Vector integrable nonlinear equations still continue to attract active attention (see, for example, [1,2,3,4,5,6,7,8,9,10]). Mainly, the vector nonlinear Schrödinger equation is considered. Much less work is devoted to the derivative form of the vector equation (see, for example, [11,12,13,14,15,16,17,18,19]). Scalar forms of the derivative nonlinear Schrödinger equation are given much more attention (see, for example, [20,21,22,23,24,25,26]). Note that for each derivative nonlinear Schrödinger equation, its vector form is obtained, and multi-soliton solutions of these vector forms are investigated. Attention to two-component variants of the nonlinear Schrodinger equation is due to the fact that with the help of double-polarized waves, twice as much information can be transmitted over an optical fiber [27,28,29,30]. In practice, it turned out that it is much more difficult to recover encoded information from a two-component signal. Apparently, this is due to the results obtained in our work. When transmitting information, it is assumed that each component is independent and carries its own part of the information. As we proved earlier [31], the spectral curve is invariant with respect to the orthogonal transformation of the solution. I.e., it does not depend on the individual components of the solution, but on their symmetric functions. This statement is also true for the equations from our current work. This is one of the possible reasons for the difficulty of recovering information from the transmitted signal. The second possible reason most likely follows from the fact that the spectral curve corresponding to a solution with linearly dependent components is greatly reduced. The correctness of this statement can be seen in the examples from this work. Therefore, when transmitting signals that differ slightly from each other, some information about the spectral characteristics of the signals may be lost. In addition, as our examples show, the genus of the spectral curve far exceeds the number of phases of the solution. Thus, part of the spectral data is redundant. Also, as we show, in the case of the vector equations, first of all, we get the law of transformation of the length of the solution vector, and then the rule of direction transformation. When replacing the components of a vector by its length and vice versa, information loss may occur. Thus, based on the results of this work, we can advise transmitting information not in Cartesian coordinates, but in polar ones.

In this paper, we use the monodromy matrix method (see, for example, [1,20,31]) to construct a hierarchy of the Gerdjikov-Ivanov vector equation and investigate the simplest solutions of equations from this hierarchy. As a rule, in the works devoted to the study of vector nonlinear equations, the individual components of the vector are analyzed. At the same time, sometimes there are works (see, for example, [7]) in which the behavior of the length and tangent of the angle of inclination of the vector is investigated. Our studies of the simplest solutions have shown that in the case of a vector nonlinear equation, the evolution of a vector can naturally be divided into two components: the evolution of the length of the vector and the evolution of its direction. Note that this statement is also true for the Manakov system, which can be seen by looking at the calculations in [1]. For example, assuming and $\mathbf{q}=\sigma {\mathbf{p}}^{*}$, where $\sigma =\pm 1$, we have and If the reduction has the form then the angle $\varphi$ becomes purely imaginary $\varphi =i\widehat{\varphi }$, where $\widehat{\varphi }\in \mathbb{R}$. In this case, the “direction” of the vector $\mathbf{p}$ is defined by the function $\widehat{v}=cosh\left(2\widehat{\varphi }\right)\ge 1$. Thus, if $\widehat{v}<1$, then it is possible to construct solutions that satisfy the reduction $\mathbf{q}=\sigma {\mathbf{p}}^{*}$. If $\widehat{v}>1$, then the solutions will satisfy the reduction (3). When $\widehat{v}=1$, the second component of the vector $\mathbf{p}$ is missing ($u_2=0$). The reduction sign $\sigma$ is determined by the sign of the function u: Note that the functions u, v and $\widehat{v}$ naturally appear during calculations. Also note that from the equations (1) it follows that $|p_j|=\sqrt{|u_j|}$. Therefore, to plot the amplitudes of the individual components $p_j$ of the vector $\mathbf{p}$, it is enough to find $u_j$. The analysis of the examples showed that when the direction of the vector $\mathbf{p}$ is independent of the coordinate and time ($\widehat{v}=const$), the spectral curve splits into two separate components, and the dynamics of the solution is determined by a spectral curve of a smaller kind than in the case when the direction of the vector $\mathbf{p}$ changes depending on the coordinate and time.

The presented article consists of an introduction, four sections and concluding remarks. In the first section, we define the Lax operator, define the monodromy matrix, find recurrent relations between its elements, and derive the equation of spectral curves associated with multiphase solutions. In section 2, we define the second Lax pair operators and obtain vector integrable nonlinear differential equations from the hierarchy of the Gerdjikov-Ivanov vector equation. The first equations from this hierarchy have the form and If we replace vectors with scalars in these equations, we get the Gerdjikov-Ivanov equation and one of the forms of the mKdV equation.

In section 3, we consider solutions in the form of plane waves. We show that there are two types of plane waves that differ in the properties of their spectral curves. If $\mathbf{p}(x,t)=p(x,t)\mathbf{k}$, where $\mathbf{k}$ is a constant vector, then the equation of the spectral curve does not depend on the direction of the vector $\mathbf{p}$, in another case, the equation of the spectral curve depends on the direction of the vector $\mathbf{p}$. In the case when the direction of the vector $\mathbf{p}$ is fixed, the corresponding spectral curve splits into separate components.

