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Integration of a non-linear Hirota type equation with additional terms A. B. Khasanovab  , R. Kh. Eshbekova, T. G. Hasanovc a Samarkand State University b V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent c Urgench State University named after Al-Khorezmi
Bibliographic databases:
    § 1. IntroductionInverse spectral problems, which play a significant role in the integration of some important evolutionary equations of mathematical physics, can be considered as one of the most important achievements of the 20th century. An important breakthrough was made in the papers of Gardner, Green, Kruskal and Miura of 1967, where they discovered a deep connection between the well-known non-linear Korteweg–de Vries (KdV) equation
$$
\begin{equation*}
q_t = 6qq_x-q_{xxx},\quad q(x,t)|_{t=0} = q_0 (x),\qquad x \in \mathbb{R},\quad t>0,
\end{equation*}
\notag
$$
and the spectral theory of the Sturm–Liouville operator
$$
\begin{equation*}
\mathfrak{L}(t)y \equiv -y''+ q(x,t)y=\lambda y,\qquad x \in \mathbb{R},\quad t>0.
\end{equation*}
\notag
$$
Gardner, Greene, Kruskal, and Miura [1] found a global solution of the Cauchy problem for the KdV equation by reducing it to the inverse scattering problem (ISPM). The ISP for the Sturm–Liouville operator on the entire line was studied by Faddeev [2], Marchenko [3], Levitan [4], and others. In [5], Lax verified universality of the ISPM and of the generalized the KdV equation by introducing the concept of a higher KdV equation. Zakharov and Shabat [6] showed that the non-linear Schrodinger equation (NLS) can also be included in the ISPM formalism. Using the machinery proposed by Lax, they found a solution to the NLS equation for given initial functions $u(x,0)$ that decay sufficiently quickly as $|x| \to \infty$. Soon after, Wadati [7], based on the ideas of Zakharov and Shabat, proposed a method for solving the Cauchy problem for the modified KdV equation (mKdV):
$$
\begin{equation*}
u_t \pm 6 u^2 u_x + u_{xxx} = 0, \qquad u_t \pm 6 |u|^2 u_x + u_{xxx} = 0.
\end{equation*}
\notag
$$
By combining the focusing non-linear Schrödinger equation (FNLS) and the complex modified Korteweg–de Vries equation (cmKdV), Hirota [8] in 1973 proposed a method for solution of the Hirota equation (FNLS–cmKdV)
$$
\begin{equation*}
iu_t + \alpha (u_{xx}+2|u|^2 u) + i \beta(u_{xxx} + 6|u|^2 u_x) = 0, \qquad \alpha, \beta \in \mathbb{R},\quad x\in \mathbb{R},\quad t>0.
\end{equation*}
\notag
$$
Zakharov, Takhtadzhyan, and Faddeev [9], as well as Ablowitz, Kaup, Newell, and Segur [10] showed that the ISPM can also be applied to solving the sine–Gordon equation (sG) In [11]–[17], the modified Korteweg–de Vries–sine Gordon equation (mKdV–sG)
$$
\begin{equation*}
u_{xt} + a\biggl\{\frac{3}{2}\, u_x^2 u_{xx} + u_{xxxx} \biggr\} = b \sin u,
\end{equation*}
\notag
$$
where $a,b=\mathrm{const}$, was integrated by a direct method in the class of rapidly decreasing functions. The mKdV–sG equation was proposed in [11] as a model equation for describing non-linear vibrations in an atomic lattice. Along with the KdV and sG equations, the mKdV–sG equation has numerous applications. Using its solutions, it proved possible to construct two-dimensional surfaces with specified conditions imposed on the curvature. An important achievement in the theory of the mKdV–sG equation was the determination of analytical soliton and multisoliton solutions. An application of the ISPM to the NLS, mKdV, sG, FNLS–cmKdV and mKdV–sG equations is based on the scattering problem for the following Dirac operator on the entire axis:
$$
\begin{equation*}
\mathfrak{L}=i \begin{pmatrix} \dfrac{d}{dx} & -q(x) \\ r(x) & -\dfrac{d}{dx} \end{pmatrix},\qquad x\in \mathbb{R}.
\end{equation*}
\notag
$$
The inverse scattering problem for the Dirac operator on the entire axis was studied, in particular, in [18] and [19]. It is known that the operator $\mathfrak{L}$ is not self-adjoint; in the “rapidly decreasing” case it has a finite number of multiple complex eigenvalues, and can have spectral singularities that lie in the continuous spectrum. The scattering data of the non-self-adjoint Dirac operator include, in addition to the characteristics of the continuous spectrum, the discrete spectrum and spectral singularities. In [6]–[17], under the assumption that the eigenvalues of the corresponding Dirac operator $\mathfrak{L}$ are simple and there are no spectral singularities, non-linear evolution equations like NLS, mKdV, sG, FNLS–cmKdV and mKdV–sG were integrated. In this regard, it is relevant to search for solutions of non-linear evolution equations without a source and with a source corresponding to multiple eigenvalues of the Dirac operator. These problems were considered in [20]–[23]. For the Sturm–Liouville operator with periodic potential and with only a finite number of non-trivial gaps, Its and Matveev [24] and Dubrovin and Novikov [25] applied the ISPM to verify complete integrability of the KdV equation in the class of finite-gap periodic and quasi-periodic functions. The main idea of these studies is that the solution of the inverse spectral problem for the cases of finite-gap potentials was reduced to the Jacobi inversion problem of Abelian integrals on a two–sheeted compact Riemann surface with finite number of real branch points. In addition, for finite-gap potentials (that is, for solutions of the KdV equation), an explicit formula involving the Riemann theta functions was derived in [26]–[32], with the help of the ISPM for the periodic Dirac operator, the complete integrability of the NLS, mKdV, sG, FNLS–cmKdV, mKdV–sG equations was established in the class of finite-gap functions, and also, an explicit formula involving the Riemann theta functions was derived for finite-gap solutions. So, in these studie, the Cauchy problem for non-linear evolution equations was shown to be solvable for any finite-gap initial data. For a detailed account of this theory, see the books [3], [4], [33]–[35], and also the papers [36], [37]. It is well known that finding an explicit formula for solution of non-linear evolution equations like KdV, mKdV, NLS, sG, FNLS–cmKdV and mKdV–sG in the class of periodic functions depends significantly on the number of non-trivial gaps in the spectrum of the periodic Sturm–Liouville operator and the Dirac operator. In this regard, it is convenient to divide the class of periodic functions into two sets: 1) the class of periodic finite-gap functions; 2) the class of periodic infinite-gap functions. In [38] and [39], the following $n$-gap periodic Lame–Ince potentials were used: where $\wp (x)$ is the Weierstrass elliptic function.Note that in the case
$$
\begin{equation*}
q_t - 6qq_x + q_{xxx} = 0,\qquad q(x,0) = 2a\cos2x,\quad a\ne 0,
\end{equation*}
\notag
$$
the authors of the present paper have not succeeded in finding an analogue of the Its–Matveev formula [24] for the solution $q(x,t)$. It is known (see [40]) that if $q(x)=2a\cos2x, a\ne 0$, then, in the spectrum of the Sturm–Liouville operator $\mathfrak{L} y\equiv -y'' + q(x)y$, $x\in \mathbb{R}$, all the gaps are open; in other words, $q(x)$ is a periodic infinite-gap potential. Similar examples are also available for the periodic Dirac operator (see [41]). By combining the defocusing non-linear Schrödinger equation and the complex modified Korteweg–de Vries equation (DNLS–cmKdV), a method (see [42] and [43]) was proposed for construction of exact solutions of the Cauchy problem for equations of the type
$$
\begin{equation*}
\begin{gathered} \, iq_t+b(t)[q_{xx}-2|q|^2 q] - i a(t)[q_{xxx}-6|q|^2q_x]=0, \\ a(t),b(t) \in C([0,\infty)),\quad x\in \mathbb{R},\quad t>0, \\ iq_t+b(t) [q_{xx}-2(\rho^2-|q|^2)q] - i a(t) [q_{xxx}-6(\rho^2-|q|^2)q_x]=0,\qquad 0 \leqslant \rho <\infty, \end{gathered}
\end{equation*}
\notag
$$
in the class of $\pi$-periodic infinite-gap functions. Based on the ideas of [42] and [43], the Cauchy problem for the mKdV–chG and mKdV–Liouville equations of the types
$$
\begin{equation*}
\begin{gathered} \, q_{xt} = a(t) \biggl(q_{xxxx} - \frac{3}{2} q_x^2 q_{xx} \biggr) + b(t) chq, \qquad q=q(x,t),\quad x\in \mathbb{R}, \quad t>0, \\ q_{xt} = a(t) \biggl(q_{xxxx} - \frac{3}{2} q_x^2 q_{xx} \biggr) + b(t) e^q,\qquad q=q(x,t),\quad x\in \mathbb{R},\quad t>0, \\ q_{xt} = a(t) e^q,\qquad q=q(x,t),\quad x\in \mathbb{R},\quad t>0, \end{gathered}
\end{equation*}
\notag
$$
was shown to be solvable in [44]–[46] in the class of real infinite-gap functions which are $\pi$-periodic with respect to $x$. Here, $a(t),b(t) \in C([0,\infty))$ are given bounded continuous functions. Note that the Cauchy problem in the class of periodic (almost periodic) functions for non-linear evolution equations without (or with( a source and with an additional term was studied, in various settings, in [47]–[54].
