Miura transformation for the “good” Boussinesq equation

It is well known that each solution of the modified Korteveg–de Vries (mKdV) equation gives rise, via the Miura transformation, to a solution of the Korteveg–de Vries (KdV) equation. In this work, we show that a similar Miura‐type transformation exists also for the “good” Boussinesq equation. This transformation maps solutions of a second‐order equation to solutions of the fourth‐order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann–Hilbert problems and this is in fact how we construct the new transformation.

In this paper, we construct a Miura-type transformation for (1.1).Our main result is the discovery that each complex-valued function q satisfying gives rise, via an explicit non-linear transformation, to a solution u of (1.1).More precisely, we have the following.
Theorem 1.1 (Miura transformation for the "good" Boussinesq equation).If q(x, t) is a complex-valued solution of (1.2), then is a real-valued solution of the "good" Boussinesq equation (1.1).
Two concrete examples of applications of the transformation (1.3) are provided in Section 6.
The assertion of Theorem 1.1 can be verified by a long but straightforward computation.However, as we will explain in this paper, the Miura transformation (1.3) can also be derived in a systematic manner.The underlying idea is that, thanks to its integrability, the "good" Boussinesq equation has a Riemann-Hilbert (RH) problem associated to it.This RH problem is singular and, after a transformation, this singular behavior manifests itself as a double pole in the solution at k = 0, where k is the spectral parameter.Associated to the singular RH problem is a regular RH problem, which has the same jumps as the singular problem but no pole at k = 0.The integrable equation associated to this regular problem turns out to be equation (1.2).As we describe in this paper, it is possible to relate the solutions of the regular and singular RH problems explicitly.This leads to an explicit relation between the associated integrable equations, which is exactly the transformation (1.3).
1.1.Background.In 1968, Miura [28] introduced an explicit transformation that maps solutions q of the modified Korteveg-de Vries (mKdV) equation mKdV: q t + q xxx − 6q 2 q x = 0 (1.4) to solutions u of the Korteveg-de Vries (KdV) equation KdV: u t + u xxx − 6uu x = 0. (1.5) More precisely, if q satisfies (1.4), then satisfies (1.5).The mapping (1.6) from q to u is the Miura transformation.The Miura transformation was instrumental in the early days of the development of a theory for nonlinear integrable PDEs and has many applications; for example, it was used by Miura, Gardner, and Kruskal to construct infinitely many conservation laws for the KdV equation [29].In fact, the transformation (1.6) is a reflection of a much deeper correspondence between the mKdV and KdV equations that extends to the associated RH formulations.
The present paper was inspired by work of Fokas and Its [15] where this correspondence is discussed, see [15, Proposition 2.3 and Remark 2.5].Roughly speaking, the correspondence exists because the RH problem associated to KdV is singular and the associated regular RH problem (i.e., the RH problem with the same jumps, but with no singularity) is the RH problem for mKdV, see Section 7 for details.It is possible to relate the solutions of the regular and singular RH problems explicitly and this is one way to arrive at the Miura transformation (1.6).
The relationship between the RH problems for mKdV and KdV is analogous to the relationship between the RH problems for equations (1.1) and (1.2); the only difference is that whereas the solution of the RH problem for Boussinesq is a 3 × 3-matrix-valued function with a double pole at the origin, the solution of the RH problem for KdV is a 2 × 2-matrix-valued function with a simple pole at the origin.Other than that the constructions underlying the Miura transformation (1.6) and the transformation (1.3) are conceptually identical; this is why we refer to (1.3) as a Miura transformation.
Equation (1.2) was listed as an integrable system obtained by reduction in [27] and was, as far as we know, first studied in [20].In [6], a Riemann-Hilbert representation for the solution of the initial value problem for (1.2) was derived and used to obtain formulas for the long-time asymptotics.Other works analyzing aspects of 3 × 3 RH problems for integrable evolution equations include [3-5, 8, 10, 11, 18, 25, 33].
1.2.Outline of the paper.Our main result is stated in Section 2. It describes the correspondence between the RH problems associated to (1.1) and (1.2), and how it leads to the transformation (1.3).Several basic properties of the solution of the RH problems for (1.1) and (1.2) are derived in Sections 3 and 4, respectively.The proof of the main theorem is presented in Section 5 and two illustrating examples are given in Section 6.A detailed account of the correspondence between the RH problems associated to mKdV and KdV, and how it leads to the Miura transformation (1.6), is included in Section 7. Appendix A reviews some facts related to the initial value problems for the Boussinesq equation and equation (1.2) on the real line.In Appendix B, we consider soliton solutions of (1.2).