In the fourth section, the simplest nontrivial solutions of the Gerdjikov-Ivanov vector equation are investigated. In this case, the function u is an elliptic function or its degeneracy, and the function $\widehat{v}$ depends on the function u according to the following formula where ${\partial }_x\theta =\pm u^{-1}$. Note that the simplest nontrivial solutions are also divided into two types. If $A=0$, then the direction of the vector $\mathbf{p}$ is fixed, only its length changes. The spectral curve of such solution also splits into two components. If $A\ne 0$, then the vector makes small fluctuations near the direction given by the equality $\widehat{v}=B$. The amplitude of these oscillations satisfies the condition $\left|A\right|<\left|\left|B\right|-1\right|$. Therefore, if $\left|B\right|<1$, then $|\widehat{v}|<1$, and from $\left|B\right|>1$ follows the inequality $|\widehat{v}|>1$.

Let the Lax operator has the form where ${\mathbf{p}}^t=(p_1,p_2)$, ${\mathbf{q}}^t=(q_1,q_2)$.

Let us consider the equation (4), (5) with matrices (6). The monodromy matrix M is a polynomial of the spectral parameter $\lambda$, and satisfies the equation (see, for example [1,32]) From equation (7) the following structure of the matrix M follows where $V_1=\lambda U+V_1^0$, $V_{j+1}=\lambda V_j+V_{j+1}^0$, $V_{-1}=-\lambda J+Q$,

The elements of the matrix $V_k^0$ satisfy the following recurrence relations

In particular, where

From the equation (7) in addition to the recurrent relations (9), stationary equations also follow. Any m-phase solution for $m\le n$ and for all values of t and z satisfies these stationary equations. As in the case of scalar derivative nonlinear Schrödinger equations [20], stationary vector equations form two groups. For $n>1$ stationary vector equations have the form and Note that since the structure of matrices $V_n^0$ depends on parity, the scalar stationary equations for even and odd n have a different form. The compatibility of this overridden system of equations imposes restrictions on the constants $c_k$.

Another stationary equations, which are satisfied by multiphase solutions, can be obtained from the equations of the spectral curve. Recall that the equation of the spectral curve of the multiphase solution is the characteristic equation of the monodromy matrix [32] From the formula (8) it follows that the equation of the spectral curve $\mathsf{\Gamma }$ has the form where

Let ’s define the second equation of the Lax pair by the equation

Then the following integrable nonlinear evolutionary equations follow from the Lax pair compatibility condition.

Thus, the first equations from this hierarchy have the form and For $p_j(x,t_1)=k_{j}p(x,t_1)$ and $q_j(x,t_1)=k_{j}q(x,t_1)$, where $k_j\in \mathbb{R}$, $k_1^2+k_2^2=1/2$, the equations (13) and (14) transform to coupled Gerdjikov-Ivanov equations and to coupled complex mKdV equations Since any solutions of the equations from the Gerdjikov-Ivanov hierarchy after multiplying them by a constant vector ${(k_1,k_2)}^t$ will satisfy the equations (13) and (14), then these equations can be considered as vector forms of the Gerdjikov-Ivanov and mKdV equations. These equations, as well as the Manakov [1], Kundu-Eckhaus [31] and Kulish-Sklyanin equations, are invariant with respect to the orthogonal transformation T of solutions. The proof can be found in [31]. Since the transformation T is simultaneously a transformation of the similarity of the monodromy matrix M, we can assume that the matrix $J_n$ is diagonal. Solutions with a non-diagonal matrix $J_n$ can be obtained by orthogonal transformation of solutions corresponding to the diagonal matrix $J_n$. Note that the equations of spectral curves of multiphase solutions of equations from this hierarchy are also invariant with respect to this transformation.

Let $n=0$. Then $M=U+c_1{V}_{-1}+J_0$, where $J_0=\mathrm{diag}(-2c_2,c_2+c_3,c_2-c_3)$, $c_k\in \mathbb{R}$. The first set of stationary equations has the form Solving these equations, we have It follows from the equations (1) and (15) that the functions $u_k(x,t)$ do not depend on x:

Substituting (15) into the second set of stationary equations, we obtain the following equalities Therefore, the system of stationary equations is compatible only if one of the two conditions is met. Or $c_1=0$, or $c_2=-2u/3$ and $c_3=0$.

From equations (13) the equalities ${\partial }_{t_1}{u}_k=0$ and follow. Hence (see (2)), ${\partial }_{t_1}u=0$, ${\partial }_{t_1}v=0$, and It is not difficult to see that the solutions (16) satisfy the reduction

Thus, for $n=0$, the solution of the equation (13) are plane waves of constant amplitude $\left|\mathbf{p}\right|$ and constant direction. But there can be two types of plane waves.

