16)

are met, the elements of the monodromy matrix satisfy the following relations

$$G_k=\sigma H_k^{*},F_k^{*}=F_k.$$

(17)

3. Conservation Laws

Conservation laws are described by the following stationary nonlinear differential equations

$$\begin{array}{cc}\hfill & H_{n+1}+\sum _{j=1}^n{c}_j{H}_{n+1-j}+c_{n+1}u+c_{n+2}p=0,\hfill \\ \hfill & G_{n+1}+\sum _{j=1}^n{c}_j{G}_{n+1-j}+c_{n+1}v+c_{n+2}q=0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & H_{n+2}-p{F}_{n+1}+\sum _{j=1}^{n-1}{c}_j\left(H_{n+2-j}-p{F}_{n+1-j}\right)\hfill \\ \hfill & -{\displaystyle \frac{i{c}_n}{2}}{u}_x-{\displaystyle \frac{c_{n+1}}{2}}\left(p^{2}q+i{p}_x\right)+c_{n+2}u=0,\hfill \\ \hfill & G_{n+2}-q{F}_{n+1}+\sum _{j=1}^{n-1}{c}_j\left(G_{n+2-j}-q{F}_{n+1-j}\right)\hfill \\ \hfill & +{\displaystyle \frac{i{c}_n}{2}}{v}_x-{\displaystyle \frac{c_{n+1}}{2}}\left(p{q}^2-i{q}_x\right)+c_{n+2}v=0.\hfill \end{array}$$

These equations also follow from equation (13).

Any m-phase solution for $m\le n$ and for all values of t and z satisfies these stationary equations. The parameters of the corresponding multiphase solution depend on the constants $c_k$ and coefficients $f_k$ of the spectral curve (see below). It follows from the realness conditions (17) that stationary equations admit reduction (16).

In particular, for $n=0$, the stationary equations have the form

$$\begin{array}{cc}\hfill & i{p}_t-2c_{1}u-2c_{2}p=0,\hfill \\ \hfill & i{q}_t+2c_{1}v+2c_{2}q=0,\hfill \\ \hfill & i{u}_t+c_1(p^{2}q+i{p}_x)-2c_{2}u=0,\hfill \\ \hfill & i{v}_t-c_1(p{q}^2-i{q}_x)+2c_{2}v=0.\hfill \end{array}$$

(18)

It is easy to see that for $c_1\ne 0$, the components u and v are connected to the components p and q using the following relations

$$u=-{\displaystyle \frac{1}{2c_1}}(2c_{2}p-i{p}_x),v=-{\displaystyle \frac{1}{2c_1}}(2c_{2}q+i{q}_x).$$

As will be shown below, in this case all components are expressed in terms of elliptic functions or their degenerations. Note that for $c_1=0$, the system of stationary equations (18) splits into two separate identical systems, the solutions of which are plane waves. Accordingly, when $c_1=0$, additional components can be removed from the model by putting $u=v=0$.

Recall that the characteristic equation of the monodromy matrix M is the equation of the corresponding spectral curve

$$\Gamma :\mathcal{R}(\mu ,\lambda )=det(M-\mu I)=0.$$

(19)

Here I is identity matrix. It follows from the equations (14) and (19) that the equation of the spectral curve has the form

$${\mu }^2={\lambda }^{2n+4}+\sum _{k=1}^{2n+4}{f}_k{\lambda }^{2n+4-k},$$

(20)

where $f_k$ are constants (integrals). Therefore, the spectral curve $\Gamma$ is a hyperelliptic curve of genus $g=n+1$. Naturally, this statement is true only for equations corresponding to non-degenerate connected curves.

For n=0 (g=1), the equation of the spectral curve has the form

$${\mu }^2={\lambda }^4+2c_1{\lambda }^3+(c_1^2+2c_2){\lambda }^2+f_3\lambda +f_4,$$

where the integrals $f_3$ and $f_4$ are equal

$$\begin{array}{cc}\hfill & f_3=2c_1{c}_2-c_{1}pq-uq-pv,\hfill \\ \hfill & f_4=c_2^2+(c_2-c_1^2)pq+{\displaystyle \frac{1}{4}}{p}^2{q}^2-uv-c_1(pv+uq).\hfill \end{array}$$

I.e., for $n=0$, the nondegenerate spectral curve is elliptic.

