Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov.

A reader who is interested in the physical meaning of this ibv problem can refer to [16]. In [27] the Riemann-Hilbert problem formalism for the ibv problem (1.1), (1.2), (1.3) (in more general setting) was formulated, and in [29], [30] the long-time asymptotic analysis was done.
They obtained the following results, which we summarize in Theorems 1.1-1.3 (full formulation of their result in the region ω 2 0 t < x < ω 2 t is too involved, and we give here a shortened version (Theorem 1.2). An interested reader can find a full formulation in the appendix (Theorem 4.1)): [30]. A plane wave of finite amplitude). Let ω 0 > 0 be defined by the formula Here where λ ± (ξ) are defined in (2.11), and A(k), X(k) are defined in (2.6), (2.10).
Remark 1.1. Universality. It is claimed in [30] that the result of the theorem 1.1 is true also for more general initial data u 0 (x) → 0 as x → +∞ under assumption of the absense of the discrete spectrum of the corresponding Lax pair operator. We see that the parameters that determines the asymptotics are determined only by the constants l, p, ω, and the actual form of the initial function u 0 (x) does not influence the leading-order asymptotic terms. Theorem 1.2 (Kotlyarov-Moskovchenko [30]. Shortened formulation. A modulated elliptic wave of finite amplitude). In the region ω 2 0 t < x < ω 2 t the solution of the ibv problem (1.1), (1.2), (1.3) for t → ∞ takes the form of a modulated elliptic wave Here ξ = t 4x is a slow variable, and are some functions given explicitly in terms of the initial data.
Remark 1.2. Universality. It is claimed in [30] that the result of the theorems 1.2, 4.1 is true also for more general initial data u 0 (x) → 0 as x → +∞ under assumption of the absence of the discrete spectrum of the Lax pair operator. In the latter case all the Θ− functions are determined by the parameters B g , B ζ ∆ (see Theorem 4.1). Further, the parameters B g , B ζ are fully determined solely by the constants p, l, ω, and do not require knowledge of the actual form of the initial function. The only quantity that depends on the actual form of the initial function u 0 (x) is ∆. This shows that the solution of the ibv problem behaves universally with respect to the initial function.
Theorem 1.2 (4.1) was proved under an implicit assumption that the parameters of the corresponding g-function exist in the specified region. However, this was not proved in [30]. The goal of this paper is to establish the existence of such parameters, and thus justify the results in [30]. We establich this in Theorem 3.1, which is based on the lemma 3.1.
Let us mention, that the points d(ξ), d(ξ) in equations (3.16) play role of branch points of Riemann surfaces in the Whitham modulation theory, and hence the equations (3.16) are Whitham type equations.
There is a lot of bibliography devoted to the Whitham modulation theory approach to asymptotics of solutions of integrable equations, including cases with different finite-gap boundary conditions as x → ±∞, see (see [2]- [9], [31], [17], [12], [1], [18], [19] and the bibliography therein). Most of these results were devoted to initial-value problems associated with self-adjoint Lax operators, but there are also a few results for problems associated with non-self-adjoint Lax operators( [2], [3], [31]). We would like to mention that in our case the associated Lax operator is also nonself-adjoint, which indicates modulation instability, caused by possible presence of small-norm breathers.
In the Whitham modulation theory the question of unique solvability of Whitham equations plays the central role and the main attention is devoted to analysis of complex Whitham deformations of the corresponding Riemann surfaces. The evolution of the branch points is governed by a transcendental system of equations, which is typically very hard to analyze, especially in the case of non-self-adjoint Lax operators ( [31]).
Recently a rigorous Riemann-Hilbert problem scheme was adjusted to the problems with nonvanishing initial data. This was achieved by introduction of the so-called g− function surgery approach. Equations for parameters of this g−function play role of Whitham equation in the Whitham modulation theory. For step-like initial data, first results were obtained by Buckingham, Venakides [13] and independently in Boutet de Monvel, Its, Kotlyarov [11] for Nonlinear Schrodinger equation (NS). In [13] the construction of the corresponding g− function was done with the help of Cauchy integrals, but the system of equations for parameters of this g−function was too complicated to be analyzed. On the contrary, the approach in [11] employed the construction of g− function in terms of Abelian integrals in the corresponding cutted complex plane. In this approach the existence of parameters for the corresponding g−function followed from positiveness of a polynomial of two real variables of degree 2 in a given domain.
