Existence of Global Solution and Traveling Wave of the Modified Short-Wave Equation

The nonlinear propagation of waves of short wavelength in dispersive systems was discussed in [1, 2]. They proposed the simple nonlinear short-wave equation that is likely to describe the asymptotic behaviour of the Benjamin-BonaMahony-Peregrine equation [3, 4]. The short-wave equation was studied in [1] numerically for periodic and nonperiodic boundary conditions. The following characteristic initial value problem for the periodic short-wave equation was discussed in [5]:


Introduction
The nonlinear propagation of waves of short wavelength in dispersive systems was discussed in [1,2]. They proposed the simple nonlinear short-wave equation that is likely to describe the asymptotic behaviour of the Benjamin-Bona-Mahony-Peregrine equation [3,4]. The short-wave equation was studied in [1] numerically for periodic and nonperiodic boundary conditions. The following characteristic initial value problem for the periodic short-wave equation was discussed in [5]: with initial data where the real valued function u satisfies the periodic boundary uð0, tÞ = uðL, tÞ and describes a small amplitude wave depending on space variable x and time variable t. The solution of the Fourier series was considered in [5]: where u −n = u n for all n ∈ Z. Integrating (1) on ½0, L, they obtained which implies, combined with (3), Then, they obtained from which a restriction on the initial data is imposed to guarantee ∑ n∈Z,n≠0 n 2 ju n j 2 < 1/36. Let us consider a homogeneous solution uðx, tÞ = uðtÞ. To satisfy (4), possible homogeneous solutions are uðtÞ = 0 and uðtÞ = 1/3 which gives a serious constraint on the initial data. The present work is motivated by the question of whether the restriction of the initial data can be removed. In order for the initial value problem to make sense for a large range of initial data, it seems to be essential to modify the differential equation (1). We consider the following modified short-wave equation: with initial and boundary conditions where we assume compatibility condition f ð0Þ = f ðLÞ = gð0Þ.
We impose the second condition in (8) to guarantee uniqueness. Note that the initial data (2) only is not sufficient for the uniqueness of solution to the initial value problem [6][7][8][9][10][11]. Even for the linear equation we have solutions vðx, tÞ = f ðxÞ + hðtÞ, where h is any C 1 function with hð0Þ = 0. For the solution to (7) satisfying (8), we have a compatibility condition, considering uð0, tÞ = ∑ n∈Z u n ðtÞ, We refer to Section 2 for more information of (7) and (8).
We consider the solution of Fourier series (3). Let us denote by H the space of complex sequences v = fv n g n∈Z : where the norm is defined by The first result is concerned with the global existence of solution.
Then, we can show several regularity properties by applying the same argument as Proposition 2.3 in [5]. In fact, for all t ≥ 0, Fourier series solution (3) converges uniformly in x. Its sum is differentiable in x for almost all x ∈ ½0, L. The derivative satisfies the condition u x ð·, tÞ ∈ L 2 ½0, L and u x ðx, ·Þ ∈ C½0, ∞Þ. Moreover, u x is differentiable in t, and (7) holds for almost all x ∈ ½0, L (2) It is an interesting problem to consider an initial value problem of (1) on the whole line x ∈ R. We refer to [7,10,11] for more information Our next result is concerned with the existence of the traveling wave solutions of the form Note that any constant function u = C is a steady solution of (7). We know, for L periodic function u, where m is a constant. Substituting the ansatz (13) in (7), we obtain where ξ = x + ct.
Theorem 3. There are nontrivial traveling wave solutions uð x, tÞ = uðx + ctÞ to (7) for m < 0 or 0 < m/L < 1/16. In fact, we have solutions of the elliptic function We refer to Section 3 for precise values of a, b, c, λ, and k.
With the change of variable t = 1/2ðT + XÞ and x = 1/2ð T − XÞ, equation (1) becomes a semilinear wave equation and initial condition (2) becomes data on characteristic line T + X = 0. The Cauchy problem on the torus T n for the semilinear wave equation v tt − Δv + f ðvÞ = 0 with initial data vðx , 0Þ = v 0 ðxÞ, v t ðx, 0Þ = v 1 ðxÞ was studied in [12,13]. Stability of periodic waves of KdV, Schrödinger, Klein-Gordon equations was studied in [14][15][16]. It is a quite interesting problem to study the stability or instability of the above traveling wave solution to (7). In Section 2, Theorem 1 is proved. In Section 3, we find traveling wave solutions of (7) to prove Theorem 3. We will use A ≲ B to denote an estimate of the form A ≤ CB, where C is a positive constant.