For $n=1$ ($g=2$), the stationary equations have a more complex form

$$\begin{array}{cc}\hfill & i{u}_t-(pv+qu)p+i{c}_1{p}_t-2c_{2}u-2c_{3}p=0,\hfill \\ \hfill & i{v}_t+(pv+qu)q+i{c}_1{q}_t+2c_{2}v+2c_{3}q=0,\hfill \\ \hfill & p_{tt}-2(pv+qu)u-ipq{p}_t+2i{c}_1{u}_t+2c_2(p^{2}q+i{p}_t)-4c_{3}u=0,\hfill \\ \hfill & q_{tt}-2(qu+pv)v+ipq{q}_t-2i{c}_1{v}_t+2c_2(p{q}^2-i{q}_t)-4c_{3}v=0.\hfill \end{array}$$

(21)

It follows from the first two equations of system (21) that in the case $g=2$, the dependence of the components u and v on p and q can be found from the solution of the following linear matrix differential equation

$$i{\partial }_t\left(\genfrac{}{}{0pt}{}{u}{v}\right)-\left(\begin{array}{cc}pq+2c_2& p^2\\ -q^2& -pq-2c_2\end{array}\right)\left(\genfrac{}{}{0pt}{}{u}{v}\right)=-i{c}_1{\partial }_t\left(\genfrac{}{}{0pt}{}{p}{q}\right)+\left(\begin{array}{cc}2c_3& 0\\ 0& -2c_3\end{array}\right)\left(\genfrac{}{}{0pt}{}{p}{q}\right).$$

It is easy to see that for $c_1=c_3=0$, this equation admits solutions of the form $u=v=0$. In this case, the stationary equations for the components p and q will take a simpler form

$$\begin{array}{cc}\hfill & p_{tt}-ipq{p}_t+2c_2(p^{2}q+i{p}_t)=0,\hfill \\ \hfill & q_{tt}+ipq{q}_t+2c_2(p{q}^2-i{q}_t)=0.\hfill \end{array}$$

(22)

At the same time, it is not difficult to see that for $c_1=c_3=0$ there are solutions of stationary equations (21) that satisfy the condition $u^2+v^2\neg \equiv 0$.

The equation of the spectral curve for $n=1$ has the form

$${\mu }^2={\lambda }^6+2c_1{\lambda }^5+(c_1^2+2c_2){\lambda }^4+2(c_1{c}_2+c_3){\lambda }^3+f_4{\lambda }^2+f_5\lambda +f_6,$$

where the integrals $f_4$, $f_5$ and $f_6$ are equal

$$\begin{array}{cc}\hfill & f_4=c_2^2+2c_1{c}_3-c_1(pv+qu)-c_{2}pq+{\displaystyle \frac{1}{4}}{p}^2{q}^2-uv-{\displaystyle \frac{i}{2}}(p{q}_t-q{p}_t),\hfill \\ \hfill & f_5=2c_2{c}_3+(c_3-c_1{c}_2)pq-c_1^2(pv+qu)+{\displaystyle \frac{1}{2}}{c}_1{p}^2{q}^2-2c_{1}uv+{\displaystyle \frac{1}{2}}pq(pv+qu)\hfill \\ \hfill & -{\displaystyle \frac{i}{2}}{c}_1(p{q}_t-q{p}_t)-{\displaystyle \frac{i}{2}}(u{q}_t-v{p}_t),\hfill \\ \hfill & f_6=c_3^2+(c_1{c}_3-c_2^2)pq+{\displaystyle \frac{1}{4}}{c}_1^2{p}^2{q}^2-c_1^{2}uv+(c_3-c_1{c}_2)(pv+qu)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{c}_{1}pq(pv+qu)+{\displaystyle \frac{1}{4}}(p^2{v}^2+q^2{u}^2)+{\displaystyle \frac{1}{2}}pquv\hfill \\ \hfill & -{\displaystyle \frac{i}{2}}{c}_2(p{q}_t-q{p}_t)-{\displaystyle \frac{i}{2}}{c}_1(u{q}_t-v{p}_t)-{\displaystyle \frac{1}{4}}{p}_t{q}_t.\hfill \end{array}$$