§ 2. Statement of the problemConsider the Cauchy problem for a Hirota-type equation with additional terms of the form
$$
\begin{equation}
\begin{cases} p_t = a(t) [p_{xxx}-6p_x (p^2+q^2)] + b(t) [-q_{xx}+2q (p^2+q^2)] + c(t) p_x + d(t) q, \ \\ q_t = a(t) [q_{xxx}-6q_x (p^2+q^2)] + b(t) [p_{xx}-2p (p^2+q^2)] + c(t) q_x - d(t) p \end{cases}
\end{equation}
\tag{2.1}
$$
and with the initial conditions
$$
\begin{equation}
\begin{alignedat}{2} p(x,t) |_{t=0} &= p_0 (x), &\qquad p_0 (x+\pi) &= p_0 (x)\in C^6 (\mathbb{R}), \\ q(x,t) |_{t=0} &= q_0 (x), &\qquad q_0 (x+\pi) &= q_0 (x)\in C^6(\mathbb{R}), \end{alignedat}
\end{equation}
\tag{2.2}
$$
in the class of real infinite-gap functions which are $\pi$-periodic with respect to $x$,
$$
\begin{equation}
\begin{gathered} \, p(x+\pi,t) = p (x,t),\quad q(x+\pi,t) = q (x,t),\qquad x\in \mathbb{R},\quad t>0, \\ p(x,t), q(x,t) \in C_x^3 (t>0) \cap C_t^1 (t > 0) \cap C(t \geqslant 0). \end{gathered}
\end{equation}
\tag{2.3}
$$
Here, $a(t), b(t),c(t), d(t) \in C([0;\infty))$ are given continuous bounded functions. The system of equations (2.1) is obtained from Hirota-type equations with additional terms of the form
$$
\begin{equation*}
iu_t = -b(t) (u_{xx} - 2|u|^2 u) + i a(t) (u_{xxx} - 6|u|^2 u_x) + ic(t) u_x + d(t) u,\qquad i=\sqrt{-1}
\end{equation*}
\notag
$$
with $u(x,t)=q(x,t)-ip(x,t)$. In the present paper, we propose an algorithm for construction of infinite-gap solutions $p(x,t)$, $q(x,t)$, $x\in \mathbb{R}$, $t>0$, of problem (2.1)–(2.3) by reducing it to the inverse spectral problem for the self-adjoint Dirac operator
$$
\begin{equation}
\mathfrak{L}(\tau,t)y\equiv B \, \frac{dy}{dx} + \Omega(x+\tau,t)y=\lambda y,\qquad x\in \mathbb{R},\quad \tau \in \mathbb{R},\quad t>0,
\end{equation}
\tag{2.4}
$$
where
$$
\begin{equation*}
B \,{=} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\qquad \Omega(x+\tau,t) \,{=} \begin{pmatrix} p(x+\tau,t) & q(x+\tau,t) \\ q(x+\tau,t) & -p(x+\tau,t) \end{pmatrix},\qquad y(x) \,{=} \begin{pmatrix} y_1 (x) \\ y_2 (x) \end{pmatrix}.
\end{equation*}
\notag
$$
The inverse problem for the operator $\mathfrak{L}(0) = \mathfrak{L}(0,0)$
$$
\begin{equation*}
p_0 (x)=p_0 (x+\pi),\quad q_0 (x)=q_0 (x+\pi),\qquad x\in \mathbb{R},
\end{equation*}
\notag
$$
with periodic coefficients were studied, in various settings, in [55]–[60] and [61]–[63]. The inverse problem for the Hill operator was studied in [65]–[67]. By $c(x,\lambda,\tau,t)=(c_1 (x,\lambda,\tau,t),c_2 (x,\lambda,\tau,t))^\top $ and $s(x,\lambda,\tau,t) = (s_1 (x,\lambda,\tau,t), s_2 (x,\lambda,\tau,t ))^\top $ we denote, respectively, the solutions of equation (2.4) with initial conditions $c(0,\lambda,\tau,t)=(1,0)^\top $ and $s(0,\lambda,\tau,t)=(0,1)^\top$. The function $\Delta (\lambda,\tau,t)=c_1 (\pi,\lambda,\tau,t)+s_2 (\pi,\lambda,\tau,t)$ is called the Lyapunov function for equation (2.4). The spectrum of the operator $\mathfrak{L}(\tau,t)$, which is purely continuous, is as follows:
$$
\begin{equation*}
\sigma (\mathfrak{L})\equiv \{\lambda \in \mathbb{R}\colon |\Delta (\lambda )|\leqslant 2\} =\mathbb{R}\setminus \biggl(\bigcup_{n=-\infty }^{+\infty }(\lambda_{2n-1},\lambda_{2n})\biggr).
\end{equation*}
\notag
$$
The intervals $(\lambda_{2n-1},\lambda_{2n} )$, $n\in \mathbb{Z}$, are called gaps, where $\lambda_n$ are the roots of the equation $\Delta(\lambda) \mp 2 = 0$. These roots coincide with the eigenvalues of the periodic or antiperiodic ($y(0) = \pm y(\pi)$) problem for equation (2.4). The roots of the equation $s_1 (\pi,\lambda,\tau,t)=0$, which we denote by $\xi_n(\tau,t)$, $n\in \mathbb{Z}$, coincide with the eigenvalues of the Dirichlet problem for system (2.4) with boundary conditions $y_1 (0,\lambda,\tau,t)=0$, $y_1 (\pi,\lambda,\tau,t )=0$, and, in addition, $\xi_n(\tau,t) \in [\lambda_{2n-1},\lambda_{2n}]$, $n\in \mathbb{Z}$. The numbers $\xi_n(\tau,t)$, $n\in \mathbb{Z}$, and the signs $\sigma_n(\tau,t) = \operatorname{sgn}\{ s_2 (\pi,\xi_n,\tau,t )-c_1 (\pi,\xi_n,\tau,t)\}$, $n\in \mathbb{Z}$, are called the spectral parameters of the operator $\mathfrak{L}(\tau,t)$. The spectral parameters $\xi_n(\tau,t)$, $\sigma_n(\tau,t) =\pm 1$, $n\in \mathbb{Z}$, and the spectrum boundaries $\lambda_n(\tau,t)$, $n\in \mathbb{Z}$, are called the spectral data of the Dirac operator $\mathfrak{L}(\tau,t)$. Definition 2.1. The coefficients $p(x,t)$ and $q(x,t)$ of the periodic Dirac operator $\mathfrak{L}(\tau,t)$ are called infinite-gap functions if the boundaries of the gap $(\lambda_{2n-1}, \lambda_{2n})$, $n\in \mathbb{Z}$, satisfy the conditions Definition 2.2. The coefficients $p(x,t)$ and $q(x,t)$ of the periodic Dirac operator $\mathfrak{L}(\tau,t)$ are called finite-gap functions if there exists a finite number $N$ such that $\lambda_{2n-1} = \lambda_{2n} = \xi_n$, $n=\pm(N+1), \pm(N+2),\dots$ for all $|n|>N$. The problem of recovery of the coefficient $\Omega(x,t)$ of the operator $\mathfrak{L}(\tau,t)$ from spectral data is known as the inverse problem. The coefficients $p(x + \tau,t)$ and $q(x+\tau,t)$ of the operator $\mathfrak{L}(\tau,t)$ are determined uniquely from the spectral data $\lambda_n (\tau,t)$, $\xi_n (\tau,t)$, $\sigma_n (\tau,t) = \pm 1$, $n \in \mathbb{Z}$. If, using the initial function $p_0 (x+\tau)$ and $q_0 (x+\tau)$, $\tau \in \mathbb{R}$, we construct the Dirac operator $\mathfrak{L}(\tau,0)$ of the form
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{L}(\tau,0)y\equiv B \, \frac{dy}{dx} + \Omega_0(x+\tau)y=\lambda y,\qquad x\in \mathbb{R},\quad \tau \in \mathbb{R}, \\ B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\qquad \Omega_0(x+\tau) = \begin{pmatrix} p_0(x+\tau) & q_0(x+\tau) \\ q_0(x+\tau) & -p_0(x+\tau) \end{pmatrix},\!\qquad y(x) = \begin{pmatrix} y_1 (x) \\ y_2 (x) \end{pmatrix}, \end{gathered}
\end{equation*}
\notag
$$
then we will see that the boundaries of the spectrum $\lambda_n (\tau)$, $n \in \mathbb{Z}$, of the resulting problem are independent of the parameter $\tau \in \mathbb{R}$, that is, $\lambda_n (\tau)= \lambda_n$, $n\in \mathbb{Z}$, while the spectral parameters depend on the parameter $\tau$, that is, $\xi_n^0 = \xi_n^0 (\tau)$, $\sigma_n^0 = \sigma_n^0 (\tau) = \pm 1$, and are periodic functions, that is, $\xi_n^0 (\tau + \pi) = \xi_n^0 (\tau)$, $\sigma_n^0 (\tau + \pi) = \sigma_n^0 (\tau)$, $\tau \in \mathbb{R}$, $n \in \mathbb{Z}$. By solving the direct problem, we will find the spectral data $\lambda_n$, $\xi_n^0(\tau)$, $\sigma_n^0 (\tau) = \pm 1$, $n \in \mathbb{Z}$, of the operator $\mathfrak{L}(\tau,0)$.
§ 3. The main result and discussionThe main result of this paper is given by the following theorem. Theorem 3.1. Let $p(x,t)$, $q(x,t)$, $x\in \mathbb{R}$, $t>0$, be solutions of problem (2.1)–(2.3). Then the boundaries of the spectrum $\lambda_n (\tau,t)$, $n\in \mathbb{Z}$, of the operator $\mathfrak{L}(\tau,t)$ are independent of the parameter $\tau$ and $t$, that is, $\lambda_n (\tau,t)=\lambda_n$, $n\in \mathbb{Z}$, and the spectral parameters $\xi_n (\tau,t)$, $\sigma_n (\tau,t)=\pm 1$, $n\in \mathbb{Z}$, satisfy an analogue of the system of Dubrovin equations,
$$
\begin{equation}
\frac{\partial \xi_n (\tau,t)}{\partial \tau} = 2 (-1)^{n-1} \sigma_n (\tau,t) h_n (\xi(\tau,t)) \{p(\tau,t) + \xi_n (\tau,t)\},\qquad n\in \mathbb{Z},
\end{equation}
\tag{3.1}
$$
$$
\begin{equation}
\frac{\partial \xi_n (\tau,t)}{\partial t} = 2 (-1)^{n} \sigma_n (\tau,t) h_n (\xi(\tau,t)) g_n (\xi (\tau,t)),\qquad n\in \mathbb{Z}.