T.Claeys [14] studied the long-time asymptotics for the solution of the Korteweg-de Vries equation with a particular choice of unbounded initial datum which grows as 3 √ −x for large x. In this case the solvability of the corresponding Whitham type equations was established in [32].
Further the approach in [11] was extended to the modified Korteweg-de Vries equation (MKdV) [22]- [26] (the degree of the corresponding polynomial was 2), and to Camassa-Holm equation [21], [20]. The step-like problem for the Korteweg-de Vries equation [15] employs the same g−function, as MKdV. Initial value problem for NS with more general type of initial function ( [10]) also employed the same g−function, as in [11].
In this article we show, that the Whitham type equations for the parameters of the corresponding g−function for the SRS model is reduced to prove positiveness of a polynomial of degree 12 in a given domain. This approach also produces a nice elementary problem (lemma 3.1), which is however far from being trivial and can be offered to students in some mathematical olympiads. An interested reader might try to solve it first before looking in the solution.
The structure of the paper is the following: in section 2 we recall the Riemann-Hilbert problem formulation for the ibv problem (1.1), (1.2), (1.3) from [27], [30], and the definition of the corresponding phase g-functions, which are used in the asymptotic analysis of oscillatory Riemann-Hilbert problems via the nonlinear steepest descet method. In section 3 we prove our main Theorem 3.1 and the underlying lemma 3.1. It is remarkable that the values of α 0 (3.21), x 0 (3.22) from lemma 3.1 are given for free within the framework of the RH problem analysis via the g−function surgery approach.
Acknowledgements. A.M. would like to thank Vladimir Kotlyarov for useful discussions and Koen van den Dungen for careful reading a version of this manuscript and giving valuable comments and suggestions.
2 Original Riemann-Hilbert problem and g−functions It was shown in [27] - [30] that the solution of the ibv problem (1.1), (1.2), (1.3) can be obtained from the solution of the following RH problem: Riemann-Hilbert problem 1. Find a 2 × 2 matrix-valued function M (x, t; k) that is sectionally analytic in k ∈ C \ Σ and satisfies the following properties: 2. end points behavior: M (x, t; k) is bounded in the vicinities of the points k = E, 0, E, and the contour (see Figure 1) where consists of the real line Im k = 0 and the circle arc γ ∪ γ, which is defined by equations ( [30]) The subarcs γ and γ are symmetric with respect to the real line and are divided by it.
Thus, to find an asymptotics of the the solution of the ibv problem it is enough to find an asymptotics of the solution of the Riemann-Hilbert problem. The authors in [27] - [30] then perform a series of transformations of this Riemann-Hilbert problem in the spirit of Deift-Zhou steepest descent method in order to reduce the problem to some explicitly solvable model problem and small-norm problems.
In turn, the crucial role in the asymptotic analysis of the Rieman-Hilbert problem 1 is played by the decomposition of the complex plane into regions where Im θ ≷ 0. In the long time asymptotics it is convenient to introduce a slow changing variable and a regularized phase function For different values of ξ, depending on the mutual location of the lines Im θ(k, ξ) = 0 and the contour Σ, V. Kotlyarov and A. Moscovchenko [30] introduced new phase functions g(k, ξ), g(k, ξ), which mimic some properties of θ, such as behavior as k → ∞ and k → 0, distribution of signs of Im θ, but also have different properties on the arcs γ, γ.
The necessity of new functions g, g appear when the curves Im θ = 0 start to intersect the contour Σ of the original RHP 1. Below we briefly describe the properties of θ, g, g, and then prove the solvability of equations for parameters of g.
In this region the original phase function θ = 1 4k + k x t is appropriate in asymptotic analysis of the corresponding Riemann-Hilbert problem 1. Lines where Im θ = 0 are as follows [30] (see Figure  2): This function is appropriate in asymptotic analysis until the bold circle reaches end points E, E of the arcs γ, γ. This happens when ξ = |E|, i.e. x = ω 2 t. 2.2 Region 0 < x < ω 2 0 t. A plane wave of finite amplitude.