For $n=2$ ($g=3$), the stationary equations become even more complicated

$$\begin{array}{cc}\hfill & u_{tt}+i{p}^2{v}_t+2i(pv+qu)p_t-2u^{2}v+p^{3}qv+{\displaystyle \frac{3}{2}}{p}^2{q}^{2}u\hfill \\ \hfill & +c_1(p_{tt}-ipq{p}_t-2(pv+qu)u)+2i{c}_2{u}_t+2c_3(p^{2}q+i{p}_t)-4c_{4}u=0,\hfill \\ \hfill & v_{tt}-i{q}^2{u}_t-2i(pv+qu)q_t-2u{v}^2+p{q}^{3}u+{\displaystyle \frac{3}{2}}{p}^2{q}^{2}v\hfill \\ \hfill & +c_1(q_{tt}+ipq{q}_t-2(pv+qu)v)-2i{c}_2{v}_t+2c_3(p{q}^2-i{q}_t)-4c_{4}v=0,\hfill \\ \hfill & p_{tt}-i{p}^2{q}_t-4uvp-2q{u}^2+{\displaystyle \frac{1}{2}}{p}^3{q}^2+2c_1(i{u}_t-(pv+qu)p)\hfill \\ \hfill & +2i{c}_2{p}_t-4c_{3}u-4c_{4}p=0,\hfill \\ \hfill & q_{tt}+i{q}^2{p}_t-4uvq-2p{v}^2+{\displaystyle \frac{1}{2}}{p}^2{q}^3-2c_1(i{v}_t+(pv+qu)q)\hfill \\ \hfill & -2i{c}_2{q}_t-4c_{3}v-4c_{4}q=0.\hfill \end{array}$$

It is easy to see that for $n=2$, the dependence between the components $(u,v)$ and $(p,q)$ is described by nonlinear equations. At the same time, these equations also allow solutions of the form $u=v=0$ for $c_1=c_3=0$. The equations for the components p and q in this case have the form

$$\begin{array}{cc}\hfill & p_{tt}-i{p}^2{q}_t+{\displaystyle \frac{1}{2}}{p}^3{q}^2+2i{c}_2{p}_t-4c_{4}p=0,\hfill \\ \hfill & q_{tt}+i{q}^2{p}_t+{\displaystyle \frac{1}{2}}{p}^2{q}^3-2i{c}_2{q}_t-4c_{4}q=0.\hfill \end{array}$$

(23)

We will omit the values of the coefficients in the equation of the spectral curve for $n=2$, since they are very cumbersome and are not interesting at the moment.

Let us note that the nonlinear stationary equations (22) and (23) are very different, despite the fact that they have the same order.

4. Integrable Nonlinear Evolutionary Equations

Let the second equation of the Lax pair have the form

$$i{\Psi }_{t_k}+V_k\Psi =0,$$

(24)

where $V_k=2^m{W}_k$, $k=2m$ or $k=2m-1$.

Then from the compatibility condition of the equations (2) and (24) the following evolutionary equations

$$\begin{array}{cc}\hfill i& p_{z_k}+2^{m+1}{H}_{k+1}=0,\hfill \\ \hfill -i& q_{z_k}+2^{m+1}{G}_{k+1}=0,\hfill \\ \hfill i& u_{z_k}+2^{m+1}(H_{k+2}-p{F}_{k+1})=0,\hfill \\ \hfill -i& v_{z_k}+2^{m+1}(G_{k+2}-q{F}_{k+1})=0\hfill \end{array}$$

(25)

follow.

It is not difficult to see that the evolutionary equations (25), as well as the stationary equations, admit reduction (16). Accordingly, after replacing (16), an integrable nonlinear two-component equation with a different dependence of the components on the variables $z_k$ will be obtained from the system of equations (25). Note that the well-known two-component integrable nonlinear equations [37,38,40,41,42,51,52,53,54,55,56,57,58,59,60,61] describe models with identical dependencies of components on variables $z_k$.