\end{equation}
\tag{3.2}
$$
Here, the signs $\sigma_n (\tau,t)=\pm 1$, $n\in \mathbb{Z}$, are reversed with each collision of the point $\xi_n (\tau,t)$, $n\in \mathbb{Z}$, with boundaries of its gap $[\lambda_{2n-1},\lambda_{2n}]$. In addition, the initial conditions
$$
\begin{equation}
\xi_n (\tau,t)|_{t=0} =\xi_n^0(\tau),\qquad \sigma_n (\tau,t)|_{t=0} =\sigma_n^0(\tau),\quad n\in \mathbb{Z},
\end{equation}
\tag{3.3}
$$
are satisfied, where $\xi_n^0(\tau), \sigma_n^0(\tau) =\pm 1$, $n\in \mathbb{Z}$, are the spectral parameters of the Dirac operator $\mathfrak{L}(\tau,0)$. The sequences $h_n (\xi)$ and $g_n(\xi)$, $n\in \mathbb{Z}$, in equation (3.2) are given by
$$
\begin{equation}
\begin{gathered} \, h_n (\xi) =\sqrt{(\xi_n - \lambda_{2n-1})(\lambda_{2n} -\xi_n)} \cdot f_n (\xi), \nonumber \\ f_n (\xi) =\sqrt{\prod_{\substack{k=-\infty\\k\ne n}}^{+\infty} \frac{(\lambda_{2k-1} -\xi_n )(\lambda_{2k} -\xi_n)}{(\xi_k -\xi_n)^2}}, \nonumber \\ \begin{aligned} \, g_n (\xi) &=a(t)[4\xi_n^3 + 4p\xi_n^2 + 2\xi_n(p^2+q^2+q_\tau) - p_{\tau \tau} + 2(pq_\tau - p_\tau q) + 2p(p^2+q^2)] \nonumber \\ &\qquad + b(t)[2\xi_n^2+2p\xi_n + p^2 +q^2 +q_\tau] + c(t)[-\xi_n-p] + \frac12\, d(t),\qquad n\in \mathbb{Z}, \end{aligned} \end{gathered}
\end{equation}
\tag{3.4}
$$
where $p=p(\tau,t)$, $q=q(\tau,t)$, $\xi\equiv \xi(\tau,t) = (\dots,\xi_{-1}(\tau,t),\xi_0(\tau,t),\xi_1(\tau,t),\dots)$, and $\sigma \equiv \sigma(\tau,t) = (\dots,\sigma_{-1}(\tau,t),\sigma_0(\tau,t),\sigma_1(\tau,t),\dots)$. Proof. Let functions $p(x,t)$, $q(x,t)$, $x\in \mathbb{R}$, $t>0$, which are $\pi $-periodic with respect to $x$, satisfy equation (2.1).
By $y_n = (y_{n,1} (x,\tau,t),y_{n,2} (x,\tau,t))^\top$, $n\in \mathbb{Z}$, we denote the orthonormal eigenvector-functions of the operator $\mathfrak{L}(\tau,t)$ on $[0,\pi]$ with Dirichlet boundary conditions corresponding to the eigenvalues $\xi_n =\xi_n (\tau,t)$, $n\in \mathbb{Z}$. Differentiating the identity
$$
\begin{equation*}
\xi_n (\tau,t)=(\mathfrak{L}(\tau,t)y_n ,y_n),\qquad n\in \mathbb{Z},
\end{equation*}
\notag
$$
with respect to $t$, we have, since the operator $\mathfrak{L}(\tau,t)$ is symmetric,
$$
\begin{equation}
\frac{\partial \xi_n (\tau,t)}{\partial t} = \bigl(\dot{\Omega }(x+\tau,t)y_n ,y_n \bigr),\qquad n\in \mathbb{Z}.
\end{equation}
\tag{3.5}
$$
Using the explicit form of the inner product
$$
\begin{equation*}
(y,z)=\int_0^{\pi }\bigl[y_1 (x)\overline{z_1 (x)} + y_2 (x)\overline{z_2 (x)}\,\bigr]\, dx,\qquad y=\begin{pmatrix} y_1 (x) \\ y_2(x) \end{pmatrix},\quad z=\begin{pmatrix} z_1 (x) \\ z_2(x) \end{pmatrix},
\end{equation*}
\notag
$$
and recalling the definition of matrix multiplication, we can write (3.5) in the form
$$
\begin{equation}
\begin{aligned} \, \frac{\partial \xi_n (\tau,t)}{\partial t} &= \int_0^{\pi} \bigl(y_{n,1}^2(x,\tau,t) - y_{n,2}^2(x,\tau,t) \bigr)p_t(x+\tau,t)\, dx \nonumber \\ &\qquad+\int_0^{\pi} 2y_{n,1}(x,\tau,t)y_{n,2}(x,\tau,t) q_t(x+\tau,t)\,dx. \end{aligned}
\end{equation}
\tag{3.6}
$$
Substituting (2.1) into (3.6), we obtain
$$
\begin{equation*}
\frac{\partial \xi_n (\tau,t)}{\partial t} = a(t) I_1 (\tau,t) + b(t) I_2 (\tau,t) + c(t) I_3 (\tau,t) + d(t) I_4(\tau,t).
\end{equation*}
\notag
$$
Here,
$$
\begin{equation*}
\begin{aligned} \, I_1 (\tau,t) &= 2 \int_0^{\pi} [(y_{n,1}^2 - y_{n,2}^2)(p_{xxx} - 6p_x(p^2+q^2)) \\ &\qquad\qquad+2y_{n,1}y_{n,2} (q_{xxx} - 6q_x(p^2+q^2))]\, dx, \\ I_2 (\tau,t) &= \int_0^{\pi} [(y_{n,1}^2 - y_{n,2}^2)(-q_{xx}+2q(p^2+q^2)) \\ &\qquad\qquad+2y_{n,1}y_{n,2} (p_{xx}-2p(p^2+q^2))]\, dx, \\ I_3 (\tau,t) &= \int_0^{\pi} [(y_{n,1}^2 - y_{n,2}^2) p_x + 2y_{n,1}y_{n,2} q_x]\, dx, \\ I_4 (\tau,t) &= \int_0^{\pi} [(y_{n,1}^2 - y_{n,2}^2)q - 2y_{n,1}y_{n,2} p]\, dx, \end{aligned}
\end{equation*}
\notag
$$
and $y_{n,1} = y_{n,1}(x,\tau,t)$, $y_{n,2} = y_{n,2}(x,\tau,t)$, $p=p(x+\tau,t)$, $q=q(x+\tau,t)$.
Using the identities
$$
\begin{equation}
\begin{cases} y'_{n,1} = q y_{n,1}-p y_{n,2} -\xi_n y_{n,2}, \\ y'_{n,2} = \xi_n y_{n,1}-p y_{n,1} -q y_{n,2}, \end{cases}
\end{equation}
\tag{3.7}
$$
one can easily check that
$$
\begin{equation*}
\begin{cases} (y_{n,1}\cdot y_{n,2})' = \xi_n(y_{n,1}^2 - y_{n,2}^2)-p(y_{n,1}^2 + y_{n,2}^2), \\ (y_{n,1}^2 - y_{n,2}^2)' = 2q(y_{n,1}^2 + y_{n,2}^2)-4\xi_n y_{n,1} y_{n,2}, \\ (y_{n,1}^2 + y_{n,2}^2)' = 2q(y_{n,1}^2 - y_{n,2}^2)-4p y_{n,1} y_{n,2}. \end{cases}
\end{equation*}
\notag
$$
Using the recurrence relations
$$
\begin{equation*}
\begin{aligned} \, r_k (x+\tau,t) &=-2\int_0^{x}[p(s+\tau,t)a_k (s+\tau,t)+q(s+\tau,t)b_k (s+\tau,t)]\, ds +d_k (\tau,t), \\ a_{k+1} (x+\tau,t) &=\frac12\, b'_k (x+\tau,t)+q(x+\tau,t)r_k (x+\tau,t), \\ b_{k+1} (x+\tau,t) &=-\frac12\, a'_k (x+\tau,t)-p(x+\tau,t)r_k (x+\tau,t),\qquad k=0,1,2,3,4 \end{aligned}
\end{equation*}
\notag
$$
(see [67]), setting
$$
\begin{equation*}
\begin{alignedat}{2} a(x+\tau,t,\xi_n) &=\sum_{k=0}^3 a_k (x+\tau,t)\xi_n^{3-k} (t), &\qquad b(x+\tau,t,\xi_n ) &=\sum_{k=0}^3b_k (x+\tau,t)\xi_n^{3-k} (t), \\ r(x+\tau,t,\xi_n ) &=\sum_{k=0}^3 r_k (x+\tau,t)\xi_n^{3-k} (t), &\qquad d(\tau,t,\xi_n) &=\sum_{k=0}^3d_k (\tau,t)\xi_n^{3-k} (t), \end{alignedat}
\end{equation*}
\notag
$$
and putting
$$
\begin{equation*}
\begin{gathered} \, r_0=d_0=4,\quad r_1=d_1=0,\quad r_2 = d_2 = 2(p^2+q^2), \quad r_3 = d_3 = 2(p q_x-p_x q), \\ \begin{alignedat}{4} a_0 &= 0, &\qquad a_1 &= 4q, &\qquad a_2 &= -2p_x, &\qquad a_3 &= -q_{xx}+2q(p^2+q^2), \\ b_0 &= 0, &\qquad b_1 &= -4p, &\qquad b_2 &= -2q_x, &\qquad b_3 &= p_{xx}-2p(p^2+q^2), \end{alignedat} \end{gathered}
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
\begin{gathered} \, a= 4q\xi_n^2 - 2p_x \xi_n-q_{xx}+2q(p^2+q^2), \qquad b=-4p\xi_n^2 - 2q_x \xi_n+p_{xx}-2p(p^2+q^2), \\ r = d = 4\xi_n^3+2\xi_n(p^2+q^2) + 2(p q_x - p_x q). \end{gathered}
\end{equation*}
\notag
$$
Using the above and employing (3.7), the first integral is evaluated as follows:
$$
\begin{equation*}
I_1 (\tau,t)=\int_0^{\pi}[(y_{n,1}^2 -y_{n,2}^2)(b'+2qr-2 a\xi_n)-2y_{n,1} y_{n,2} (a'+2pr+2b \xi_n)]\, dx.