Main result
Before we can state our main result, we need to introduce RH problems associated to (1.1) and (1.2).Instead of considering a RH problem directly related to (1.1), it is convenient to first transform (1.1) into the equation by means of the transformation It is easy to see that if u satisfies (2.1), then û satisfies (1.1).Following [11,35], we rewrite (2.1) as the system (2.3) A Riemann-Hilbert approach to the system (2.3) was developed in [7] and we will work with the RH problem for (2.3) constructed in [7].This RH problem has a jump contour Γ consisting of six rays oriented away from the origin as in Figure 2.1.Let ω := e 2πi 3 and define {l j (k), Given two functions r 1 : (0, ∞) → C and r 2 : (−∞, 0) → C, we define the jump matrix v(x, t, k) for k ∈ Γ by where v j denotes the restriction of v to the subcontour of Γ labeled by j in (a) M (x, t, The limits of M (x, t, k) as k approaches Γ \ {0} from the left (+) and right (−) exist, are continuous on Γ \ {0}, and satisfy the jump condition where v is defined in terms of r 1 and r 2 by (2.5).
where the matrices M (1) and M (2) depend on x and t but not on k, and satisfy (2.9) Moreover, there exist complex coefficients α, β, γ, δ, ϵ depending on x and t such that and the third column of (2.12) (e) M satisfies the symmetries where Remark 2.3.We show in Section 3 that the solution M of RH problem 2.2 is unique if it exists.Moreover, we review in Appendix A how the initial value problem for (2.3) can be solved in terms of the solution M of RH problem 2.2; in the context of the initial value problem, the functions r 1 (k) and r 2 (k) are defined in terms of the given initial data q 0 (x) = q(x, 0) and can be thought of as nonlinear Fourier transforms of q 0 .For the present purposes, there is no need to assume that r 1 (k) and r 2 (k) originate from any initial data.
Note that M (x, t, k) has a double pole at k = 0 as a consequence of (2.9) in the generic case when α is non-zero.The RH problem associated to (1.2) has the same form as RH problem 2.2 except that its solution m(x, t, k) does not have a singularity at the origin.
RH problem 2.4 (RH problem for m).Let r 1 : (0, ∞) → C and r 2 : (−∞, 0) → C be some given functions.Find a 3 × 3-matrix valued function m(x, t, k) with the following properties: The limits of m(x, t, k) as k approaches Γ \ {0} from the left (+) and right (−) exist, are continuous on Γ \ {0}, and satisfy the jump condition where v is defined in terms of r 1 and r 2 by (2.5).