We present the first evolutionary integrable equations from the corresponding hierarchy. For $k=1$, the structure of the first two equations of the system differs from the structure of the last two:

$$\begin{array}{cc}\hfill i{p}_{z_1}=& 2i{u}_t-2(pv+qu)p,\hfill \\ \hfill -i{q}_{z_1}=& -2i{v}_t-2(pv+qu)q,\hfill \\ \hfill i{u}_{z_1}=& p_{tt}-ipq{p}_t-2(pv+qu)u,\hfill \\ \hfill -i{v}_{z_1}=& q_{tt}+ipq{q}_t-2(pv+qu)v.\hfill \end{array}$$

(26)

Note that the last two equations of the system (26) when performing reduction (16) are analogous to the derivative nonlinear Schrödinger equation. At the same time, the first two equations describe a fairly simple relationship between the components.

For $k=2$, the evolutionary equations have the form

$$\begin{array}{cc}\hfill i{p}_{z_2}=& p_{tt}-i{p}^2{q}_t+{\displaystyle \frac{1}{2}}(p^3{q}^2-8uvp-4u^{2}q),\hfill \\ \hfill -i{q}_{z_2}=& q_{tt}+i{q}^2{p}_t+{\displaystyle \frac{1}{2}}(p^2{q}^3-8uvq-4v^{2}p),\hfill \\ \hfill i{u}_{z_2}=& u_{tt}+i{p}^2{v}_t+2i(pv+qu)p_t+{\displaystyle \frac{1}{2}}(3p^2{q}^{2}u+2p^{3}qv-4u^{2}v),\hfill \\ \hfill -i{v}_{z_2}=& v_{tt}-i2q^2{u}_t-2i(pv+qu)q_t+{\displaystyle \frac{1}{2}}(3p^2{q}^{2}v+2p{q}^{3}u-4u{v}^2).\hfill \end{array}$$

(27)

It is easy to see that equations (27) are analogs of the two-component coupled derivative nonlinear Schrödinger equation. Assuming $q=-p^{*}$ and $v=-u^{*}$, we obtain from equation (27) a new two-component derivative nonlinear Schrödinger equation

$$\begin{array}{cc}\hfill i{p}_{z_2}=& p_{tt}+i{p}^2{p}_t^{*}+{\displaystyle \frac{1}{2}}\left({\left|p\right|}^{4}p+8{\left|u\right|}^{2}p+4u^2{p}^{*}\right),\hfill \\ \hfill i{u}_{z_2}=& u_{tt}-i{p}^2{u}_t^{*}-2i(p{u}^{*}+u{p}^{*})p_t+{\displaystyle \frac{1}{2}}{\left(3\right|p|}^{4}u+{2\left|p\right|}^2{p}^2{u}^{*}+{4\left|u\right|}^{2}u).\hfill \end{array}$$

(28)

Unlike the usual vector derivative nonlinear Schrödinger equation [57,58,59,60,61], in this case the evolution of the components p and u is described by different equations.

For $k=3$, the evolutionary equations again divided into two groups. The first two equations are analogs of the nonlinear Schrodinger equation, and the second are analogs of the modified Korteweg–de Vries equation

$$\begin{array}{cc}\hfill i{p}_{z_3}=& 2u_{tt}+2i(q{u}_t-u{q}_t)p+i(6pv+4qu)p_t+(p^2{q}^2-4uv)u,\hfill \\ \hfill -i{q}_{z_3}=& 2v_{tt}-2i(p{v}_t-v{p}_t)q-i(6qu+4pv)q_t+(p^2{q}^2-4uv)v,\hfill \\ \hfill u_{z_3}=& -p_{ttt}+i(q{p}_{tt}+p{q}_{tt})p+6(v{p}_t+q{u}_t)u+(4v{u}_t+2u{v}_t)p\hfill \\ \hfill & +2ip{p}_t{q}_t-{\displaystyle \frac{3}{2}}{p}^2{q}^2{p}_t-2i{p}^{2}quv+{\displaystyle \frac{i}{2}}{p}^4{q}^3,\hfill \\ \hfill v_{z_3}=& -q_{ttt}-i4(q{p}_{tt}+p{q}_{tt})q+6(p{v}_t+u{q}_t)v+(2v{u}_t+4u{v}_t)q\hfill \\ \hfill & -2iq{p}_t{q}_t-{\displaystyle \frac{3}{2}}{p}^2{q}^2{q}_t+2ip{q}^{2}uv-{\displaystyle \frac{i}{2}}{p}^3{q}^4.\hfill \end{array}$$