\end{equation*}
\notag
$$
It is easily checked that
$$
\begin{equation*}
\begin{aligned} \, &(y_{n,1}^2 -y_{n,2}^2)(b'+2qr-2 a\xi_n)-2y_{n,1} y_{n,2} (a'+2pr+2b \xi_n) \\ &\qquad= [(r+b)y_{n,1}^2-2ay_{n,1}y_{n,2} + (r-b)y_{n,2}^2]'. \end{aligned}
\end{equation*}
\notag
$$
Hence, since $y_{n,1} (0,\tau,t) = y_{n,1}(\pi,\tau,t)=0$, we have
$$
\begin{equation*}
\begin{aligned} \, I_1 (\tau,t) &= \{r(\tau,t,\xi_n) - b(\tau,t,\xi_n) \} [y_{n,2}^2 (\pi,\tau,t) - y_{n,2}^2(0,\tau,t)] \\ &= \bigl\{4\xi_n^3 (\tau,t) + 2\xi_n(\tau,t)\bigl(p^2(\tau,t)+q^2(\tau,t)+q_\tau (\tau,t) \bigr) \\ &\qquad+2\bigl(p(\tau,t) q_\tau(\tau,t) - p_\tau(\tau,t) q(\tau,t)\bigr)+4p(\tau,t)\xi_n^2 (\tau,t) - p_{\tau \tau} (\tau,t) \\ &\qquad+ 2p(\tau,t)(p^2(\tau,t)+q^2(\tau,t)) \bigr\}[y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)]. \end{aligned}
\end{equation*}
\notag
$$
Setting $a_0=0$, $a_1=2q$, $a_2=-p_x$, $b_0=0$, $b_1=-2p$, $b_2=-q_x$, $r_0=d_0=2$, $r_1=d_1=0$, $r_2=d_2=p^2+q^2$, and proceeding as above, we have
$$
\begin{equation*}
\begin{aligned} \, I_2(\tau,t) &= \{2\xi_n^2(\tau,t) + 2\xi_n(\tau,t)p(\tau,t)+p^2(\tau,t)+q^2(\tau,t)+q_\tau (\tau,t)\} \\ &\qquad \times [y_{n,2}^2 (\pi ,t)-y_{n,2}^2 (0,t)]. \end{aligned}
\end{equation*}
\notag
$$
The third and fourth integrals $I_3 (\tau,t)$ and $I_4 (\tau,t)$ are evaluated as follows:
$$
\begin{equation*}
\begin{aligned} \, I_3 (\tau,t) &= \int_0^{\pi} [(y_{n,1}^2 - y_{n,2}^2) p_x + 2y_{n,1}y_{n,2} q_x]\, dx \\ &= \int_0^{\pi} (y_{n,1}^2 - y_{n,2}^2)\, dp + \int_0^{\pi} 2 y_{n,1} y_{n,2}\, dq \\ & = [(y_{n,1}^2 - y_{n,2}^2)p + 2y_{n,1}y_{n,2} q] \big|_{x=0}^{x=\pi} - \!\int_0^{\pi} [2pq (y_{n,1}^2 + y_{n,2}^2) - 4p\xi_n y_{n,1}y_{n,2}]\, dx \\ &\qquad-\int_0^{\pi} [2q\xi_n(y_{n,1}^2 - y_{n,2}^2) - 2pq(y_{n,1}^2 + y_{n,2}^2)]\, dx \\ &= -p[y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)]- \xi_n \int_0^{\pi} [2q(y_{n,1}^2 - y_{n,2}^2) - 4py_{n,1} y_{n,2}]\, dx \\ &= \{-p-\xi_n\} [y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)], \\ I_4 (\tau,t) &= \frac12\int_0^{\pi} [2q(y_{n,1}^2 - y_{n,2}^2) - 4p y_{n,1}y_{n,2}] \, dx = \frac12 [y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)]. \end{aligned}
\end{equation*}
\notag
$$
So, we have the equation
$$
\begin{equation}
\begin{aligned} \, \frac{\partial \xi_n (\tau,t)}{\partial t} &= [y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)] \biggl\{a(t)[4\xi_n^3+4p\xi_n^2+2\xi_n(p^2+q^2+q_\tau) \nonumber \\ &\!\qquad -p_{\tau \tau}+2(pq_\tau - p_\tau q) + 2p(p^2 +q^2)] + b(t)[2\xi_n^2 + 2\xi_n p + p^2 + q^2 + q_\tau] \nonumber \\ &\!\qquad+ c(t)[-\xi_n-p]+\frac12 \, d(t)\biggr\}. \end{aligned}
\end{equation}
\tag{3.8}
$$
Now let us calculate the difference $[y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)]$. The eigenvalues $\xi_n (\tau,t)$ of the Dirichlet problem for equation (2.3) are simple, and so we have
$$
\begin{equation*}
y_n (x,\tau,t)=\frac1{c_n (\tau,t)} s(x,\xi_n,\tau,t),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
c_n^2 (\tau,t)=\int_0^{\pi}[s_1^2 (x,\xi_n,\tau,t)+s_2^2 (x,\xi_n,\tau,t)]\, dx =-\frac{\partial s_1 (\pi ,\xi_n,\tau,t)}{\partial \lambda } \, s_2 (\pi,\xi_n,\tau,t).
\end{equation*}
\notag
$$
Using this equality, we have
$$
\begin{equation*}
y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)=-\frac{s_2 (\pi,\xi_n,\tau,t)-(s_2 (\pi,\xi_n,\tau,t))^{-1}}{\partial s_1 (\pi ,\xi_n,\tau,t)/\partial \lambda}.
\end{equation*}
\notag
$$
Putting
$$
\begin{equation*}
s_2 (\pi,\xi_n,\tau,t)-\frac1{s_2 (\pi,\xi_n,\tau,t)} =\sigma_n (\tau,t)\sqrt{\Delta^2 (\xi_n)-4}
\end{equation*}
\notag
$$
in this equality, we obtain
$$
\begin{equation}
y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)=-\frac{\sigma_n (\tau,t)\sqrt{\Delta^2 (\xi_n )-4} }{\partial s_1 (\pi,\xi_n,\tau,t)/\partial \lambda}.
\end{equation}
\tag{3.9}
$$
Next, we have
$$
\begin{equation*}
\begin{gathered} \, \Delta^2 (\lambda)-4=-4\pi^2 \prod_{k=-\infty }^{+\infty }\frac{(\lambda -\lambda_{2k-1} )(\lambda -\lambda_{2k} )}{a_k^2}, \\ s_1 (\pi,\lambda,\tau,t)=\pi \prod_{k=-\infty }^{+\infty }\frac{\xi_k(\tau,t) -\lambda }{a_k}, \end{gathered}
\end{equation*}
\notag
$$
where $a_0 =1$ and $a_k =k$ for $k\ne 0$, and now we can write equality (3.9) as
$$
\begin{equation*}
y_{n,2}^2 (\pi,\tau,t)-y_{n,2}^2 (0,\tau,t)=2(-1)^{n} \sigma_n (\tau,t) h_n (\xi(\tau,t)).
\end{equation*}
\notag
$$
Putting this expression into identity (3.8), we obtain (3.2). A similar argument proves (3.1).
Replacing the Dirichlet boundary conditions by periodic ($y(0,t)=y(\pi,t)$) or antiperiodic ($y(0,t)=-y(\pi,t)$) boundary conditions, we obtain, instead of equation (3.8),
$$
\begin{equation*}
\frac{\partial \lambda_n (\tau,t)}{\partial t}=0,\quad \text{that is, } \ \lambda_n (\tau,t) = \lambda_n (\tau,0), \quad n\in \mathbb{Z}.
\end{equation*}
\notag
$$
Now we put $t=0$ in the equation $\mathfrak{L}(\tau,t)\nu_n =\lambda_n (\tau,t)\nu_n$, $n\in \mathbb{Z}$. The eigenvalues $\lambda_n (\tau )=\lambda_n (\tau,0)$, $n\in \mathbb{Z}$, for the periodic or antiperiodic problems for the equation $\mathfrak{L}(\tau,0)\nu_n =\lambda_n (\tau )\nu_n$, $n\in \mathbb{Z}$, are independent of the parameter $\tau \in \mathbb{R}$, and hence $\lambda_n (\tau ,t)=\lambda_n (\tau )=\lambda_n$, $n\in \mathbb{Z}$. This proves Theorem 3.1. Now we use the trace formulas
$$
\begin{equation}
\begin{aligned} \, p(\tau,t) &= \sum_{k=-\infty}^{+\infty} \biggl(\frac{\lambda_{2k-1} + \lambda_{2k}}{2} - \xi_k (\tau,t) \biggr), \end{aligned}
\end{equation}
\tag{3.10}
$$
$$
\begin{equation}
\begin{gathered} \, q(\tau,t) = \sum_{k=-\infty}^{+\infty} (-1)^{k-1} \sigma_k (\tau,t) h_k (\xi(\tau,t)), \\ q^2(\tau,t) + q_\tau(\tau,t) = \sum_{k=-\infty}^{+\infty} \biggl(\frac{\lambda_{2k-1}^2 + \lambda_{2k}^2}{2} - \xi_k^2 (\tau,t) \biggr). \nonumber \end{gathered}
\end{equation}
\tag{3.11}
$$
We also change of the variables to
$$
\begin{equation}
\xi_n (\tau ,t)=\lambda_{2n-1} +(\lambda_{2n} -\lambda_{2n-1} )\sin^2(x_n (\tau ,t)),\qquad n\in \mathbb{Z}.