.16)
(d) There exist matrices {m Our main result is formulated under the the following two assumptions on the spectral functions r 1 (k) and r 2 (k).Assumption 2.6.Assume r 1 and r 2 satisfy the following properties: (ii) The functions r 1 (k), r 2 (k), and their derivatives ∂ j k r l (k) have continuous boundary values at k = 0 for l = 1, 2 and for all j = 0, 1, 2, . .., and there exist expansions which can be differentiated termwise any number of times.
(iii) r 1 (k) and r 2 (k) are rapidly decreasing as k → ±∞, i.e., for each integer N ≥ 0, max j=0,1,...,N sup k∈(0,∞) Assumption 2.7.Assume r 1 and r 2 satisfy Remark 2.8.The RH problems 2.2 and 2.4 can be formulated for any choice of the functions {r j (k)} 2 1 and they give rise to solutions of (1.1) and (1.2), respectively, for a very large class of such functions.Here, we will restrict ourselves to functions {r j (k)} 2 1 satisfying Assumptions 2.6 and 2.7.These assumptions are natural from the point of view of the initial value problem for the "good" Boussinesq equation (2.3), in the sense that they are fulfilled for generic solitonless Schwartz class initial data, see Appendix A.
The following is our main result.Theorem 2.9 (Miura correspondence for "good" Boussinesq).Let r 1 (k) and r 2 (k) be such that Assumptions 2.6 and 2.7 hold.Define the 3 × 3-matrix valued jump matrix v(x, t, k) in terms of r 1 and r 2 by (2.5).Let N be an open subset of R × [0, +∞) and suppose RH problem 2.4 has a (necessarily unique) solution m(x, t, •) for each (x, t) ∈ N .(a) The function q defined by q(x, t) = lim k→∞ k m(x, t, k) 13 (2.20) where {y j (x, t)} 3 1 are real-valued and given by Lemma 3.1 (Asymptotics of M as k → ∞).For each (x, t) ∈ N , M (x, t, k) admits an expansion of the following form as k → ∞: where M (1) 33 (x, t), and M (2) 13 (x, t) are real-valued functions of (x, t) ∈ N .
Proof.Combining the large k asymptotics (2.7) with the symmetries (2.13), we obtain Recalling the definitions (2.14) of A and B, and using the conditions in (2.8), we find that and that M (1) 33 are real-valued.The last condition in (2.8) implies that ω 2 M (2) 13 is real-valued and the desired conclusion follows.□ Lemma 3.2 (Reality of α, β, γ, δ, ϵ).If r 1 and r 2 satisfy Assumptions 2.6 and 2.7, then the functions α, β, γ, δ, ϵ in (2.10)-(2.12)are real-valued and Proof.We fix (x, t) ∈ N and omit the (x, t)-dependence for convenience.The main idea is to combine the jump relation (2.6) with the symmetries where M n , n = 1, . . ., 6, denotes the restriction of M to D n .The jumps for M are not analytic in k, but they can be Taylor expanded as k → 0. The existence of the all-order expansion (2.9) of M 1 implies together with (2.6) the existence of similar expansions for M 2 , . . ., M 6 .We define the following formal series: The matrix M j coincides to any order with M L j as k → 0, k ∈ Dj , and the matrix v j coincides to any order with v L j as k → 0, k ∈ Γ j .Using the symmetries (3.4), we obtain Using (3.6) and considering the terms of O(k −2 ) and O(k −1 ), we obtain On the other hand, Taylor expanding the jump condition M + = M − v along each of the six rays of Γ, we find the relations These identities give rise to relations between the values of r 1 , r 2 , and their derivatives at k = 0.In particular, since r 1 and r 2 satisfy Assumption 2.7, (3.10) is satisfied with j = 1 and j = 2 if and only if ) Substituting (3.11) into the first column of (3.9), we deduce (3.3).On the other hand, the first three orders of (3.7) yield From the first column of (3.14), we deduce that α ∈ R. The second column of (3.15) implies that δ ∈ R, and then the first column of (3.15), together with (3.3), shows that β ∈ R and γ ∈ R. From the third column of (3.16), we infer that ϵ ∈ R. □ Lemma 3.3.M has unit determinant.
Proof.Fix (x, t) ∈ N .The determinant det M (x, t, •) is analytic in C\{0} and approaches 1 as k → ∞.The behavior (2.9) of M 1 (x, t, k) as k → 0 implies by a direct computation that det M has at most a simple pole at k = 0. Thus, for some function f (x, t).On the other hand, by Proof.Suppose M and N are two solutions.By Lemma 3.3, det M and det N are identically equal to one.In particular, the inverse transpose N A := (N −1 ) T of the matrix N can be expressed in terms of its minors.Expanding this expression for N A as k → 0 in D 1 and using (2.9), we find, as k ∈ D 1 approaches 0, Similarly, expanding the expression for N A as k → ∞ and using (3.1), we find where N (2) 13 .By (2.9) and (3.17), we have, as k ∈ D1 approaches 0, showing that M N −1 has at most a double pole at k = 0. Since M N −1 is analytic for k ∈ C \ {0} and approaches the identity matrix as k → ∞, we conclude that for some matrices P (x, t) and Q(x, t).In fact, keeping track of the terms of order k −2 in (3.19), we see that Furthermore, the symmetries (2.13) hold for M and N and hence also for M N −1 .Thus, P and Q have the form ω 2 P 33 ωP 13 P 13 ω 2 P 13 ωP 33 P 13 ω 2 P 13 ωP 13 P 33 where P 33 , Q 33 are real-valued and P 13 , Q 13 are complex-valued functions.Together with (3.21), this implies that On the other hand, substituting the expansion (3.1) of M and the transpose of the expansion (3.18) of N A into the left-hand side of (3.20) and identifying coefficients of k −1 and k −2 in the resulting equation, we conclude that 33 N (1) In order to reconcile the expressions (3.24) and (3.22) for Q, we must have Q = 0, and then (3.20) becomes Since the determinant of the left-hand side is identically equal to one, we conclude that M We establish several properties of the solution m of RH problem 2.4.Throughout this section, we assume that N is a (not necessarily open) subset of R × [0, +∞) and that m(x, t, •) satisfies RH problem 2.4 for each (x, t) ∈ N .Proof.The determinant det m is analytic in C \ {0} with a removable singularity at 0 and approaches 1 as k → ∞.Hence det m = 1.Thus, if m and n are two solutions, then n is invertible and The solution m obeys the symmetries Proof.Since the jump matrix v obeys the symmetries the functions Am(x, t, ωk)A such that m(x, t, k) coincides with m L j (x, t, k) to any order as k ∈ Dj tends to 0. More precisely, for any N ≥ 0, (x, t) ∈ N , and j = 1, . . ., 6, The formal power series where {v L j } 6 j=1 are defined in (3.5).In particular, the first coefficients satisfy the relations (1) If r 1 and r 2 also satisfy Assumption 2.7, then 1 has the form for some real-valued functions m (4.8) Proof.For notational convenience, we omit the (x, t)-dependence of several quantities.The existence of the formal power series m L 1 follows directly from condition (d) of RH problem 2.4.The existence of the other formal series m L j , j = 2, . . ., 6 then follows from the symmetries (4.1) of m.Since the jumps v j can be Taylor expanded as k → 0, a Taylor expansion of (4.1) together with (4.2) implies that , which proves (4.4) and (4.5), and in particular the relations (4.6).