For $n=0$, $c_3\ne 0$ and $c_1=0$, the coefficients of the equation of the spectral curve (10) are equal Since the discriminant of the polynomial $\mathcal{R}\left(\mu \right)$ with coefficients (17) is a polynomial of $\lambda$ of degree 8 then the curve (10), (17) has 8 branching points. Using the Riemann-Hurvitz formula, we obtain that the genus of the spectral curve $\mathsf{\Gamma }$ is equal to 2. Therefore, in this case, the coefficients $p_{k0}$ are functions of the constants determined by the parameters of the curve $\mathsf{\Gamma }$ of genus $g=2$, invariant under the involution. So, apparently, the solution is determined by the parameters of the curve $\mathsf{\Gamma }/\tau$.

Note that in this case, the complex phases ${\alpha }_j$ of the components $p_j$ depend on v, i.e., on the direction of the vector $\mathbf{p}$.

For $n=0$, $c_1\ne 0$, and $c_3=0$, $c_2=-2u/3$, the equation of the spectral curve (10) takes the form: Therefore, in this case, the spectral curve decomposes into two components. These components are descibed by the solutions of equation (18): Hence, in this case, the genus of both components is zero.

Note that in this situation, the complex phases ${\alpha }_j$ of the components $p_j$ coincide and do not depend on the direction of the vector $\mathbf{p}$. That is, when $c_3=0$, the solution to the vector equation of Gerdjikov-Ivanov is a product of the solution to the scalar Gerdjikov-Ivanov equation and a constant vector.

Also, these two types of plane waves differ in the dependence of the spectral curve equation on the direction of the vector $\mathbf{p}$. When $c_3\ne 0$, the equation of the spectral curve depends on the direction of the vector $\mathbf{p}$, while when $c_3=0$, the equation of the spectral curve does not depend on the direction of the vector $\mathbf{p}$.

Let $n=1$. Then $M=V_1+c_{1}U+c_2{V}_{-1}+J_1$, where $J_1=\mathrm{diag}(-2c_3,c_3+c_4,c_3-c_4)$, $c_k\in \mathbb{R}$.

The first set of stationary equations has the form: Solving these equations for $c_1\ne 0$, we obtain: where ${\tilde{c}}_{3,4}=c_{3,4}/c_1$.

Substituting (19) into the second set of stationary equations, we obtain the conditions: ${\tilde{c}}_4=0$ and ${\tilde{c}}_3=c_2/3$ (${\tilde{c}}_3=-2u/3$). Since this case is analogous to the second case from the previous paragraph, we will omit it.

For $c_1=0$, the first set of stationary equations is satisfied when $c_3=c_4=0$, and the second set takes the form: or Let’s make the substitution in the equations (20): where $u_j=p_j{q}_j$, $w_j=p_j{\partial }_x{q}_j-q_j{\partial }_x{p}_j$.

After simplification, we obtain: and where $c_5,c_6\in \mathbb{R}$ are constants of integration.

The transformation of (23) using relations (2) give us the following equalities:

To obtain additional relations for the functions u and v, let’s consider the coefficients of the spectral curve equation (10), which in this case are equal to: where It’s easy to see that the spectral curve (10), (25) possesses a holomorphic involution: That is, this spectral curve has symmetry.

Usage the additional integrals (26) allows us to proceed from equations (24) to equations: and Integrating (27), we obtain: where $c_7\in \mathbb{R}$ is a constant of integration.

Therefore, the function $u\left(x\right)\equiv {\sigma \left|\mathbf{p}\right|}^2$ is an elliptic function or its degeneration. From equations (24), (26), and (29), it follows that:

Let’s replace the function v with $\widehat{v}=v/u$ in equation (28). From relations (24), (27), (28), and (29), it follows that the function $\widehat{v}$ satisfies the equation:

From this equation, it follows that the function $\widehat{v}$ has the form: where ${\partial }_x\theta =\pm u^{-1}$, $c_7=-{\kappa }^2$.

It’s easy to see that if ${\kappa }^2={(c_5+c_6)}^2$ or ${\kappa }^2={(c_5-c_6)}^2$, the direction of the vector $\mathbf{p}$ is fixed ($\widehat{v}=(c_5^2-c_6^2)/{\kappa }^2$). In other cases, it depends on its length $\left|\mathbf{p}\right|=\sqrt{\sigma u}$ according to the formula (30).

It is obvious that the coefficients in the equation (30) are real in one of the two cases.

In the first case: Then ${\kappa }^2>\left|c_5^2-c_6^2\right|$, which implies that $|\widehat{v}|<1$ for continuous real $\theta$.

In the second case: and ${\kappa }^2<\left|c_5^2-c_6^2\right|$. Therefore, in this case, for continuous real $\theta$, the inequality $|\widehat{v}|>1$ holds.

From the equation (13), it follows that for $n=1$, the dynamics of the functions $u(x,t_1)$ and $v(x,t_1)$ is described by the following relations: Therefore, where $X_1=x-c_2{t}_1$, and the function $f\left(x\right)$ satisfies the equation (29). Substituting (30) into (31), we get: Therefore, where $\widehat{\theta }\left(x\right)$ is a solution to the equation ${\partial }_x\theta =\pm u^{-1}$. Thus, if $\widehat{v}\ne \mathrm{const}$, then the dynamics of the vector direction differ from the dynamics of its length.