\end{equation}
\tag{3.12}
$$
Now the system of Dubrovin equations (3.2)–(3.3) can be written as a single equation
$$
\begin{equation}
\frac{dx(\tau ,t)}{dt} =H(x(\tau,t)),\qquad x(\tau ,t)\big|_{t=0} =x^0 (\tau),\quad x^0 (\tau)\in \mathbb{K},
\end{equation}
\tag{3.13}
$$
in the Banach space $\mathbb{K}$, where
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \mathbb{K} &=\biggl\{ x=\bigl(\dots,x_{-1}(\tau,t), x_0(\tau,t), x_1(\tau,t),\dots\bigr)\colon \\ &\qquad\qquad\qquad \|x\| =\sum_{n=-\infty}^{+\infty } (1+|n|^2)(\lambda_{2n} -\lambda_{2n-1})|x_n|<\infty \biggr\}, \end{aligned} \\ H(x) = \bigl(\dots,H_{-1}(x), H_1(x),\dots\bigr), \\ \begin{aligned} \, H_n (x(\tau,t)) &=(-1)^{n} \sigma_n^0 (\tau)f_n \bigl(\dots,\lambda_1 + (\lambda_2-\lambda_1)\sin^2 (x_1(\tau,t)),\dots\bigr) \\ &\qquad\times g_n \bigl(\dots,\lambda_1 + (\lambda_2-\lambda_1)\sin^2 (x_1(\tau,t)),\dots\bigr) \\ &=(-1)^n \sigma_n^0 (\tau) f_n (x(\tau,t)) g_n (x(\tau,t)). \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
From the conditions $p_0 (x+\pi )=p_0 (x) \in C^6 (\mathbb{R})$, $q_0 (x+\pi )=q_0 (x) \in C^6 (\mathbb{R})$, and the asymptotics of the eigenvalues of the periodic and antiperiodic problem for the Dirac system of equations, we obtain the equalities
$$
\begin{equation}
\begin{cases} {\displaystyle\lambda_{2k-1},\ \lambda_{2k} = k+\sum_{j=1}^7c_j k^{-j} \pm 2^{-6} |k|^{-6} |q_{2k}^6| +|k|^{-7} \varepsilon_k^{\pm}}, \\ \gamma_k \equiv \lambda_{2k} -\lambda_{2k-1}=\dfrac{|q_{2k}^6|}{2^5 |k|^6}+\dfrac{\delta_k}{|k|^7}, \\ {\displaystyle\sum_{k=-\infty}^{+\infty}|q_{2k}^5|^2 <\infty,\ \sum_{k=-\infty}^{+\infty}(\varepsilon_k^{\pm} )^2 <\infty,\ \delta_k =\varepsilon_k^+ -\varepsilon_k^-} \end{cases}
\end{equation}
\tag{3.14}
$$
(see [57]). Since $\xi_n (\tau,t)\in [\lambda_{2n-1},\lambda_{2n}]$, we have
$$
\begin{equation}
\inf_{k\ne n} |\xi_n (\tau,t)-\xi_k (\tau ,t)|\geqslant a> 0.
\end{equation}
\tag{3.15}
$$
Now we will use (3.14) and (3.15) to estimate the functions
$$
\begin{equation*}
|f_n(x(\tau,t))|,\quad \biggl|\frac{\partial f_n(x(\tau,t))}{\partial x_m}\biggr| \quad \text{and} \quad |g_n(x(\tau,t))|,\quad \biggl|\frac{\partial g_n(x(\tau,t))}{\partial x_m}\biggr|.
\end{equation*}
\notag
$$
where $C_j>0$, $j=1,\dots,5$, are independent of the parameters $m$ and $n$, and $x = (\dots,x_{-1}, x_0, x_1,\dots)$. Proof. According to [47],
$$
\begin{equation}
c_1 \leqslant |f_n (\xi)|\leqslant c_2,\qquad \biggl|\frac{\partial f_n (\xi)}{\partial \xi_m} \biggr| \leqslant c_3,
\end{equation}
\tag{3.18}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \xi &= (\dots,\xi_{-1},\xi_0,\xi_1,\dots)= \bigl(\dots,\lambda_{-3}+(\lambda_{-2}-\lambda_{-3}) \sin^2 x_{-1}, \\ &\qquad \lambda_{-1}+(\lambda_0-\lambda_{-1}) \sin^2 x_0, \lambda_1+(\lambda_2-\lambda_1) \sin^2 x_1,\dots \bigr), \end{aligned}
\end{equation*}
\notag
$$
and $C_1$, $C_2$, $C_3$ are constants. Using inequalities (3.18) and the change of variables (3.12), we have
$$
\begin{equation*}
\begin{gathered} \, C_1 \leqslant |f_n(x)| \leqslant C_2, \\ \biggl|\frac{\partial f_n (x)}{\partial x_m} \biggr| = \biggl|\frac{\partial f_n (\xi)}{\partial \xi_m} \biggr| \cdot \biggl|\frac{\partial \xi_m}{\partial x_m} \biggr| =\biggl|\frac{\partial f_n (\xi)}{\partial \xi_m} \biggr| \cdot |\gamma_m \sin2x_m|\leqslant C_3 \gamma_m, \end{gathered}
\end{equation*}
\notag
$$
which verifies (3.16). Now let us prove (3.17). Using the change of variables (3.12), and the trace formulas (3.10) and (3.11), we have
$$
\begin{equation}
\begin{aligned} \, p(\tau,t) &= \frac12 \sum_{k=-\infty}^{+\infty} \gamma_k (1-2\sin^2 x_k),\qquad\! p_\tau (\tau,t) = - \sum_{k=-\infty}^{+\infty} \gamma_k \sin 2x_k \, \frac{\partial x_k}{\partial \tau}, \\ p_{\tau \tau}(\tau,t) &= -2\sum_{k=-\infty}^{+\infty} \gamma_k \cos 2x_k \biggl(\frac{\partial x_k}{\partial \tau}\biggr)^2 + \sum_{k=-\infty}^{+\infty} \gamma_k \sin 2x_k\, \frac{\partial^2 x_k}{\partial \tau^2}, \\ q(\tau,t) &= \frac12 \sum_{k=-\infty}^{+\infty} (-1)^{k-1} \sigma_k^0 (\tau) \gamma_k \sin 2x_k f_k(x), \\ q_\tau (\tau,t) &= \sum_{k=-\infty}^{+\infty} (-1)^{k-1} \sigma_k^0 (\tau) \cos 2x_k \, \frac{\partial x_k}{\partial \tau}\, f_k(x) \\ &\qquad + \frac12 \sum_{k=-\infty}^{+\infty} (-1)^{k-1} \sigma_k^0 (\tau) \gamma_k \sin 2x_k \, \frac{\partial f_k (x)}{\partial \tau}, \\ q^2 (\tau,t) + q_\tau (\tau,t) &= \sum_{k=-\infty}^{+\infty} \biggl\{\frac{\lambda^2_{2k}-\lambda^2_{2k-1}}{2} + \gamma_k \sin^2 x_k (2\lambda_{2k-1}-\gamma_k \sin^2 x_k) \biggr\}, \end{aligned}
\end{equation}
\tag{3.19}
$$
where $x_k = x_k(\tau,t)$, $k\in \mathbb{Z}$.