5.
Proof of Theorem 2.9 Suppose that r 1 (k) and r 2 (k) satisfy Assumptions 2.6 and 2.7 and define v(x, t, k) by (2.5).Assume that m(x, t, k) solves RH problem 2.4 for all (x, t) ∈ N , where N is an open subset of R × [0, +∞).
By Lemma 4.1, the solution m is unique.By Lemma 4.4, the function q(x, t) is welldefined by (2.20).Viewing (4.11) as a singular integral equation for m − and using that r 1 (k) and r 2 (k) have rapid decay as k → ∞, standard arguments imply that m(x, t, k) and q(x, t) are smooth functions of (x, t) ∈ N , see e.g.[21,Section 5] for details in a similar situation.Assertion (a) of Theorem 2.9 then follows from the following lemma.
Proof.Since det m = 1, the inverse m −1 is well defined.Recalling the definition (2.5) of the jump matrix v, we can write with a matrix v 0 (k) independent of x and t.This yields from which we deduce or, equivalently, Therefore, the functions , respectively, and as k → 0 they are O(1).Hence there exist 3 × 3-matrix valued functions {F j (x, t)} 1 j=0 and {G j (x, t)} 2 j=0 such that The Lax pair equations (5.1) follow from (5.2) if we can show that ) where U and V are given by (A.2) and (A.3), respectively.
To prove (5.3), we note that the terms of O(k 2 ) as k → ∞ of (5.2) show that G 2 = J 2 , and the terms of O(k) and O(1) then yield and ) , J 2 ]m (1) . (5.4b) An explicit computation of these equations gives F 0 = U and where u = m (2) 13 − m 33 q + q2 .After substituting these expressions for {F j } 1 0 and {G j } 2 0 into the compatibility condition m xt = m tx and simplifying, we find the relation The terms of order k in (5.7) yield (5.8) Using the explicit formulas for F 0 , F 1 , and G 1 , we find that (5.8) is equivalent to (5.9) The relations (5.3) follow by substituting (5.9) into (5.6).
The terms of order 1 in (5.7) yield or, equivalently, where we have used (5.9) in the last step.Since ω = e 2πi 3 , it follows that q satisfies (1.2). □ Assertion (b) follows from Lemma 4.4 and (2.20).Let y 1 , y 2 , and y 3 be the real-valued functions defined in terms of q and m by (2.23).Let where A(x, t) and B(x, t) are given in terms of y 1 , y 2 , and y 3 by (2.22), and define M by Assertion (c) is a consequence of the next lemma.
Proof.Since M = T m, the function M is analytic for k ∈ C \ Γ and satisfies the same jump conditions as m on Γ.Moreover, since y 1 , y 2 , and y 3 are real-valued functions, it is easy to check that the factor T has unit determinant and obeys the symmetries for (x, t) ∈ N .Since m obeys the same symmetries by Lemma 4.2, M does as well.
Let us prove that M satisfies (2.9) as k → 0 with coefficients fulfilling (2.10)-(2.12)for some choice of the functions α, β, γ, δ, ϵ.We deduce from Lemma 4.3 and the definition (5.12) of M that M has an expansion of the form (2.9) as k ∈ D1 approaches 0. The first few coefficients are given by where {m 32 , m 1 .This shows that M has the desired behavior as k → 0.
We finally show that the asymptotics (2.7) hold as k → ∞ with matrix-coefficients M (1) and M (2) satisfying (2.8).As k → ∞, m obeys the expansion (4.9).Hence M satisfies where = B + Am (1) + m (2) . (5.14) Using that {y j } 3 1 are given by (2.23), we conclude that the three conditions in (2.8) are satisfied.This completes the proof.□ Using (5.14), we can write the definition (2.25) of u(x, t) as By considering the terms of order k −1 of the ( 33)-entry of the first equation in (5.1), we infer that m (1) Substituting this into (5.15) and using the definition (2.22) of A(x, t), we obtain which is the Miura transformation (2.27).Since q is smooth, it follows from (2.27) that u is smooth and real-valued on N .Since q solves (1.2), it also follows from (2.27) that u satisfies the "good" Boussinesq equation (2.1) and that {u, v}, with v defined by (2.26), satisfy the system (2.3) on N .For completeness, we give in the next lemma an alternative proof of the latter two facts; this lemma also shows that M satisfies the Lax pair equations associated to (2.3).
Lemma 5.3.For (x, t) ∈ N , define u(x, t) by (2.25) and define v(x, t) by (2.26).Then M satisfies the Lax pair equations where U and V are given in terms of u and v by (A.7).In particular, for (x, t) ∈ N , u and v satisfy (2.3) and u satisfies (2.1).
Proof.Since T and m have unit determinant, so does M ; hence the inverse M −1 is well defined.Since M = T m, we have where {F j } 1 j=0 and {G j } 2 j=0 are the functions in (5.2).It follows from the definition of T that the functions in (5.18a) and (5.18b) are analytic for k ∈ C \ {0}.Moreover, as k → ∞, and we also verify from a direct computation that Hence there exist 3 × 3-matrix valued functions {f j (x, t)} 0 j=−2 and {g j (x, t)} 1 j=−2 such that (5.19) The Lax pair equations (5.17) are a consequence of (5.18) and (5.19) if we can show that where U and V are given by (A.7).To prove (5.20), we substitute (5.3) and the definition (5.11) of T into (5.19) and identify coefficients of powers of k.Recalling the definitions of u and v, the identities in (5.20) follow from long but straightforward computations in which we use the relation (5.9).The facts that u and v satisfy (2.3) and that u satisfies (2.1) follow from the compatibility of the Lax pair equations (5.17).□ The proof of Theorem 2.9 is complete.