Proof of Theorem 1
Let us introduce the following main result of [5].

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Two restrictions on the initial data are imposed in Theorem 4. Let us review the derivation of equation (1). The Benjamin-Bona-Mahony-Peregrine equation reads as Taking change of variables Vðx, tÞ = Uðξ, τÞ with ξ = x/ε and τ = εt, equation (19) becomes Considering small ε, an asymptotic equation U ξ − U ξξτ − 3ðU 2 Þ ξ = 0 is obtained which can be integrated in ξ to give where HðτÞ is integration constant with respect to variable ξ. The zero condition HðτÞ = 0 was considered in [1].
To remove restrictions on the initial data in Theorem 4 and allow more solutions like traveling wave solutions, we consider the following modified short-wave equation: with conditions where we assume f ∈ H, g ∈ C 1 ½0,∞Þ and compatibility condition f ð0Þ = f ðLÞ = gð0Þ. We can check that Ð L 0 u tx dx = 0 in (22). Note that the initial data (23) only is not sufficient for the uniqueness, and additional condition (24) is needed for the characteristic initial value problem [6,7,17].
Substituting (3) into (22), we obtain a system of ordinary differential equations For n = 0, the left side of (22) is zero. Let us calculate the right-hand side of (22). Considering the solution of Fourier series (3), we have Then, the right-hand side of (22) becomes for n = 0 where u −α = u α is used. Relation (24) implies Therefore, we arrive at the following system of ODEs: where f ðxÞ = ∑ n∈Z f n e i2πnx/L . We say that a function u ∈ Cð½0, T, HÞ is a solution to problem (22)-(24), if the Fourier coefficients u n satisfy (29) for all n. For v ∈ Cð½0, TÞ, HÞ, we define an operator Φ : Cð½ 0, TÞ, HÞ → Cð½0, TÞ, HÞ: We will prove a local existence part of Theorem 1 by applying a standard contraction mapping theorem. Denote by F g ∈ Cð½0, T, HÞ the function F g ðx, tÞ = f ðxÞ + gðtÞ and consider a space where T > 0 and M > 0.
Proposition 5. Let f ∈ H and g ∈ C 1 . Then, the mapping (30) is a contraction mapping from S TM to S TM for a sufficiently small T.
Proof. We follow the argument of Proposition 2.2 in [5] with little modifications which come from the different definition 3 Advances in Mathematical Physics of Φ 0 ðvÞ in (30) from (6). For v, ω ∈ S TM , we have We also have where (32) is used. Combining (32) and (33) and considering v, ω ∈ S TM , we have which is contraction mapping for sufficiently small T > 0.
Let u ∈ Cð½0, T, HÞ be a solution of the equation u = Φð uÞ. Then, we can show several regularity properties in Remark 2 by applying the same argument as Proposition 2.3 in [5]. We skip the proof. We will prove the conservation of H norm.
Multiplying ∂ x u on both sides of (22) and integrating on ½0, L, we have d dt which implies Moreover, a direct calculation implies that from which we can derive (35). From Proposition 5, we have a local solution u ∈ Cð½0, T Þ, HÞ of (22)-(24) for a sufficiently small T > 0. By Proposition 6, we can extend a local solution to a global one which completes the proof of Theorem 1.

Traveling Waves
Here, we consider a traveling wave solution to (7) of the form where a positive constant c will be determined later. Note that we have, for L periodic function u, where m is a constant. Substituting the ansatz (39) in (7), we obtain where ξ = x + ct and A = m/L. We integrate (41) to obtain c 2 du dξ We will consider the cases of 0 < A < 1/16 or A < 0.