By (3.14) and (3.15), the function series (3.19) converge uniformly. The functions $a(t)$, $b(t)$, $c(t)$, $d(t)$, $t \in [0,\infty)$, are bounded, and hence there exist numbers $M_j>0$, $j=1,2,3,4$, such that $|a(t)| \leqslant M_1$, $|b(t)| \leqslant M_2$, $|c(t)| \leqslant M_3$, $|d(t)| \leqslant M_4$. Hence by (3.14) For a proof of the second inequality in (3.17), we will use the asymptotics (3.14), equation(3.1), and equalities (3.19). We have
$$
\begin{equation*}
\begin{aligned} \, &1)\ \ \biggl|\frac{\partial x_k}{\partial \tau}\biggr| = \biggl|\frac{\partial x_k}{\partial \xi_k} \biggr|\cdot \biggl|\frac{\partial \xi_k}{\partial \tau} \biggr| = \biggl|\frac1{\gamma_k \sin2x_k} \biggr|\cdot \biggl|\frac{\partial \xi_k}{\partial \tau} \biggr| \\ &\qquad= \Biggl| (-1)^{k-1} \sigma_k^0 (\tau) f_k (x) \Biggl\{ \sum_{\substack{j=-\infty\\j\ne k}}^{+\infty} \frac{\gamma_j (1-2\sin^2 x_j)}{2} + \frac{\lambda_{2k}+\lambda_{2k-1}}{2} \Biggr\}\Biggr| \\ &\qquad\leqslant A_1 |k|, \\ &2)\ \ \biggl|\frac{\partial f_k (x)}{\partial \tau}\biggr| = \biggl|\sum_{i=-\infty}^{+\infty} \frac{\partial f_k (x)}{\partial x_i} \frac{\partial x_i}{\partial \tau} \biggr| \leqslant A_2, \\ &3)\ \ \biggl|\frac{\partial^2 x_k}{\partial \tau^2}\biggr| =\Biggl| (-1)^{k-1} \sigma_k^0 (\tau) \sum_{i=-\infty}^{+\infty} \biggl\{\frac{\partial f_k (x)}{\partial x_i} \, \frac{\partial x_i}{\partial \tau} \biggr\} \\ &\qquad\qquad\qquad \times \Biggl\{ \sum_{\substack{j=-\infty\\j\ne k}}^{+\infty} \frac{\gamma_j (1-2\sin^2 x_j)}{2} + \frac{\lambda_{2k}+\lambda_{2k-1}}{2} \Biggr\} \\ &\qquad\qquad+(-1)^k \sigma_k^0 (\tau) f_k (x) \sum_{\substack{j=-\infty\\j\ne k}}^{+\infty} \gamma_j \sin 2x_j \, \frac{\partial x_j}{\partial \tau}\Biggr| \leqslant A_3 |k|, \\ &4)\ \ \biggl|\frac{\partial f_k(x)}{\partial x_m} \biggr| = \biggl|\sum_{i=-\infty}^{+\infty} \frac{\partial f_k }{\partial \xi_i} \, \frac{\partial \xi_i}{\partial x_m} \biggr| =\frac{\partial f_k }{\partial \xi_m} \, \frac{\partial \xi_m}{\partial x_m} \leqslant A_4 \gamma_m, \\ &5)\ \ \biggl|\frac{\partial}{\partial x_m}\biggl(\frac{\partial x_k}{\partial \tau} \biggr) \biggr| \\ &\qquad = \Biggl|(-1)^{k-1} \sigma_k^0(\tau)\cdot \frac{\partial}{\partial x_m} \Biggl(f_k(x)\Biggl\{ \sum_{\substack{j=-\infty\\j\ne k}}^{+\infty} \frac{\gamma_j (1-2\sin^2 x_j)}{2}+ \frac{\lambda_{2k}+\lambda_{2k-1}}{2} \Biggr\} \Biggr) \Biggr| \\ &\qquad \leqslant A_5 |k| \gamma_m, \\ &6)\ \ \biggl|\frac{\partial}{\partial x_m}\biggl(\frac{\partial^2 x_k}{\partial \tau^2} \biggr) \biggr| \leqslant A_6 \gamma_m. \end{aligned}
\end{equation*}
\notag
$$
Using these inequalities, we have
$$
\begin{equation*}
\begin{aligned} \, &7)\ \ \biggl|\frac{\partial}{\partial x_m}\{4\xi_n^3\} \biggr| = \biggl|\frac{\partial}{\partial x_m}\{\xi_n\} \biggr| = \biggl|\frac{\partial}{\partial x_m}\{2\xi_n^2\} \biggr| \equiv 0,\qquad m \ne n, \\ &8)\ \ \biggl|\frac{\partial}{\partial x_m}\{p\xi_n^2\} \biggr| \leqslant B_1 \gamma_m |n|^2, \\ &9)\ \ \biggl|\xi_n \frac{\partial}{\partial x_m} \{p^2+q^2+q_\tau \} \biggr| \leqslant B_2 \gamma_m |n|(|m|+1), \\ &10)\ \ \biggl|\frac{\partial}{\partial x_m}\{pq_\tau - p_\tau q\} \biggr| \leqslant B_3 \gamma_m (|m|+1), \\ &11)\ \ \biggl|\frac{\partial}{\partial x_m}\{p^3+pq^2\} \biggr| \leqslant B_4 \gamma_m, \\ &12)\ \ \biggl|\frac{\partial p_{\tau \tau}}{\partial x_m}\biggr| \leqslant B_5 \gamma_m |m|(|m|+1), \\ &13)\ \ \biggl|\frac{\partial p}{\partial x_m}\biggr| \leqslant B_6 \gamma_m. \end{aligned}
\end{equation*}
\notag
$$
In 1)–13), $A_i = \mathrm{const}$, $B_j=\mathrm{const}$, $i=1,\dots,6$, $j=1,\dots,6$, $p=p(\tau,t)$, and $q=q(\tau,t)$.
From 7)–13), we have the estimate
$$
\begin{equation*}
\begin{aligned} \, \biggl|\frac{\partial g_n (x)}{\partial x_m} \biggr| &=\bigg|a(t) \biggl[4\bigl(\lambda_{2n-1} + (\lambda_{2n}-\lambda_{2n-1})\sin^2 x_n\bigr)^2 \, \frac{\partial p}{\partial x_m} \\ &\qquad +2\bigl(\lambda_{2n-1} + (\lambda_{2n}-\lambda_{2n-1})\sin^2 x_n\bigr)\, \frac{\partial}{\partial x_m} \{p^2 +q^2 + q_\tau \} - \frac{\partial p_{\tau \tau}}{\partial x_m} \\ &\qquad +2 \, \frac{\partial}{\partial x_m} \{pq_\tau + p_\tau q\} + 2 \, \frac{\partial}{\partial x_m}\{p^3 + p q^2 \} \biggr] \\ &\qquad + b(t) \biggl[2\bigl(\lambda_{2n-1} + (\lambda_{2n}-\lambda_{2n-1})\sin^2 x_n\bigr) \, \frac{\partial p}{\partial x_m} \\ &\qquad +\frac{\partial}{\partial x_m}\{p^2+q^2 + q_\tau \} \biggr] - c(t) \, \frac{\partial p}{\partial x_m} + \frac12\, d(t) \biggr| \leqslant C_5 \gamma_m |n|^2 (|m|^2+1). \end{aligned}
\end{equation*}
\notag
$$
Lemma 3.1 is proved. Lemma 3.2. If the initial functions $p_0 (x)$ and $q_0 (x)$ satisfy then the vector function $H(x)$ is Lipschitz continuous in the Banach space $\mathbb{K}$, that is, there is a constant $L=\mathrm{const}>0$ such that, for all $x,y \in \mathbb{K}$, where Proof. By Lemma 3.1, the derivative with respect to $x_m (\tau,t)$ of $F_n (x) = f_n (x) g_n (x)$ is estimated as
$$
\begin{equation*}
\begin{aligned} \, \biggl|\frac{\partial F_n (x)}{\partial x_m} \biggr| &\leqslant \biggl|\frac{\partial f_n (x)}{\partial x_m} \biggr| |g_n(x)|+|f_n(x)| \biggl|\frac{\partial g_n (x)}{\partial x_m} \biggr| \\ &\leqslant C_3 C_4 \gamma_m |n|^3 + C_2 C_5 \gamma_m |n|^2 (|m|^2+1) \\ &\leqslant C_3 C_4 \gamma_m |n|^3 (|m|^2+1)+C_2 C_5 \gamma_m |n|^2 (|m|^2+1) \\ &\leqslant \gamma_m |n|^3 (|m|^2+1)\biggl\{C_3 C_4 + \frac{C_2 C_5}{|n|} \biggr\} \leqslant \gamma_m |n|^3 (|m|^2+1)\{C_3 C_4 + 1\} \\ &= A\gamma_m |n|^3 (|m|^2+1), \end{aligned}
\end{equation*}
\notag
$$
where $A=C_3 C_4+1$ is independent of the parameters $m$ and $n$.
Next, since
$$
\begin{equation*}
H_n (x) = (-1)^n \sigma_n^0 (\tau) F_n (x(\tau,t)),\qquad n\in \mathbb{Z},
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
|H_n(x(\tau,t)) - H_n(y(\tau,t))| = |F_n(x(\tau,t)) - F_n(y(\tau,t))|.
\end{equation*}
\notag
$$
Applying the Lagrange mean value theorem to the function on $t \in [0,1]$, we have
$$
\begin{equation*}
f(1)-f(0) = f'(t^*),\qquad F_n(x)-F_n(y) = \sum_{k=-\infty}^{+\infty} \frac{\partial F_n (\theta)}{\partial x_k}\cdot (x_k-y_k),
\end{equation*}
\notag
$$
where $\theta = x+t^* (y-x)$. As a result,
$$
\begin{equation*}
\begin{aligned} \, &|H_n(x(\tau,t)) - H_n(y(\tau,t))| = |F_n(x(\tau,t)) - F_n(y(\tau,t))| \\ &\qquad\leqslant \sum_{k=-\infty}^{+\infty} \biggl|\frac{\partial F_n (\theta)}{\partial x_k}\biggr|\cdot |x_k(\tau,t)-y_k (\tau,t)| \\ &\qquad\leqslant A|n|^3 \sum_{k=-\infty}^{+\infty} \gamma_k (1+|k|^2)|x_k(\tau,t)-y_k (\tau,t)| \leqslant A|n|^3 \|x-y\|. \end{aligned}
\end{equation*}
\notag
$$
Now let us estimate the norm
$$
\begin{equation*}
\begin{aligned} \, \|H(x) - H(y)\| &= \sum_{n=-\infty}^{+\infty} \gamma_n (1+|n|^2)|H_n(x)-H_n (y)| \\ &\leqslant \sum_{n=-\infty}^{+\infty} \gamma_n (1+|n|^2) A|n|^3 \|x-y\| \\ &= \|x-y\| \sum_{n=-\infty}^{+\infty} A\gamma_n |n|^3 (1+|n|^2) = L \|x-y\|. \end{aligned}
\end{equation*}
\notag
$$
Here,
$$
\begin{equation}
L = A\sum_{n=-\infty}^{+\infty} \gamma_n |n|^3 (1+|n|^2)<\infty.
\end{equation}
\tag{3.20}
$$
That the series in (3.20) is convergent follows from (3.14).
