A note on solitary waves solutions of classical wave equations
aa r X i v : . [ m a t h . A P ] A p r A note on solitary waves solutions of classicalwave equations
Claire David †∗ Universit´e Pierre et Marie Curie-Paris 6 † Institut Jean Le Rond d’Alembert, UMR CNRS 7190,Boˆıte courrier n November 4, 2018
Abstract
The goal of this work is to determine whole classes of solitary wave solu-tions general for wave equations.
Consider a differential equation of the form: F ( u, ∂u r ∂x r , ∂u s ∂t s ) = 0 , (1)The determination of travelling wave solutions of specific cases of (1), such as theBurgers or Burgers-Korteweg-De Vries equations, for instance, has been a majortopic in the past few years, and play a crucial role in the study of wave equa-tions. We presently aim at extending previous results, and give the whole classesof solitary wave solutions general for (1).The paper is organized as follows. The general method is exposed in Section 2. Aspecific case is studied in section 3. ∗ Corresponding author: [email protected]; fax number: (+33) 1.44.27.52.59. Solitary waves
Following Feng [4] and our previous work [5], in which travelling wave solutionsof the
CBKDV equation were exhibited as combinations of bell-profile waves andkink-profile waves, we aim at determining travelling wave solutions of (1) (see [8],[9],[10], [11], [12], [13], [14], [15], [16]).Following [4], we assume that equation (1) has travelling wave solutions of the form u ( x, t ) = u ( ξ ) , ξ = x − v t (2)where v is the wave velocity. Substituting (2) into equation (1) leads to: F ( u, u ( r ) , ( − v ) s u ( s ) ) = 0 , (3)Performing an integration of (3) with respect to ξ leads to an equation of the form: F P ξ ( u, u ( r ) , ( − v ) s u ( s ) ) = C, (4)where C is an arbitrary integration constant, which will be the starting point forthe determination of solitary waves solutions.In the previous works, this integration constant is usually taken equal to zero. Yet,it should not be so, since it can lead to a loss of solutions, as we are going to showit in the following. The discussion in the preceding section provides us useful information when weconstruct travelling solitary wave solutions for equation (3). Based on these results,in this section, a class of travelling wave solutions is searched as a combination ofbell-profile waves and kink-profile waves of the form˜ u (˜ x, ˜ t ) = n X i =1 (cid:0) U i tanh i (cid:2) C i (˜ x − v ˜ t ) (cid:3) + V i sech i (cid:2) C i (˜ x − v ˜ t + x ) (cid:3)(cid:1) + V (5)where the U ′ i s , V ′ i s , C ′ i s , ( i = 1 , · · · , n ), V and v are constants to be determined.In the following, c is taken equal to 1. 2 .2 Theoretical analysis Substitution of (5) into equation (4) leads to an equation of the form X i, j, k A i tanh i (cid:0) C i ξ (cid:1) sech j (cid:0) C i ξ (cid:1) sinh k (cid:0) C i ξ (cid:1) = C (6)the A i being real constants.The difficulty for solving equation (6) lies in finding the values of the constants U i , V i , C i , V and v by solving the over-determined algebraic equations. Following [4],after balancing the higher-order derivative term and the leading nonlinear term,we deduce n = 1.Then, following [5] we replace sech( C ξ ) by e C ξ + e − C ξ , sinh( C ξ ) by e C ξ − e − C ξ ,tanh( C ξ ) by e C ξ − e − C ξ e C ξ + e − C ξ , and multiply both sides by (1 + e ξ C ) , so that equation(6) can be rewritten in the following form: X k =0 P k ( U , V , C , v, V ) e k C ξ = 0 , (7)where the P k ( k = 0 , ..., U , V , C , V and v .Depending wether (6) admits or no consistent solutions, spurious solitary wavessolutions may, or not, appear. Consider the specific case when (1) is the equivalent equation of a
DRP scheme,the coefficients of which will be denoted by γ k , k ∈ {− m, m } (see [1]): − u t − σ u tt + 2 σµ Re h m X k =1 k γ k u x = 0 (8) where Re h denotes the mesh Reynolds number, σ , the cf l coefficient, and µ , theviscosity.Equation (3) is then given by: − v ˜ u ′ ( ξ ) − v σ u ′′ ( ξ ) + 2 σµ Re h m X k =1 k γ k ˜ u ′ ( ξ ) = 0 (9) Performing an integration of (9) with respect to ξ yields: − v ˜ u ( ξ ) − v σ u ′ ( ξ ) + 2 σµ Re h m X k =1 k γ k ˜ u ( ξ ) = C (10)
3. e.: ( σµ Re h m X k =1 k γ k − v ) ˜ u ( ξ ) − v σ u ′ ( ξ ) = C (11) where C is an arbitrary integration constant.Substitution of (5) for n = 1 into equation (11) leads to: ( σµ Re h m X k =1 k γ k − v ) { U tanh [ C ξ ] + V sech [ C ξ ] + V } − v σ U sech [ C ξ ] − V sinh [ C ξ ]cosh [ C ξ ] ff = C (12) i. e.: ( σµ Re h m X k =1 k γ k − v ) U e C ξ − e − C ξ e C ξ + e − C ξ + 2 V e C ξ + e − C ξ + V ff − v σ ( U „ e C ξ + e − C ξ « − V e C ξ − e − C ξ ` e C ξ + e − C ξ ´ ) = C (13) Multiplying both sides by (cid:0) e C ξ (cid:1) yields: ( σµ Re h m X k =1 k γ k − v ) U “ e C ξ − ” + 2 V “ e C ξ + e C ξ ” + V “ e C ξ ” ff − v C σ C n U − V “ e C ξ − ”o = C (14) which is a fourth-order equation in e C ξ . This equation being satisfied for any realvalue of ξ , one therefore deduces that the coefficients of e k C ξ , k = 0 , . . . , ( σµ Re h m X k =1 k γ k − v ) {− U + V } − v C σ { U + 2 V } = C ( σµ Re h m X k =1 k γ k − v ) V = 02 ( σµ Re h m X k =1 k γ k − v ) V = 02 ( σµ Re h m X k =1 k γ k − v ) V + v C σ V = 0 ( σµ Re h m X k =1 k γ k − v ) { U + V } = 0 (15) v = σµ Re h m X k =1 k γ k , V = 0 leads to the trivial null solution. Therefore, V isnecessarily equal to zero, which implies: 4 >>>>>><>>>>>>: v = 2 σµ Re h m X k =1 k γ k U = − C C v σV ∈ IR , C ∈ IR (16)
It is easy to note that, if the integration constant C had been taken equal to zero,the solitary waves of the considered equation would have been loss. The importance of choosing an integration constant which is not equal to zero, inthe determination of solitary wave solutions of wave equations, has been carriedout. We show that taking this constant equal to zero leads to a loss of solutions.
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