A nonlinear inverse problem of the Korteweg-de Vries equation

In this paper we prove the existence and uniqueness of solutions of an inverse problem of the simultaneous recovery of the evolution of two coefficients in the Korteweg-de Vries equation.

In this paper, we want to study the nonlinear inverse problem consists of finding a set of the functions {u, α, f } satisfying (1.1)- (1.4). This kind of inverse problem of incompressible Naiver-Stokes equations and Ginzburg-Landau equations in superconductivity has been studied in [3][4][5].
Throughout this paper, we will assume By a similar proof as that in [6], we can prove an existence and uniqueness result of the direct problem (1.1)-(1.3) and we here omit the details.

Theorem 1.1 Let (H1)-(H3) be satisfied. Then there exists a unique solution u satisfying
Based on Theorem 1.1, we can define the nonlinear operator acting on every vector χ = {α(t), f (t)} as follows: Solving the above system and using (H6) and N is a large integer to be determined in the following calculations.
We will use · := · L 2 ( ) . In the next Sect. 2, we give some preliminaries to the proof of Theorem 1.3, which is given in the final Sect. 3. (2.1) Here T e γ T α C ≤ 1 2 and T e γ T f C ≤ 1 2 .
Proof Multiplying (1.1) by u and integrating by parts lead to and thus Dividing both sides by u , we get Integrating this inequality gives which gives (2.1).

Lemma 2.2 Let N ≥ 8.
Then Here T e γ T α C ≤ 1 8 and T e γ T f C ≤ 1 8 .
Integrating this inequality gives Here T e γ T α C ≤ 1 8 and T e γ T f C ≤ 1 8 . We used the embedding inequality Testing (1.1) by We obtain Thus we have (2.7) Here we take N ≥ 8 and γ > 2C 2 and C 2 ≥ u 1x + u 2x L ∞ ( ×(0,T )) . We used u 1x dx = u 2x dx = 0, and the embedding inequality (2.8) Multiplying (2.8) by u 1 − u 2 and integrating by parts, and noting that Integrating the above inequality, we derive which leads to An application of Gronwall's inequality gives (2.7).
Proof of Theorem 1.3 The proof of Theorem 1.3 follows easily from Lemma 3.1 and Lemma 3.2 by employing the contraction mapping principle.