This verifies the Lipschitz continuity of $H(x)$. Hence, for all $t>0$ and $\tau \in \mathbb{R}$, the solution of the Cauchy problem (3.2), (3.3) exists and is unique. Lemma 3.2 is proved. Remark 3.1. Theorem 3.1 and Lemma 3.2 give a method for finding a solution to problem (2.1)–(2.3). 1. We first find the spectral data $\lambda_n$, $\xi_n^0 (\tau )$, $\sigma_n^0 (\tau )=\pm 1$, $n\in \mathbb{Z}$, of the Dirac operator $\mathfrak{L}(\tau ,0)$. 2. Denote the spectral data of the operator $\mathfrak{L}(\tau ,t)$ by $\lambda_n$, $\xi_n (\tau ,t)$, $\sigma_n (\tau ,t )=\pm 1$, $n\in \mathbb{Z}$. Solving the Cauchy problem (3.2), (3.3) with arbitrary $\tau$, we find $\xi_n (\tau ,t)$, $\sigma_n (\tau ,t)$, $n\in \mathbb{Z}$. 3. The functions $p(\tau,t)$ and $q(\tau,t)$, that is, the solutions to problem (2.1)–(2.3), are found from the trace formula. Thus, we have proved the following theorem. Theorem 3.2. If the initial functions $p_{0} (x)$, $q_{0} (x)$ satisfy the conditions
$$
\begin{equation*}
p_0 (x+\pi )=p_0 (x)\in C^6 (\mathbb{R}),\qquad q_0 (x+\pi )=q_0 (x)\in C^6 (\mathbb{R}),
\end{equation*}
\notag
$$
Then problem (2.1)–(2.3) has unique solutions $p(x,t)$ and $q(x,t)$ from the class $C_x^3 (t>0) \cap C_t^1(t>0) \cap C(t \geqslant 0)$. These solutions are given, respectively, by the sum of series (3.10) and (3.11). The next result follows from [59] and [66]. Theorem 3.3. If the initial functions $p_0 (x)$ and $q_0 (x)$ are real analytical $\pi$-periodic functions, then the solutions $p(x,t)$, $q(x,t)$, $x \in \mathbb{R}$, $t>0$, to the Cauchy problem (2.1)–(2.3) are real analytic functions with respect to $x$. The next result follows from [58] and [60]. Theorem 3.4. If $\frac{\pi}{2}$ is a period (antiperiod) of the initial functions $p_0 (x)$ and $q_0 (x)$, then all the roots of the equation $\Delta(\lambda) + 2 = 0$ ($\Delta(\lambda) - 2 = 0$ are double. The Lyapunov function $\Delta(\lambda,0,t)$ for the $\mathcal L(0,t)y=\lambda y$ corresponding to the coefficients $p(x,t)$, $q(x,t)$, $x \in \mathbb{R}$, $t>0$, coincides with $\Delta(\lambda)$, and hence the number $\frac{\pi}{2}$ is also a period (antiperiod) of the solutions $p(x,t)$ and $q(x, t)$ with respect to $x$. Now let us consider the finite-gap case. Here, the solutions $p(\tau,t)$ and $q(\tau,t)$, $\tau \in \mathbb{R}$, $t>0$, to problem (2.1)–(2.3) are determined by
$$
\begin{equation*}
p(\tau,t) \,{=} \sum_{k=-N}^{N} \biggl(\frac{\lambda_{2k-1} + \lambda_{2k}}{2} - \xi_k (\tau,t) \biggr), \quad q(\tau,t) \,{=} \sum_{k=-N}^{N} (-1)^{k-1} \sigma_k (\tau,t) h_k (\xi(\tau,t)).
\end{equation*}
\notag
$$
Here, the coordinates $\xi \equiv \xi(\tau,t) = (\xi_{-N}(\tau,t),\dots,\xi_{-1}(\tau,t), \xi_0(\tau,t),\xi_1(\tau,t),\dots, \xi_N(\tau,t))$, $\sigma \equiv \sigma(\tau,t) = (\sigma_{-N}(\tau,t),\dots,\sigma_{-1}(\tau,t),\sigma_0(\tau,t),\sigma_1(\tau,t),\dots,\sigma_N (\tau,t))$ obey the system of Dubrovin equations
$$
\begin{equation*}
\begin{aligned} \, &\frac{\partial \xi_n (\tau,t)}{\partial t} \\ &= \begin{cases} 2 (-1)^{n} \sqrt{(\xi_n(\tau,t) - \lambda_{2n-1})(\lambda_{2n} -\xi_n(\tau,t))}\, f_n(\xi(\tau,t)) g_n(\xi(\tau,t)), &\!|n| \leqslant N, \\ 0, &\!|n| \geqslant N+1, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
with initial conditions
$$
\begin{equation*}
\xi_n (\tau,0) = \xi_n^0(\tau),\qquad \sigma_n (\tau,0) = \sigma_n^0(\tau),\quad n=0, \pm 1, \pm 2, \dots, \pm N,
\end{equation*}
\notag
$$
where $\xi_n^0(\tau), \sigma_n^0(\tau) = \pm 1$, $n=0, \pm 1, \pm 2, \dots, \pm N$, are the spectral parameters of the finite-gap Dirac operator of the form
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{L}(\tau,0)y &\equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} y_1' \\ y_2' \end{pmatrix} + \begin{pmatrix} p_0(x+\tau) & q_0(x+\tau) \\ q_0(x+\tau) & -p_0(x+\tau) \end{pmatrix}\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \\ &=\lambda \begin{pmatrix} y_1 \\ y_2 \end{pmatrix},\qquad x\in \mathbb{R},\quad \tau \in \mathbb{R}. \end{aligned}
\end{equation*}
\notag
$$
Here,
$$
\begin{equation*}
\begin{aligned} \, f_n (\xi ) &=\sqrt{\prod_{\substack{k=-N,\\ k\ne n}}^N \frac{(\lambda_{2k-1} -\xi_n )(\lambda_{2k} -\xi_n)}{(\xi_k -\xi_n)^2}},\qquad n=0, \pm 1, \pm 2, \dots, \pm N, \\ g_n (\xi) &=a(t)[4\xi_n^3 + 4p\xi_n^2 + 2\xi_n(p^2+q^2+q_\tau) - p_{\tau \tau} + 2(pq_\tau - p_\tau q) + 2p(p^2+q^2)] \\ &\qquad + b(t)[2\xi_n^2+2p\xi_n + p^2 +q^2 +q_\tau] + c(t)[-\xi_n-p] + \frac12\, d(t). \end{aligned}
\end{equation*}
\notag
$$
It is easily checked that the Cauchy problem (2.1)–(2.3) is solvable for all finite-gap initial data, since
$$
\begin{equation*}
L = A\sum_{n=-N}^{N} \gamma_n |n|^3 (1+|n|^2),\qquad \gamma_n = 0,\quad |n| \geqslant N+1.
\end{equation*}
\notag
$$
Example 3.1. Consider the single-gap potential $ u_0 (x) = q_0 (x)-ip_0 (x)$ of the Dirac operator $\mathfrak{L}(\tau,0)$ given by the spectrum $E = \mathbb{R} \setminus (\lambda_{-1}, \lambda_0)$ and the spectral parameters $\xi_0 (0) \in [\lambda_{-1}, \lambda_0]$, $\sigma_0 (0) = \pm 1$. In this case, the system of Dubrovin equations (3.1) consists of the single equation
$$
\begin{equation*}
\frac{d \xi_0 (\tau)}{d \tau} = -(\lambda_{-1}+ \lambda_0) \sigma_0 (\tau) \sqrt{(\xi_0(\tau) - \lambda_{-1})(\lambda_0 -\xi_0(\tau))}.
\end{equation*}
\notag
$$
Changing the variable to
$$
\begin{equation*}
\xi_0 (\tau) = \lambda_{-1} + (\lambda_0 - \lambda_{-1})\sin^2 \varphi (\tau)
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\frac{d \varphi (\tau)}{d \tau} = -\frac{\lambda_{-1}+ \lambda_0}{2} \sigma_0 (0),\qquad \varphi (0) = \arcsin \sqrt{\frac{\xi_0 (0)-\lambda_{-1}}{\lambda_0-\lambda_{-1}}}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\xi_0(\tau) = \frac{\lambda_{-1}+ \lambda_0}{2} - \frac{\lambda_0 - \lambda_{-1}}{2} \cos \{- (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0 (0) \varphi (0)\}.
