Solvability of One-Dimensional Semilinear Hyperbolic Systems and Sine-Gordon Equation

We study semilinear hyperbolic Eq. (1.1). We derive an explicit solution representation for some nonlinear terms F and G. For other nonlinear terms, it is shown that the solutions of the equations are related with the variable coefficient sine-Gordon equation.


Introduction
We are interested in semilinear hyperbolic equations where i = √ −1 and U, V are complex valued functions.F and G are real valued nonlinear functions of U and V.Here we consider that F and G are quadratic functionf of U and V.
The following model with F = |V| 2 , G = |U| 2 has been studied in [2,6] (1.1) 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1264-1271 which is called Thirring equation.Based on the observation of the null structure of Thirring equation, the initial value problem of (1.2) was studied in [2,6] in terms of Sobolev space H s .They showed that there exists a time T > 0 and solution U, V ∈ C([0, T], H s (ℝ)) ( s ≥ 0 ) of the Cauchy problem (1.2).Especially, the explicit solution representation was used in [6] to show that there exists a global strong solution (U, V) to the initial value problem (1.2) in the critical space C([0, ∞);L 2 (ℝ)) .The followings are the solution representation of (1.2) The following model was studied in [7] which corresponds to the nonlin F = Re (UV) , G = Re (UV).
The sine-Gordon equation is a very important equation in nonlinear solitary wave equations.The well-known solutions are and Exact solution was found in [1] and direct method was used in [10] to solve the sine-Gordon equation.
The stability of the given solition solution was investigated in [3,8,11].
A variable coefficient sine-Gordon (vSG) equation is given by where H(x, t) is a real function.Several authors [4,5,9,13] studied vSG equation.Especially, it was shown in [4] that the vSG equation has the Painlevé property if the function H(x, t) is given by where H 1 (x + t) and H 2 (x − t) are arbitrary functions.They also showed that the vSG equation, in the case of (1.9), can be transformed to the sine-Gordon equation after the following transformation.
In fact, we have We study the connection between system (1.1) and vSG Eq. (1.8).Let us write the solution of (1.1) as the polar form It is observed that the phase functions and satisfy the vSG equation.Let us take an example.For the equation (1.1) with the nonlinear term F = Im (V 2 ) and G = Im (U 2 ) , we derive the following vSG equations for + and − Note that the variable coefficients of the above equations satisfy the condition (1.9).
The connection between the sine-Gordon equation and wave map from ℝ 1+1 to 2 was exploited in [12].The derived vSG equation is where h and k are related with initial velocities.
We present some preliminaries in section 2. The main result is described in section 3, where connection between system (1.1) with several nonlinear terms and the vSG Eq. (1.8) is investigated.We conclude this section by providing a few notations.We denote space time derivatives by t = t , x = x .The standard Sobolev space

Preliminaries
Here we consider C ∞ ((0, T) × ℝ) solutions which satisfy equations in the classical sense.In fact, suppose that initial data belong to u, v ∈ H m (ℝ) for a positive integer m.Then there exists a positive constant T ∈ (0, ∞) such that the initial value problem (1.1) has the unique local solution U, V ∈ C([0, T);H m (ℝ)) .The proof can be done using the Banach contraction mapping theorem.By Sobolev embedding theorem, it is easy to see that H m solution is a classical solution for m ≥ 2.
Let U(x, t) = |U(x, t)|e i (x,t) and V(x, t) = |V(x, t)|e i (x,t) .Then we derive from (1.1) (2.1) Considering (2.2), (2.4) and (2.5), we derive Integrating (2.7) on the characteristic line, we can express in terms of initial data v.Then can be obtained by integrating (2.8) on the characteristic line.Therefore (2.6) can be solved completely in terms of initial data.For the Thirring model (1.2), we have which can be solved to be (1.3).

Relation with vSG Equation
The solution of the system (1.1) is related with the vSG Eq. (1.8).We investigate this connection for the several nonlinear terms F and G.

The Case of
Here we consider with the initial data U(x, 0) = u(x) and V(x, 0) = v(x).
The Eq. ( 2  where Journal of Nonlinear Mathematical Physics (2023) 30:1264-1271 The similar process can be applied to the case of F = Re (V 2 ) and G = Re (U 2 ).
We can derive for (3.7)The similar process can be applied to the case of F = Re (V 2 ) and G = Im (U 2 ) .We omit the detail.which was studied in [7].Making use of we can derive Adding and subtracting equations, we arrive at The Eq. (3.12) let us consider − = x g and + = − t g .Then we can derive from (3.13)For the initial data satisfying |u| = 1 = |v| , we have, with the notation h = x g, which is a perturbed sine-Gordon equation.The similar process can be applied to the case of F = (UV) and G = Im (UV) .We omit the detail.Remark 3.1 It seems difficult that we apply the similar idea to the nonlinear terms like F = Im (UV) and G = Re (UV) in which we have
Case of F = Re (UV) and G = Re (UV) Here we consider