Examples
We give two explicit examples to illustrate the transformation (1.3) of Theorem 1.1.
) is a solution of the "good" Boussinesq equation (1.1) everywhere in the xt-plane except on the lines 1  2 c 1 (x − c 1 t) ∈ πZ.The solution (6.1) is a (singular) one-soliton of (1.2) and we explain in Appendix B how it can be derived., where y := exp(3 −1/4 x).This solution is smooth whenever the denominator is nonzero.The solution (6.2) is a (singular) breather soliton of (1.2); we show in Appendix B how to construct such solitons and how we arrived at the solution in (6.2).

KdV and mKdV
The Miura transformation (1.6) maps solutions of the mKdV equation to solutions of the KdV equation.Underlying this transformation is a correspondence between the associated Riemann-Hilbert problems.In this section, we describe this correspondence; we refer to [15, Proposition 2.3 and Remark 2.5] for an earlier description.
Our goal is to give a detailed treatment of the mKdV/KdV correspondence with a particular emphasis on the analogy between that correspondence and the correspondence between the "good" Boussinesq equation and equation (1.2) presented in Sections 2-5.To make this analogy more transparent, in this section, we write M and m for the solutions of the RH problems associated with the KdV and mKdV equations, respectively (in the rest of this paper, M and m denote the solutions to the RH problems associated with the Boussinesq equation and equation (1.2), respectively).Similarly, the quantities U, V, A, B, s, etc., are defined differently in this section compared to earlier sections; but they play analogous roles here, so that the comparison between this section and the rest of the paper becomes evident.7.1.Reflection coefficients.We first define the reflection coefficients r mKdV and r KdV associated with mKdV and KdV.Let {σ j } 3 j=1 be the three Pauli matrices.The mKdV and KdV equations (1.4) and (1.5) both admit Lax pairs of the same form: where L = −ikσ 3 , Z = −4ik 3 σ 3 , and U(x, t, k) and V(x, t, k) are given by Define the 2 × 2-matrix valued function X(x, k) as the unique solution of the Volterra integral equation and define the scattering matrix s(k) by where e L UX := e L UXe −L .The mKdV reflection coefficient r mKdV (k) associated to the initial data q 0 (x) is defined by where X and s are constructed with U = −q 0 σ 2 ; the KdV reflection coefficient r KdV (k) associated to the initial data u 0 (x) is defined by the same formula (7.4) except that X and s now are constructed with U = u 0 2k (iσ 3 − σ 1 ).

Solution of the initial value problem.
In what follows, we state the RH problems associated to mKdV and KdV and recall without proofs how they can be used to solve the initial value problem on the line.For simplicity, we assume that no solitons are present; this means more precisely that we assume that (s(k)) 22 is nonzero for Im k ≥ 0 for both mKdV and KdV.Given solitonless initial data q 0 in the Schwartz class S(R), the solution q(x, t) of the initial value problem for the mKdV equation is given by where m is the unique solution of the following RH problem with r(k) taken to be the reflection coefficient r mKdV (k) associated to q 0 (x) = q(x, 0).