(29)

Assuming $q=-p^{*}$ and $v=-u^{*}$, we obtain from equation (29) a two-component mixed equation

$$\begin{array}{cc}\hfill i{p}_{z_3}=& 2u_{tt}-2i(p^{*}{u}_t-u{p}_t^{*})p-i(6p{u}^{*}+4p^{*}u)p_t+{\left(\right|p|}^4-{4\left|u\right|}^2)u,\hfill \\ \hfill u_{z_3}=& -p_{ttt}-i(p^{*}{p}_{tt}+p{p}_{tt}^{*})p-6(u^{*}{p}_t+p^{*}{u}_t)u-(4u^{*}{u}_t+2u{u}_t^{*})p\hfill \\ \hfill & -2ip|p_t{|}^2-{\displaystyle \frac{3}{2}}{\left|p\right|}^4{p}_t-{2i\left|pu\right|}^{2}p-{\displaystyle \frac{i}{2}}{\left|p\right|}^{6}p.\hfill \end{array}$$

First equation is an analogue of the derivative NLS equation, and second equation is an analogue of the modifief Korteweg-de Vries equation.

Note that further the structure of the evolutionary equations (25) will depend on the parity of the number of the equation. For odd k, the order of derivatives with respect to t will be different, for even k, the same. In particular, for $k=4$, the system of evolutionary equations will be an analogue of the vector modified Korteweg-de Vries equation. Note also that even equations admit reduction $u=v=0$, whereas odd ones do not. In this case, equations (27) under this reduction and under condition (16) pass into the Gerdjikov-Ivanov equation [25,26,27,28,29,30,32,33]. I.e., the model considered in this paper generalizes the already known integrable models of nonlinear wave propagation.

5. One-Phase Solutions

To show differences in components behaviors, we consider examples with $n=0$ and $c_1\ne 0$. To find solutions to system (26), we express u and v from its first two equations and substitute them into the rest. After simplification we have

$$\begin{array}{cc}\hfill & p_{tt}-2i(c_1^2-2c_2)p_t-2c_1^2{p}^{2}q-4c_2^{2}p=0,\hfill \\ \hfill & q_{tt}+2i(c_1^2-2c_2)q_t-2c_1^{2}p{q}^2-4c_2^{2}q=0.\hfill \end{array}$$

(30)

Let us note that equation (30) differs from equations (22) and (23).

Following [74], we will make a replacement in these equations

$$p=\sqrt{r}exp\left\{-\int {\displaystyle \frac{w}{2r}}dt\right\},q=\sqrt{r}exp\left\{\int {\displaystyle \frac{w}{2r}}dt\right\},$$

(31)

where

$$r=pq,w=p{q}_x-q{p}_x.$$

After simplification we have

$$\begin{array}{cc}\hfill & w=-2i(c_1^2-2c_2)r+i{c}_3,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & 2r{r}_{tt}-{\left(r_t\right)}^2-8c_1^2{r}^3+4c_1^2(c_1^2-4c_2)r^2-c_3^2=0.\hfill \end{array}$$

(33)

Here $c_3$ is an integration constant.

Additional relations follow from the equation of the spectral curve. Converting expressions for constants $f_3$ and $f_4$ using substitutions (31) and (32), we obtain

$$\begin{array}{cc}\hfill & f_3=2c_1{c}_2-{\displaystyle \frac{c_3}{2c_1}},\hfill \\ \hfill & {\left(r_t\right)}^2=4c_1^2{r}^3-4c_1^2(c_1^2-4c_2)r^2+4c_1^2(4c_2^2-c_3-4f_4)r-c_3^2.\hfill \end{array}$$

(34)

It is not difficult to check the compatibility of the equations (33) and (34). It follows from these equations that the function $r\left(t\right)$ is an elliptic function or one of its degenerations.