In this region the apropriate g−function surgery can be done with the help of the function where Ω(k) is determined in (2.8) and , To analyze how the lines Im g = 0 behave, it is useful to look at the differential dg : 3 where the quantities λ − ≤ λ ≤ λ + are subject to the following system of equations ( [30]): (2.11) Simple zeros of dg, i.e. λ − , λ, λ + , correspond to the points at which there are 4 emanating rays Im g = const. Points at which dg ∼ (k − k 0 ) −1/2 , i.e. the points E, E, emanate just one ray Im g = const. At the point of the second order pole, i.e. at the origin, there are infinitely many lines Im g = const passing through. Lines with positive and negative value of const are separated by the real line, at which the const is 0. The system (2.11) was analyzed in [30, p.9,10], and it was shown that the real solution λ − < λ < λ + exists for ξ ∈ (ξ 0 , +∞), where ξ 0 > 0 and (2.12) and at the boundary of the interval ξ = ξ 0 we have (2.13) At the other boundary of the region, when ξ → +∞, we have Within the region ξ 0 < ξ < +∞ we have The qualitative picture of the distribution of signs of Im g(k, ξ) is plotted in Figure 3. This function is appropriate in the asymptotic analysis of the Riemann-Hilbert problem 1 until the points λ − and λ merge.
In this region the appropriate g−function surgery can be done with the help of the function (2.14) A qualitative decomposition of the complex plane according to the distribution of signs of Im g(k, ξ) is plotted in Figure 4. Following [30], denote the part of Im g = 0, which connects E and d, by γ d , the part that connects d and λ − by γ λ , and by γ d , γ λ the corresponding parts in the lower half-plane Im k ≤ 0.
The condition 3. is well-defined, i.e. the integral does not depend on the choice of contour of integration, since by the condition 1. the residue at ∞ is 0.
(Let us notice that the first two of the above properties are also satisfied by g(k).) With these properties function g is analytic in C\(γ d ∪ γ λ ∪ γ d ∪ γ λ ). It was argued in [30] that under these three conditions the distribution of signs of Im g indeed will be as shown in Figure 4. Indeed, the local structure of the lines Im g = const can be analyzed by the same reasoning as in subsection 2.2. The only thing we need to do is to distinguish the lines where the const = 0.
Since Re g(k = E) = 0, in order to establish that Re g k = E = 0 it suffices to notice that where the order of integration is from d to λ − and from λ − to d.
The first property is due to the local analysis of the lines Im g = const. Regarding the second property, let us mention, that the property Indeed, we have γ d d g ± ∈ R, and in view of the symmetry d g(k) = d g(k), we have and then from we get the required property The three properties (2.15) listed above are equivalent to the system (3.16), and we prove its solvability in the next section.
3 Proof of existence of parameters of g-function.
Then the first two equations give us that λ ± are the roots of the equation and since −ξ 2 2rω < 0, this quadratic equation always have two distinct real solutions λ − < 0 < λ + . Now the problem is reduced to that of finding d 1 , |d|. Denote d =: r(cos ϕ + i sin ϕ).
The third condition takes the form: Hence, now the problem is reduced to that of finding r. However, afterwards we will need to check that the module of the r.-h.-s. in (3.17) is less or equal than 1.
In lemma 3.1 we prove that the above expression is nonzero for all x > 1, 1 < α < α 0 . Hence, equation F (r, ξ) uniquely determines r as a function of ξ for all 1 2ω < ξ < ξ 0 = 1 2ω 0 . Now it remains only to check that the module of the r.-h.-s. in (3.17) is less or equal than 1. In terms of parameters x, α, β (3.18) formula (3.17) becomes Hence, we need to check that for all x > 1, α > 1 we have The above inequality is obviously satisfied Let us find the minimum of the function in the r.h.s. of the above formula. The point at which the minimum is attained is among zeros of i.e.
and hence Let us notice that x min and α max coincide with x 0 (3.22), α 0 (3.21) from lemma 3.1. This finishes the proof of the theorem.
Lemma 3.1. Let β ∈ (0, 1). Consider the set where Prove that (3.21) and the minimum is attained at a single point (x 0 , α 0 ) ∈ M, where 1. 0 < z < 1 2 . In this case w > 5 2 , and hence x 2 + < 0, and x 2 − > 0. Let us check that x 2 − < 1. Indeed, (3.27) and hence is equivalent in our case to the obvious inequality Hence, the set described by inequality (3.25), which reads in this case as does not intersect with the set M (3.19).