RH problem 7.1 (RH problem for mKdV).
Let r : R → C be a given function.Find a 2 × 2-matrix valued function m(x, t, k) with the following properties: The boundary values of m(x, t, k) as k approaches R from the left and right exist, are continuous, and satisfy where v is defined in terms of r(k) by Similarly, for solitonless initial data u 0 ∈ S(R), let r(k) be the KdV reflection coefficient r KdV (k) associated to u 0 (x) = u(x, 0).In the generic case when r(0) = i, the solution u(x, t) of the initial value problem for the KdV equation is given by where M is the unique solution of the following RH problem.RH problem 7.2 (RH problem for KdV).Let r : R → C be a given function.Find a 2 × 2-matrix valued function M (x, t, k) with the following properties: The limits of M (x, t, k) as k approaches R \ {0} from the left and right exist, are continuous on R \ {0}, and satisfy where v is defined in terms of r(k) by (7.7).
, where the matrix M (1) satisfies M (1) 12 = 0. (d) There exist complex-valued functions α(x, t), β(x, t), γ(x, t), δ(x, t) such that ) where A and B are defined by 7.3.Properties of the reflection coefficients.Suppose that q 0 and u 0 are solitonless initial data in the Schwartz class S(R).Then r mKdV : R → C and r KdV : R → C are both in the Schwartz class and obey the symmetry For KdV, the simple pole of U at k = 0 implies that s(k) has an expansion at the origin of the form where, for some constant s (−1) ∈ R, For generic initial data, the coefficient s (−1) is non-zero; in this case, it is easy to see from (7.13)-(7.14) that r(0 It follows that whereas the reflection coefficient for mKdV is everywhere smaller than 1 (i.e., |r mKdV (k)| < 1 for all k ∈ R), the reflection coefficient for KdV generically satisfies This has the important implication that the Miura transformation u = q x + q 2 maps Schwartz class solutions of mKdV onto a small (measure zero) subset of all Schwartz class solutions of KdV.More precisely, only solutions u for which the associated coefficient s (−1) vanishes arise as images of mKdV Schwartz class solutions on the line.On the other hand, the RH problems 7.1 and 7.2 can be formulated for any choice of r(k) and generate solutions of mKdV and KdV, respectively, for a very large class of functions r(k).Solutions corresponding to functions r(k) which do not satisfy the above constraints, will not correspond to Schwartz class solutions, but to solutions with singularities and/or a lack of decay at spatial infinity.The Miura transformation is local in the sense that wherever a solution q of mKdV is smooth, the Miura transformation q → u = q x + q 2 is well-defined and generates a KdV solution u.
7.4.The Miura correspondence.Underlying the Miura transformation is a correspondence between RH problem 7.1 and 7.2.This correspondence exists for a large class of spectral functions r(k).We will restrict ourselves to the class of functions r ∈ S(R) such that r(0) = i and r(k) = −r(−k) for k ∈ R. In view of (7.15), this class contains the reflection coefficients relevant for generic Schwartz class solutions of KdV on the line.Our results are summarized in two theorems.The first theorem (Theorem 7.3) describes how to go from mKdV to KdV, i.e., how to construct a solution M of RH problem 7.2 given a solution m of RH problem 7.1, and how the corresponding solutions of KdV to mKdV are related by the Miura transformation.The second theorem (Theorem 7.4) describes how to go in the opposite direction, from KdV to mKdV.The analogy between Theorem 7.3 and Theorem 2.9 should be evident.
The 2 × 2-matrix valued function M (x, t, k) defined by satisfies RH problem 7.2 for each (x, t) ∈ N .(c) The function u defined by is smooth and real-valued on N and satisfies the KdV equation (1.5) for (x, t) ∈ N .(d) The solutions u and q are related by the Miura transformation u(x, t) = q x (x, t) + q(x, t) 2 , (x, t) ∈ N .(7.20) (b) The function u defined by (7.19) is smooth and real-valued on N and satisfies the KdV equation (1.5) for (x, t) ∈ N .(c) The function q defined by (7.16) is smooth and real-valued on N and satisfies the mKdV equation (1.4) for (x, t) ∈ N .(d) The solutions u and q are related by the Miura transformation u(x, t) = q x (x, t) + q(x, t) 2 , (x, t) ∈ N .(7.23) (e) For (x, t) ∈ N , it holds that δ xx (x, t) = u(x, t)δ(x, t), q(x, t) = δ x (x, t) δ(x, t) , α(x, t) = δ x (x, t).(7.24) Remark 7.5.Given u, the Miura transformation (7.23) can be viewed as a Ricatti equation for q.The relations in (7.24) can be interpreted as the standard procedure for obtaining the solution of this Ricatti equation in terms of the solution of a second-order linear ordinary differential equation.Indeed, (7.24) expresses the solution of u = q x + q 2 as q = δ x /δ where δ is the solution of the second-order equation δ xx = uδ.