In particular, if the spectral curve is given by equation

$${\mu }^2={\left({(\lambda -a)}^2+b^2\right)}^2,$$

then the constants $c_k$ and $f_4$ have the following values

$$c_1=-2a,c_2=a^2+b^2,c_3=0,f_4={\left(a^2+b^2\right)}^2.$$

For these values of constants, the function $r\left(t\right)$ satisfies the equation

$${\left(r_t\right)}^2=16a^2{r}^3+64a^2{b}^2{r}^2.$$

Solving this equation, we have

$$r\left(t\right)=-4b^2{error}^2(4abt+\alpha ),\alpha =const.$$

In this case

$$w\left(t\right)=16i{b}^2(a^2-b^2){error}^2(4abt+\alpha ).$$

Substituting this value of the functions $r\left(t\right)$ and $w\left(t\right)$ into formulas (31) and (27), we get the solution to the equations (27)

$$\begin{array}{cc}\hfill & p=2iberror\left({\varphi }_1(t,z_2)\right)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & q=2iberror\left({\varphi }_1(t,z_2)\right)e^{-i{\varphi }_2(t,z_2)},\hfill \\ \hfill & u=2berror\left({\varphi }_1(t,z_2)\right)(ia-btanh\left({\varphi }_1(t,z_2)\right))e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & v=2berror\left({\varphi }_1(t,z_2)\right)(ia+btanh\left({\varphi }_1(t,z_2)\right))e^{-i{\varphi }_2(t,z_2)},\hfill \end{array}$$

(35)

where

$$\begin{array}{cc}\hfill & {\varphi }_1(t,z_2)=4abt-16ab(a^2-b^2)z_2,\hfill \\ \hfill & {\varphi }_2(t,z_2)=2(a^2-b^2)t+4(a^4-6a^2{b}^2+b^4)z_2.\hfill \end{array}$$

It is easy to see that the components of solution (35) satisfy the reduction $q=-p^{*}$ and $v=-u^{*}$. The amplitudes of the components p and u for $a=2$, $b=3$ are shown in Figure 1.

It is not difficult to see that the shape of the component u is quite different from the shape of the component p. The component p is a classical soliton. At the same time, the component u is defined using a completely new expression.

Assuming $c_3=0$, $0<k<1$, it is possible to construct three different solutions in elliptic Jacobi functions [75,76].

If $c_2=(c_1^4-1-k^2)/\left(4c_1^2\right)$ , then the function r(t) is expressed in terms of the $error(t;k)$:

$$r\left(t\right)={\displaystyle \frac{k^2}{c_1^2}}{error}^2(t;k).$$

The spectral curve of this solution is determined by the equation

$${\mu }^2=\prod _{j=1}^4(\lambda -{\lambda }_j),$$

(36)

where

$${\lambda }_{1,2}=-{\displaystyle \frac{1+c_1^2\pm k}{2c_1}},{\lambda }_{3,4}={\displaystyle \frac{1-c_1^2\pm k}{2c_1}}.$$

In this case

$$\begin{array}{cc}\hfill & p(t,z_2)={\displaystyle \frac{k}{c_1}}error({\varphi }_1(t,z_2);k)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & u(t,z_2)=-{\displaystyle \frac{k}{2c_1^2}}\left(c_1^{2}error({\varphi }_1(t,z_2);k)-i{error}^{\prime }({\varphi }_1(t,z_2);k)\right)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & q=p^{*},v=u^{*},\hfill \end{array}$$

(37)

where

$$\begin{array}{cc}\hfill & {\varphi }_1(t,z_2)=t+{\displaystyle \frac{c_1^4+1+k^2}{c_1^2}}{z}_2,\hfill \\ \hfill & {\varphi }_2(t,z_2)={\displaystyle \frac{c_1^4+1+k^2}{2c_1^2}}t+{\displaystyle \frac{c_1^8+6(k^2+1)c_1^4+1+4k^2+k^4}{4c_1^4}}{z}_2.\hfill \end{array}$$

Since the solution (37) to the equations (27) satisfies the conditions $q=p^{*}$, $v=u^{*}$, it is not suitable for describing the propagation of nonlinear optical signals.