\end{equation*}
\notag
$$
Employing the trace formula, we find that
$$
\begin{equation*}
\begin{aligned} \, p_0(\tau) &= \frac{\lambda_0 - \lambda_{-1}}{2} \cos \{- (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0 (0) \varphi (0)\}, \\ q_0(\tau) &= -\frac{\lambda_0 - \lambda_{-1}}{2} \sin \{- (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0 (0) \varphi (0)\}. \end{aligned}
\end{equation*}
\notag
$$
Example 3.2. Consider the problem
$$
\begin{equation}
\begin{cases} p_t = a(t) [p_{xxx}-6p_x (p^2+q^2)] + b(t) [-q_{xx}+2q (p^2+q^2)] + c(t) p_x + d(t) q, \\ q_t = a(t) [q_{xxx}-6q_x (p^2+q^2)] + b(t) [p_{xx}-2p (p^2+q^2)] + c(t) q_x - d(t) p \end{cases}
\end{equation}
\tag{3.21}
$$
with initial conditions
$$
\begin{equation}
\begin{cases} p(x,t) |_{t=0} = \dfrac{\lambda_0 - \lambda_{-1}}{2} \cos \{- (\lambda_{-1}+ \lambda_0)x + 2\sigma_0 (0) \varphi (0)\}, \\ q(x,t) |_{t=0} = -\dfrac{\lambda_0 - \lambda_{-1}}{2} \sin \{- (\lambda_{-1}+ \lambda_0)x + 2\sigma_0 (0) \varphi (0)\}. \end{cases}
\end{equation}
\tag{3.22}
$$
In this single-gap case, the system of Dubrovin equations (3.2) takes the form
$$
\begin{equation}
\begin{aligned} \, \frac{\partial \xi_0 (\tau,t)}{\partial t} &= 2 \sigma_0 \sqrt{(\xi_0 - \lambda_{-1})(\lambda_0 -\xi_0)} \, \biggl\{a(t)[4\xi_0^3 + 4p\xi_0^2 + 2\xi_0(p^2+q^2+q_\tau)-p_{\tau \tau} \nonumber \\ &\qquad+ 2(pq_\tau - p_\tau q) + 2p(p^2+q^2)] + b(t)[2\xi_0^2+2p\xi_0 + p^2 +q^2 +q_\tau] \nonumber \\ &\qquad-c(t)[\xi_0 + p] + \frac12\, d(t)\biggr\}; \end{aligned}
\end{equation}
\tag{3.23}
$$
here, $\xi_0 = \xi_0 (\tau,t)$, $p=p(\tau,t)$, $q=q(\tau,t)$. For the single-gap case, using the trace formulas
$$
\begin{equation}
\begin{aligned} \, p(\tau,t) &= \frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0 (\tau,t), \end{aligned}
\end{equation}
\tag{3.24}
$$
$$
\begin{equation}
\begin{gathered} \, q(\tau,t) = -\sigma_0 (\tau,t) \sqrt{(\xi_0(\tau,t) - \lambda_{-1})(\lambda_0 -\xi_0(\tau,t))}, \\ q^2(\tau,t) + q_\tau (\tau,t) = \frac{\lambda_{-1}^2+ \lambda_0^2}{2} - \xi_0^2 (\tau,t) \nonumber \end{gathered}
\end{equation}
\tag{3.25}
$$
and their derivatives
$$
\begin{equation*}
\begin{aligned} \, p_\tau (\tau,t) &= \sigma_0(\tau,t) (\lambda_{-1}+ \lambda_0)\sqrt{(\xi_0(\tau,t) - \lambda_{-1})(\lambda_0 -\xi_0(\tau,t))}, \\ q_\tau (\tau,t) &= (\lambda_{-1}+ \lambda_0) \biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0 (\tau,t) \biggr), \\ p_{\tau \tau} &= -(\lambda_{-1}+ \lambda_0)^2 \biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0 (\tau,t) \biggr), \end{aligned}
\end{equation*}
\notag
$$
we write equation (3.23) in the closed form
$$
\begin{equation*}
\begin{aligned} \, \frac{\partial \xi_0 (\tau,t)}{\partial t} &= 2 \sigma_0 \sqrt{(\xi_0 - \lambda_{-1})(\lambda_0 -\xi_0)} \, \biggl\{a(t)\biggl[4\xi_0^3 + 4\xi_0^2\biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0\biggr) \\ &\qquad + 2\xi_0 \biggl(\biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0\biggr)^2 + \frac{\lambda_{-1}^2+ \lambda_0^2}{2} - \xi_0^2 \biggr) \\ &\qquad+ (\lambda_{-1}+ \lambda_0)^2\biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0 \biggr) + 2\biggl((\lambda_{-1}+ \lambda_0)\biggl(\frac{\lambda_{-1}+ \lambda_0}2-\xi_0\biggr)^2 \\ &\qquad-(\lambda_{-1}+ \lambda_0)(\xi_0 - \lambda_{-1})(\lambda_0 - \xi_0)\biggr) \\ &\qquad+ 2\biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0\biggr) \biggl(\biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0\biggr)^2 + (\xi_0 - \lambda_{-1})(\lambda_0 - \xi_0) \biggr)\biggr] \\ &\qquad + b(t)\biggl[2\xi_0^2+2\xi_0 \biggl(\frac{\lambda_{-1}+ \lambda_0}{2} - \xi_0\biggr) + \frac{3}{4} (\lambda_{-1}^2 + \lambda_0^2) + \frac{\lambda_{-1}\lambda_0}{2} \\ &\qquad - \xi_0(\lambda_{-1}+ \lambda_0) \biggr] - c(t)\frac{\lambda_{-1}+ \lambda_0}{2} + \frac12\, d(t)\biggr\} \\ &= 2\sigma_0\sqrt{(\xi_0 - \lambda_{-1})(\lambda_0 -\xi_0)}\, \biggl\{K\cdot a(t) + M \cdot b(t) + N \cdot c(t) + \frac12\, d(t) \biggr\}, \end{aligned}
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\begin{cases} \dfrac{\partial \xi_0 (\tau,t)}{\partial t} = 2\sigma_0\sqrt{(\xi_0 - \lambda_{-1})(\lambda_0 -\xi_0)}\, \biggl\{K\cdot a(t) + M \cdot b(t) + N \cdot c(t) \,{+}\, \dfrac12\, d(t) \biggr\},\! \\ \xi_0 (\tau,t)\big|_{t=0} = \dfrac{\lambda_{-1}+ \lambda_0}{2} - \dfrac{\lambda_0 - \lambda_{-1}}{2} \cos \{- (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0 (0) \varphi (0)\}, \\ \sigma_0 (\tau,t)\big|_{t=0} = \sigma_0(\tau); \end{cases}
\end{equation*}
\notag
$$
here,
$$
\begin{equation*}
\begin{gathered} \, \xi_0 = \xi_0 (\tau,t),\qquad \sigma_0 = \sigma_0 (\tau,t),\qquad K = \frac{5}{4}(\lambda_{-1}+ \lambda_0)^3 - 3\lambda_{-1}\lambda_0 (\lambda_{-1}+ \lambda_0), \\ M = \frac{3}{4} (\lambda_{-1}+ \lambda_0)^2 - \lambda_{-1}\lambda_0,\qquad N = -\frac{\lambda_{-1}+ \lambda_0}{2}. \end{gathered}
\end{equation*}
\notag
$$
To find the solution $\xi_0 (\tau,t)$, $\sigma_0 (\tau,t)$, we change the variables to
$$
\begin{equation}
\xi_0 (\tau,t) = \lambda_{-1} + (\lambda_0 - \lambda_{-1})\sin^2 \psi (\tau,t).
\end{equation}
\tag{3.26}
$$
Hence we have the equation
$$
\begin{equation*}
\frac{\partial \psi(\tau,t)}{\partial t} = \sigma_0(\tau)\biggl\{K\cdot a(t) + M \cdot b(t) + N \cdot c(t) + \frac12\, d(t) \biggr\}
\end{equation*}
\notag
$$
for $\psi(\tau,t)$. Consequently,
$$
\begin{equation*}
\begin{aligned} \, \psi(\tau,t) &= \sigma_0(\tau)\biggl\{K \int_0^t a(s)\, ds + M \int_0^t b(s)\, ds + N \int_0^t c(s)\, ds + \frac12\int_0^t d(s)\, ds\biggr\} \\ &\qquad+ \sigma_0(\tau)C(\tau). \end{aligned}
\end{equation*}
\notag
$$
Substituting this into (3.26), we obtain
$$
\begin{equation*}
\begin{aligned} \, &\xi_0(\tau,t) = \frac{\lambda_{-1}+ \lambda_0}{2} + \frac{\lambda_0 - \lambda_{-1}}{2} \\ &\quad\times \cos\biggl\{2K \int_0^t a(s)\, ds +2 M \int_0^t b(s)\, ds + 2N \int_0^t c(s)\, ds + \int_0^t d(s)\, ds + 2C(\tau)\biggr\}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
C(\tau) = - \frac{\lambda_{-1}+ \lambda_0}{2}\, \tau + \sigma_0(0) \psi_0(0).
\end{equation*}
\notag
$$
Using using the trace formulas (3.24), (3.25), we find the solution
$$
\begin{equation*}
\begin{aligned} \, p(\tau,t) &= \frac{\lambda_0 - \lambda_{-1}}{2} \cos \biggl\{2K \int_0^t a(s)\, ds +2 M \int_0^t b(s)\, ds + 2N \int_0^t c(s)\, ds \\ &\qquad + \int_0^t d(s)\, ds - (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0(0) \psi_0(0) \biggr\}, \\ q(\tau,t) &= -\frac{\lambda_0 - \lambda_{-1}}{2} \sin \biggl\{2K \int_0^t a(s)\, ds +2 M \int_0^t b(s)\, ds + 2N \int_0^t c(s)\, ds \\ &\qquad + \int_0^t d(s)\, ds - (\lambda_{-1}+ \lambda_0)\tau + 2\sigma_0(0) \psi_0(0) \biggr\} \end{aligned}
\end{equation*}
\notag
$$
to the Cauchy problem (3.21), (3.22). § 4. ConclusionsThe inverse spectral problem method is used to integrate a non-linear Hirota-type equation with additional terms in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proved. The solvability of the Cauchy problem for a Hirota-type equation with additional terms in the class of six times continuously differentiable periodic infinite-gap functions is established. It should be noted that, in the presence of the quite popular Hirota-type equation with additional terms, the propagation speed of a periodic traveling wave increases or decreases depending on the coefficients $a(t)$, $b(t)$, $c(t)$, $d(t)$, while the amplitude does not change. This follows from the comparison of the examples $a(t) \ne 0$, $b(t) \ne 0$, $c(t) = 0$, $d(t) = 0$ and $a(t) \ne 0$, $b(t) \ne 0$, $c(t) \ne 0$. However, it is still unknown whether problem (2.1)–(2.3) is solvable in the class $C^n (\mathbb{R})$, $0 \leqslant n \leqslant 5$, of periodic infinite-gap functions. 
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