Remark 7.6.Let us comment on the assumption in Theorem 7.4 that the function δ(x, t) be nonzero on N .It is clear from (7.24) that the inverse Miura transformation u → q may introduce singularities at the points where δ vanishes.At these points RH problem 7.1 does not have a solution.For example, as discussed above, for r(k) associated to generic Schwartz class solutions of KdV, there is a solution M (x, t, k) of the RH problem for KdV for any (x, t) ∈ R × [0, +∞).But the corresponding solution of mKdV will not be in the Schwartz class (because if it were, we would have |r(0)| < 1) and the best we can say is that it will be a smooth solution of mKdV on the set where δ(x, t) is nonzero.B, k ∈ R, so that Am(x, t, −k)A −1 and Bm(x, t, k)B also obey RH problem 7.1; by uniqueness, By Cauchy's formula, where the coefficient matrices {m (j) } 2 1 have the form 22 are real-valued and m (1) 12 are purely imaginary.In particular, q = 2m (1) 12 is well-defined by (7.16) and real-valued.Smoothness of m (and hence also of q) as k where iM (1) 22 is a real-valued function of (x, t) ∈ N .It follows from (7.32) and (7.9) that det M (x, t, k Since det M is analytic for k ∈ C \ {0}, we infer that M has unit determinant.Suppose that M and N are two solutions of RH problem 7.2.By the above argument, det M and det N are identically equal to one.In particular, the inverse N −1 exists and is given by Expanding this expression for N −1 as k → 0, Im k ≥ 0, and using (7.9), we find as k → 0, Im k ≥ 0. Similarly, expanding the expression for N −1 as k → ∞ and using (7.32), we find 22 (x, t) k By (7.9) and (7.33), we have, as k approaches 0, Im k ≥ 0, showing that M N −1 has at most a simple pole at k = 0. On the other hand, expanding Since M N −1 is entire, Liouville's theorem together with (7.34) and (7.35) show that We next show that the functions α, β, and δ in (7.9) are real-valued.Using the symmetry (7.10) and suppressing the (x, t)-dependence for conciseness, we obtain Inserting expansion (7.9), the terms of O(k −1 ) and O(1) of (7.36) yield the relations these relation imply that α, β, δ are real-valued.Assume now that the function δ(x, t) in (7.9) is nonzero on N .Then m(x, t, k) is welldefined by (7.22) for (x, t) ∈ N .Clearly, m(x, t, k) is analytic for k ∈ C \ R, obeys the jump relation (7.6) on R \ {0}, and satisfies m(x, t, k) = I + O(k −1 ) as k → ∞.As a consequence of the reality of α and δ, we have B(x, t) = −AB(x, t)A −1 = BB(x, t)B.It follows that m obeys the symmetries in (7.25).Moreover, as k → 0, Im k ≥ 0, we have 11 has a double pole.We say that the initial data {u 0 (x), v 0 (x)} is generic if the behavior of these functions is generic at k = 0 in the sense that lim It is shown in [7,Theorem 2.3] that, for any generic solitonless initial data in the Schwartz class, the spectral functions {r j (k)} 2  1 satisfy the properties in Assumptions 2.6 and 2.7.Moreover, it is shown in [7,Theorem 2.6] that if {u(x, t), v(x, t)} is a Schwartz class solution of (2.3) on R×[0, T ) with generic solitonless initial data, then RH problem 2.2 has a unique solution M (x, t, k) for any (x, t) ∈ R×[0, T ) and the solution {u(x, t), v(x, t)} can be obtained for any (x, t) ∈ R × [0, T ) from the formulas In this appendix, we construct soliton solutions of (1.2); in particular, we derive the exact solutions in (6.1) and (6.2).
Soliton solutions of (1.2) can be generated by adding poles to RH problem 2.4 and letting the jump matrix v be the identity matrix.To see how to add the poles, let X(x, t, k) be the solution of the Volterra integral equation For compactly supported initial data, the solution m(x, t, k) of RH problem 2.4 is related to where T 1 (k) is given in terms of the entries of the scattering matrix s(k) in (A.5) by with m ij (s) the (ij)th minor of the matrix s, see [6] (see also [7,Lemma 4.4] for a similar situation with more details provided).The relation (B.2) implies that the three columns [m] j , j = 1, 2, 3, of m(x, t, k) obey the following relations in D 1 : where e 1 := − s 12 (k 0 ) ṡ11 (k 0 ) ∈ C is a constant.Using the symmetry m(x, t, k) = Am(x, t, ωk)A −1 , we conclude that [m] 1 and [m] 3 have simple poles at ωk 0 and ω 2 k 0 , respectively, and that the following residue conditions hold: Together with the normalization condition m(x, t, k We evaluate the first equation at k = k 0 , the second equation at k = ω 2 k 0 , and the third equation at k = ωk 0 .The top entries of the equations in (B.5) then give This gives three algebraic equations which we can solve for the three unknowns m 11 (x, t, k 0 ), m 12 (x, t, ω 2 k 0 ), and m 13 (x, t, ωk 0 ).Similarly, the middle entries of the equations in (B.5) can be solved for m 21 (x, t, k 0 ), m 22 (x, t, ω 2 k 0 ), and m 23 (x, t, ωk 0 ).This gives explicit expressions for q(x, t) := lim k→∞ k m 13 (x, t, k) = ω 2 e 1 e θ 12 (x,t,k 0 ) m 12 (x, t, ω 2 k 0 ) and q(x, t) := lim k→∞ k m 23 (x, t, k) = ω 2 e 1 e θ 12 (x,t,k 0 ) m 22 (x, t, ω 2 k 0 ), which can be verified to satisfy (1.2) for any choice of e 1 ∈ C and k 0 ∈ R. (The above definitions of q and q follow from [6, Eq. (3.11a)].)However, in the above computation, we have not taken the B-symmetry of (4.1) into account; thus q is in general not the complex conjugate of q.A direct calculation shows that q is the complex conjugate of q if and only if |e 1 | 2 = 3k 2 0 .Writing e 1 = √ 3k 0 e i arg e 1 , this leads to the following class of (singular) one-soliton solutions of (1.

