Solutions of the two-wave interactions in quadratic nonlinear media

In this paper, we propose a reliable treatment for studying the two-wave (symbiotic) solitons of interactions in nonlinear quadratic media. We investigate Schauder's fixed point theorem for proving the existence theorem. Additionally, the uniqueness solution for this system is proved. Also, a highly accurate approximate solution is presented via an iteration algorithm.

Since the exact analytical solutions of Eqs.(I.2) cannot be found for arbitrary values of α, then the purpose of this work is to present a result of the existence and uniqueness of solutions. Also, an exact implicit solution is derived using a useful procedure at α = 1. Thus, the paper is organized as follows: in Section II, we investigate the existence and uniqueness theorem of the two-wave solitons in quadratic media, where the problem is formulated in the context of two nonlinear coupled differential equations in one dimension. Then, in Section III, we solve the coupled system by an appropriate technique with suitable boundary conditions. A systematic numerical procedure is proposed in Section IV. Finally, we conclude with some remarks in Section V.

II. AN EXISTENCE AND UNIQUENESS THEOREM
Rewrite (I.2) in the following system where f i : [l 1 , l 2 ] × R × R −→ R, i = 1, 2 are defined by f 1 (ϕ, ψ) = 1 r (ϕ − ϕψ) and f 2 (ϕ, ψ) = 1 s (αψ − 1 2 ϕ 2 ). Existence -In this section, we shall deal with the existence of solutions of the BVP (I.2) with (I.7). First off all, we shall prove the following lemmas, which are useful tools in the proof of the existence and uniqueness theorem.
Proof. From the definition of f i (ϕ, ψ), we have Hence, Similarly, we obtain The unique solution u of the following boundary value problem subject to the Dirichlet boundary conditions u(a) = u(b) = 0 is given by where G(x, y) is the Green function given by Replacing g(x), a and b by f i (ϕ, ψ), l 1 and l 2 in Lemma 2, respectively, we obtain an equivalent integral system and , consider the closed and convex set .
In the theory of differential equations, there are a lot of methods to establish the existence of solutions. Theorems concerning the existence and properties of fixed points are known as fixed-point theorems. Such theorems are the most important tools for proving the existence and uniqueness of the solution. The fundamental theorem used in this theory is Schauder' s theorem. In order to make use of this theorem, it is sufficient to prove the following lemma.
The proof is complete.

III. ON THE DECOUPLING OF THE SYSTEM (I.2)
In this section, first of all, we are concerned with the norms estimate for the functions ϕ and ψ when r = s = 1.
where . 1 is the norm defined in the Sobolev space H 1 0 (I) by Proof. Multiplying both sides of the first equation of system (I.2) by ϕ and integrating from l 1 to l 2 , we obtain From the second equation of system (I.2), we have By substitution into the last term of (III.3), we obtain Integrating by parts and taking into account the given boundary conditions, we obtain This gives (III.1).
Let us now consider the case α = 1 [7,11,14] and in view of Lemma 6 if ϕ(x) = √ 2ψ(x), then it may be shown that the two equations of system (I.2) can be separated into the following nonlinear equation The exact solution to Eq.(III.7) follows by simply multiplying both sides of Eq.(III.7) by ψ .
which can be written as follows 1 2 and integrating with respect to x, we obtain where c 1 is an arbitrary constant of integration. Thus In view of ψ (x) = dψ dx , we have ± dψ Consequently, where c 2 is also a constant of integration. The LHS of Eq.(III.13) can be evaluated direct from the integrals of irrational functions. Indeed, if we choose c 1 = 0, then (III.14) Since tanh 2 (∓x) = tanh 2 (x). Hence, a simple computation leads to the implicit solution Thus, we have Lemma 7 The system (I.2) can be decoupled without increase the order of the system into the nonlinear equation (III.7) when α = 1. Furthermore, the solution (ϕ, ψ) is given by (III.15).
In figure 1, we display the variation of the exact solutions Eqs. (III.15) in terms of the independent variables x for different values of the constant of integration c 2 (Eq. (III.13 )). it can be seen that this constant of integration shifted left or right the distribution away from the origin with negative or positive values of c 2 , respectively. Also, it does not affect the behavior of the solutions, and the maximum value of the solution remains unchanged. Thus it can be chosen c 2 = 0. where β = ϕ (l 1 ) and γ = ψ (l 1 ) are unknown constants to be determined from the second boundary conditions ϕ(l 2 ) = ψ(l 2 ) = 0. We now construct a sequence of approximation of the solution that converges to the solution. The components (ϕ n , ψ n ) can be elegantly determined by setting the recursion scheme y l 1 f 2 (ϕ n−1 (s), ψ n−1 (s))dsdy. n ≥ 1.
In view of (IV.2), the numerical solutions are then given by If we match (ϕ 1 , ψ 1 ) at x = l 2 , then we need to solve In figure 2, we present the variation of the numerical solutions Eqs. (IV.6) against of the independent variables x for different values of the rescaled soliton parameter α. Now if we insert the solutions of Eq. IV.6 in the second member of Eqs. IV.2 and performing the integrals, then using the boundary condition for (ϕ 2 , ψ 2 ) at x = l 2 , we can obtain the second solutions ϕ 2 and ψ 2 . As the mathematical expressions are more cumbersome, we plot the numerical solutions in figure. 3. It is clear from these plots that when we increase the order of the recurrence, the numerical solutions converge rapidly to the exact solutions.

V. CONCLUSION
This paper is concerned with the treatment of the interaction of two-wave solitons in nonlinear quadratic media. These kinds of problems appear in different applications of nonlinear physics and play a crucial role in the stability problems of solitary waves. The problem is presented within the framework of two coupled nonlinear differential equations, which can be solved numerically with specific boundary conditions. But the generalization to any type of boundary condition constitutes a great challenge.
With concordance to real physical problems, the boundary conditions can be chosen properly.Thus, within this framework, we have proved a theorem of existence and uniqueness for the two-wave solitons in nonlinear quadratic media. Furthermore, we have suggested a useful technique of separation of the coupled system, and we have revealed that the formalism leads to analytic solutions.
Moreover, we have explored an interesting numerical technique, and we have used it to obtain the numerical solutions of the coupled system with suitable boundary conditions. The obtained results are in good agreement with those analytically achieved.
These crucial results open a novel class of investigations, which involve solitary waves with more coupled differential equations and more coupling terms. Some other examples of two or more-coupled solitary waves can be treated with the proposed techniques, and the results will be reported elsewhere.