EXISTENCE OF PERIODIC TRAVELING WAVE SOLUTIONS TO THE GENERALIZED FORCED BOUSSINESQ EQUATION

The generalized forced Boussinesq equation, utt−uxx+[f (u)]xx+uxxxx = h0, and its periodic traveling wave solutions are considered. Using the transform z = x−ωt, the equation is converted to a nonlinear ordinary differential equationwith periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green’s function method. This integral equation generates compact operators in a Banach space of realvalued continuous periodic functions with a given period 2T . The Schauder’s fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.


Introduction.
In the 1870's, Boussinesq derived some model equations for the unidirectional propagation of small amplitude long waves on the surface of water [2].These equations possess special traveling wave solutions called solitary waves.Boussinesq's theory was the first to give a satisfactory and scientific explanation of the phenomenon of solitary waves discovered by Scott Russell [8].
The original equation obtained by Boussinesq is not the only mathematical model for small amplitude, planar, and undirectional long waves on the surface of shallow water.Different choices of the independent variables and the possibilty of modifying lower order terms by the use of the leading order relationships can lead to a whole range of equations [1].One of them is the well-known Korteweg-de Vries equation (referred to KdV equation henceforth).In a recent paper, Schneider proved that under certain conditions, solutions of the Boussinesq equation can be split up into two approximate solutions of KdV equation [7].
Twenty years ago, in an impressive survey on the KdV equation, Miura listed seven open problems on that equation [6].The seventh open problem is on the forced KdV equation.In 1995, Shen derived the 1-dimensional stationary forced KdV equation of the form λu t + αuu x + βu xxx = h x for the long nonlinear water waves flowing over long bumps, and proved the existence of positive solitary wave solutions to the equation with boundary conditions u(±∞) = u (±∞) = 0 [9].In a recent paper [3], Chen proved the existence of traveling wave solutions to the generalized forced Kadomtsev-Petviashvili equation which is a 2-dimensional generalization to the equation obtained by Shen.In this paper, we consider an equation of the form where u = u(t, x) and f (u) is a C 2 function in its argument.This equation is also called the generalized forced Boussinesq equation.We shall prove an existence theorem of periodic traveling wave solutions to this equation following the idea of Liu and Pao [5], and Chen and He [4].
The plan of this paper is as follows.In Section 2, the generalized forced Boussinesq equation is transformed to an ordinary differential equation with periodic boundary conditions.We then apply the Green's function method to derive a nonlinear integral equation equivalent to the ordinary differential equation.In Section 3, an existence theorem of solutions to the integral equation is proved.Therefore, the main result, the existence of periodic traveling wave solutions to the generalized forced Boussinesq equation is established.Furthermore, we note that the periodic traveling wave solutions are infinitely differentiable.

Formulation of the problem.
We start from the generalized forced Boussinesq equation of the form where the function f is C 2 in its argument and h 0 is a nonconstant function of t and x.We are interested in the periodic traveling wave solutions of the form u(x, t) = U(z) = U(x −ωt), where ω > 0 is the wave speed and z = x −ωt is the characteristic variable.In this paper, we only consider the case that h 0 (x, t) = h(z) is a 2T -periodic continuous function of z, where T is a preassigned positive number.Substituting the U(z) into equation (2.1) then leads to the fourth order nonlinear ordinary differential equation where C = (1 − ω 2 ).To obtain nonzero, nonconstant, periodic solutions of this equation, we impose the following boundary conditions It is obvious that any solution U(z) of the boundary value problem consisting of equations (2.2), (2.3), and (2.4) can be extended to a 2T -periodic traveling wave solution to the original Boussinesq equation (2.1).
From now on we denote the function f (U(z)) − H(z) on the right hand side of equation (2.5) by F(U(z)) and only consider the two cases: (1) C > 0, (2) C < 0, but −C ≠ (kπ /T ) 2 with k being any integer.Treating the right hand side of equation (2.5) as a forcing term and using the Green's function method [5,11,10], the boundary value problem equations (2.5) and (2.6) can be converted to a nonlinear integral equation of the form ( where the kernels K i , i = 1, 2, are defined as follows: (1) When C > 0, we denote (2) When C < 0 but −C ≠ (kπ /T ) 2 with k being any integer, let (2.9) Lemma 1.The kernels K 1 and K 2 have the following properties: ) (2.12) Proof.Straightforward computations from the definitions of the K 1 (z, s) and K 2 (z, s) given in equations (2.8) and (2.9).

Theorem 2. A function U(z) is a solution of the boundary value problem consisting of equations (2.5) and (2.6) if and only if it is a solution of the integral equation (2.7).
Proof.The if part can be proved by direct differentiations of equation (2.7) and the only if part is based on the Green's function method by treating the right hand side of equation (2.5) as a nonhomogeneous term.

Existence theorem.
It is seen from Theorems 1 and 2 that U(z) is a solution to the integral equation (2.7) if and only if it is a solution to the boundary value problem consisting of equations (2.2), (2.3), and (2.4).Therefore, to show the existence of 2Tperiodic traveling wave solutions to the generalized forced Boussinesq equation it is sufficient to show that solutions to equation (2.7) exist.
We now define the operators Ꮽ i ,i = 1, 2, on C 2T as where the kernels K i ,i = 1, 2, are defined in equations (2.8), (2.9), and Notice that the operator Ꮽ i depends on T and λ i ,i = 1, 2. We shall show that there exists an r > 0 such that Ꮽ i v ≤ r for any nontrivial function v ∈ B(0,r ) ⊆ C 2T , i = 1, 2. This implies that the equation Ꮽ i v = v has at least one solution in B(0,r ).And hence, the existence of solutions to equation (2.7) is therefore established.This, in turn, leads to the existence of 2T -periodic traveling wave solution U(z) to the generalized forced Boussinesq equation.
A consequence of Lemma 1 can be stated now.
We define B(0,r ) as a bounded ball in C 2T with r > 0, then there exists an M > 0 such that M = sup[ F(v) : v ∈ B(0,r )].We are now ready to prove the following theorem:

Proof. First we show that Ꮽ
Let v be an element in C 2T , we have The existence of dᏭ 1 v/dz and dᏭ 2 v/dz implies that both Ꮽ 1 v and Ꮽ 2 v are continuous on [0, 2T ], and hence, we have proved Let S be any bounded subset of C 2T , i.e., there exists an L 0 > 0 such that v ≤ L 0 for all v ∈ S. Then since f is C 2 in its argument, there exists an M 0 > 0 such that Since sin λ 2 T ≠ 0 and max 0≤z≤2T 2T we can obtain the following results from equations (3.1), (3.2), and (3.3) ) where τ 1 = 1 and τ 2 = |sin λ 2 T |.Thus, Ꮽ i S, i = 1, 2, is uniformly bounded and equicontinuous, and by the Ascoli-Arzela theorem both Ꮽ 1 and Ꮽ 2 are compact operators from C 2T into C 2T .
It is worth nothing that as long as 2T 0 K i (z, s)H(s)ds ≠ 0, i = 1, 2, by Theorem 3, we see that the equation Ꮽ i v = v, i = 1, 2, has at least one nonconstant solution v(z) in B(0,r ).This solution v(z) is infinitely differentiable on (0, 2T ) since Ꮽ i v is differentiable on (0, 2T ).The extension of the v(z) to a 2T -periodic function V (z) provides an infinitely differentiable nonconstant 2T -periodic traveling wave solution to the generalized forced Boussinesq equation.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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and H(z) is a 2T -periodicfunction of z such that H"(z) = h(z).