4 .
. This shows that M = N and completes the proof of the lemma.□ Properties of the solution m of RH problem 2.4

Lemma 4 . 1 .
The solution m of RH problem 2.4 is unique.

2
l=0 are the coefficient in (4.3).We substitute the expression (4.7) for m (0) 1 into (4.6b) and solve the six entries in the first two rows of this equation for m (1) 31 , m

6. 1 . 3 √ 3 2
Example 1. Equation (1.2) admits the family of traveling wave solutions q(x, t) wave speed c ∈ R. The solution in (6.1) is smooth in the xt-plane except on the lines c(x − ct) ∈ πZ.Applying Theorem 1.1, we find that, for any

Theorem 7 . 3 (
From mKdV to KdV).Let r ∈ S(R) be such that r(k) = −r(−k) for k ∈ R and r(0) = i.Define the 2 × 2-matrix valued jump matrix v(x, t, k) in terms of r(k) by (7.7).Let N be an open subset of R × [0, +∞) and suppose RH problem 7.1 has a (necessarily unique) solution m(x, t, •) for each (x, t) ∈ N .(a) The function q defined by q(x, t) = 2 lim k→∞ k m(x, t, k) 12 (7.16) is smooth and real-valued on N and satisfies the mKdV equation (1.4) for (x, t) ∈ N .(b) Define A(x, t) by

7. 5 .
Proof of Theorem 7.3.Suppose r ∈ S(R) satisfies r(k) = −r(−k) for k ∈ R and r(0) = i, and assume that m(x, t, •) is a solution of RH problem 7.1 for all (x, t) in some open subset N of R×[0, +∞).Since det v = 1, standard arguments show that the solution m is unique.Moreover, since r(k) = −r(−k) for k ∈ R, we have v(x, t, k) = Av(x, t, −k) −1 A −1 = Bv(x, t, k) −1 i.e., if M N −1 = I.This proves that the solution M is unique.

1 .
One-solitons.One-soliton solutions arise when s 11 (k) has a simple zero at some point k 0 > 0.Suppose s 11 has a simple zero at k 0 > 0 and that s 12 (k 0 ) ̸ = 0.According to (B.3b), it follows that [m(x, t, k)] 2 has a simple pole at k 0 satisfying the residue condition Res k=k 0
Define the 2 × 2-matrix valued jump matrix v(x, t, k) in terms of r(k) by (7.7).Let N be an open subset of R × [0, +∞) and suppose RH problem 7.2 has a (necessarily unique) solution M (x, t, •) for each (x, t) ∈ N and that the function δ(x, t) in (7.9) is nonzero on N .
12 has a double pole if and only if s 11 has a double pole, and s A 12 has a double pole if and only if s A t) −i(γ(x, t) + δ(x, t)) iβ(x, t) γ(x, t) − δ(x, t) + O(k) s