When $c_2=(c_1^4+2-k^2)/\left(4c_1^2\right)$ the function $r\left(t\right)$ has the form

$$r\left(t\right)=-{\displaystyle \frac{1}{c_1^2}}{error}^2(t;k).$$

The spectral curve is again determined by equation (36). Only the branching points in this case are not real, but complex conjugate

$${\lambda }_{1,2}=-{\displaystyle \frac{c_1}{2}}\pm i{\displaystyle \frac{1+\sqrt{1-k^2}}{2c_1}},{\lambda }_{3,4}=-{\displaystyle \frac{c_1}{2}}\pm i{\displaystyle \frac{1-\sqrt{1-k^2}}{2c_1}}.$$

The following periodic solution to the equations (27) corresponds to this curve:

$$\begin{array}{cc}\hfill & p(t,z_2)={\displaystyle \frac{i}{c_1}}error({\varphi }_1(t,z_2);k)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & u(t,z_2)=-{\displaystyle \frac{i}{2c_1^2}}\left(c_1^{2}error({\varphi }_1(t,z_2);k)-i{error}^{\prime }({\varphi }_1(t,z_2);k)\right)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & q=-p^{*},v=-u^{*},\hfill \end{array}$$

(38)

where

$$\begin{array}{cc}\hfill & {\varphi }_1(t,z_2)=t+{\displaystyle \frac{c_1^4-2+k^2}{c_1^2}}{z}_2,\hfill \\ \hfill & {\varphi }_2(t,z_2)={\displaystyle \frac{c_1^4-2+k^2}{2c_1^2}}t+{\displaystyle \frac{c_1^8+6(k^2-2)c_1^4+6-6k^2+k^4}{4c_1^4}}{z}_2.\hfill \end{array}$$

Since reductions $q=-p^{*}$, $v=-u^{*}$ are performed, solution (38) can be used to describe the propagation of a periodic nonlinear two-component wave. The amplitudes of the components p and u for $c_1=0.5$, $k=0.7$ are shown in Figure 2. It is easy to see that the component p is an ordinary “dnoidal” wave, while component u has a non-standard shape.

The third elliptic solution to the equations (27) for $c_2=(c_1^4-1+2k^2)/\left(4c_1^4\right)$ is expressed in terms of the $error(t;k)$ [75,76]:

$$r\left(t\right)=-{\displaystyle \frac{k^2}{c_1^2}}{error}^2(t;k),$$

and

$$\begin{array}{cc}\hfill & p(t,z_2)={\displaystyle \frac{ik}{c_1}}error({\varphi }_1(t,z_2);k)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & u(t,z_2)=-{\displaystyle \frac{ik}{2c_1^2}}\left(c_1^{2}error({\varphi }_1(t,z_2);k)-i{error}^{\prime }({\varphi }_1(t,z_2);k)\right)e^{i{\varphi }_2(t,z_2)},\hfill \\ \hfill & q=-p^{*},v=-u^{*},\hfill \end{array}$$

(39)

where

$$\begin{array}{cc}\hfill & {\varphi }_1(t,z_2)=t+{\displaystyle \frac{c_1^4+1-2k^2}{c_1^2}}{z}_2,\hfill \\ \hfill & {\varphi }_2(t,z_2)={\displaystyle \frac{c_1^4+1-2k^2}{2c_1^2}}t+{\displaystyle \frac{c_1^8+6(1-2k^2)c_1^4+1-6k^2+6k^4}{4c_1^4}}{z}_2.\hfill \end{array}$$

The spectral curve of the solution (39) is given by equation (36), where

$${\lambda }_{1,2}=-{\displaystyle \frac{c_1}{2}}+{\displaystyle \frac{\sqrt{1-k^2}\pm ik}{2c_1}},{\lambda }_{3,4}=-{\displaystyle \frac{c_1}{2}}-{\displaystyle \frac{\sqrt{1-k^2}\pm ik}{2c_1}}.$$

The examples we have considered have shown that, as in the case of the standard nonlinear Schrödinger equation, the location of the branching points of the spectral curve corresponds to the sign of reduction. If $q=p^{*}$, then the branching points are on the real axis. If $q=-p^{*}$, then the branching points form complex conjugate pairs.