Classical Solutions for the Generalized Korteweg-de Vries Equation

: The Korteweg-de Vries equation models the formation of solitary waves in the context of shallow water in a channel. In Equation (1), f or p = 2 and p = 3 (Korteweg-de Vries equations (KdV)) and (modiﬁed Korteweg-de Vries (mKdV) respectively), these equations have many applications in Physics. (gKdV) is a Hamiltonian system. In this article we investigate the generalized Korteweg-de Vries (gKdV) Equation (3). A new topological approach is applied to prove the existence of at least one classical solution. The arguments are based upon recent theoretical results.


Introduction
If p is integer, the Korteweg-de Vries Equation [1] is as follows ∂ t u + (u xx + |u| p ) x = 0. (1) It is particularly very important as a prototypical example of an exactly solvable nonlinear system (that is, completely integrable infinite dimensional system). The generalized Korteweg-de Vries equation (gKdV) is a Hamiltonian system. In particular, three quantities are conserved, at least formally udx = u 0 dx, The natural energy space for the study of this equation is therefore H 1 . Note however that the first conservation law is little used, because it is not a signed quantity, and moreover it is not in the energy space. Moreover, the equation admits a scale invariance: if u is a solution of (gKdV), we have u λ (x, t) = λ 2/(p−1) u(λx, λ 3 t).
The stability of these solutions was investigated in [2], whereas asymptotic stability has been studied in [3,4].
In this paper, we investigate the Cauchy problem for the generalized Korteweg-de Vries equation under the next hypothesis Let us now suppose In the last section we will give examples for g, A, B and B 1 that satisfy (Hyp2) and (Hyp3).
The present paper is marshaled as follows. In the second Section 2, we state some useful auxiliary results and needed tools. In Section 3, we present and prove some needed results. In Section 4 we prove the main Theorem 1 and the second main result Theorem 2 will be shown in Section 5. The last Section 6 will be an example illustrating our main results.

Preliminary Results
The first continuation theorems applicable to nonlinear problems were due to Leray and Schauder (1934) [5] (Theorem 10.3.10). This result is the most famous and most general result of the continuation theorems (see [5] pages 28,29). In [6] (1955), Scheafer formulated a special case of Leray-Schauder continuation theorem in the form of an alternative, and proves it as a consequence of Schauder fixed point theorem. In this paper, we will use some nonlinear alternatives, in one hand, to develop a new fixed point theorem and in another hand to study the existence of solutions for Problem Equation (3). In what follows we recall these alternatives. Proposition 1. (Leray-Schauder nonlinear alternative [7]) Let C ⊂ E be a convex, closed subset in a Banach space E, 0 ∈ V ⊂ C where V is an open set. Let f : V → C be a compact, continuous map. Then (a) either f has a fixed point in V, (b) or there exist x ∈ ∂V, and λ ∈ (0, 1) such that x = λ f (x).
As a consequence, we obtain Proposition 2. (Schaefer's Theorem or Leray-Schauder alternative, [8], p124 or [5], p 29) Let E be a Banach space and f : E → E be completely continuous map. Then, Another version of Scheafer's Theorem is given by: Proposition 3. (Scheafer's Theorem [6]) Let E be a Banach space and f : E → E be completely continuous map. Then The following theorem will be used to prove Theorems 1 and 2.
Theorem 3. Let E be a Banach space, Z a closed, convex subset of E, with R > 0. Consider two operators W and G, where for ε ∈ R, and G : V → E be such that Then there exists x * ∈ V such that Proof. We have that the operator 1 ε (I − G) : V → Z is continuous and compact. Suppose that ∃x 0 ∈ ∂V and µ 0 ∈ (0, 1) such that that is This contradicts the condition (ii). From Leray-Schauder nonlinear alternative, it follows that there exists x * ∈ V so that or

Auxiliary Results
Let exists. ∀u ∈ X, we define then u is solution of Equation (3).
Proof. We have t ∈ [0, ∞), x ∈ R, which we differentiate with respect to t and we have We put t = 0 in Equation (12) and we obtain Then, the functionu is solution to the initial value problem Equation (3), which completes our proof. (Hyp1) holds. If u ∈ X and B ≥ u , then we have

Lemma 2. Suppose
Proof. We have This completes the proof.
For u ∈ X, define the operator for some constant c, then u is solution of Equation (3).
Proof. We differentiate two times in t and four times in x the Equation (18) to get Then, As G 1 (u)(·, ·) is a continuous on [0, ∞) × R, we have Therefore By using Lemma 1, the desired result is obtained.

Proof of Theorem 1
Below, assume that the hypotheses (Hyp1) and (Hyp2) are satisfied. Let Z denote the set of all equi-continuous families in X with respect to · . Let also, Z = Z be the closure of Z, For u ∈ V and > 0, define the operators For u ∈ V, we have Thus, G : V → X is continuous and (I − G)(V) resides in a compact subset of Z. Now, suppose that there is a u ∈ Z so that u = B and for some λ ∈ 0, 1 . Note that (Z, · ) is a Banach space. Assume that the set is bounded. By Equation (33), it follows that the set A is not empty set. Then, by the Schaefer's Theorem, it follows that there is a u * ∈ Z such that or i.e., u * is solution to Equation (3). Assume that the set A is unbounded. Then, by Schaefer's Theorem, it follows that the equation has at least one small solution u * ∈ Z for any µ ∈ [0, 1]. In particular, for µ = 1, there is a u * ∈ Z such that Equation (35) holds and then it is solution to Equation (3). Let now, {u ∈ Z : u = λ 1 (I − G)(u), u = B} = ∅ ∀λ 1 ∈ 0, 1 . Then, by Theorem 3, the operator W + G has a fixed point u * ∈ Z. Then immediately after which Then, u * is solution to the problem Equation (3). The proof is now completed.

Proof of Theorem 2
Below, assume that the hypotheses (Hyp1), (Hyp2) and (Hyp3) are satisfied. Let Z denote the set of all equi-continuous families in X with respect to · . Let also, Z = Z be the closure of Z, so (Z, · ) is a Banach space. Denote We have that Z is a closed, convex subset in Z. Let Note that Ω is a compact set in Z. For u ∈ Ω and > 0, define the operators Thus, I − G : Ω → Z is continuous and (I − G)(Ω) resides in a compact subset of Z.
Let us suppose that there is a u ∈ Ω so that u = B and for some λ ∈ 0, 1 . Hence, we find From the assumption (Hyp3), we get G 2 (u) ≤ B.
Hence, we have u ≥ 0 and which is contradicts our claim. Then, from Theorem 3, it follows that the operator W + G has a fixed point u * ∈ Ω. Then immediately after which Then, u * is a nonnegative bounded solution to the problem Equation (3). This completes the proof.

Conclusions
This paper concerning the problem of existence of solutions of the generalized Korteweg-de Vries equation. The considered work represents a variant of classical question about the structure of solutions of partial derivative system. It adds more to previous results. The obtained theorems are very interesting, and the model is important and finds applications, such as physical, chemical, biological, thermal and economics. Here, a new topological approach is applied to prove the exis tence of at least one classical solution. The arguments are based upon recent several axiomatic theoretical results. Our results are illustrated by example.