Abstract
The modified short-wave equation is considered under periodic boundary condition. We prove the global existence of solution with finite energy. We also find traveling wave solutions which is the form of elliptic function.
1. 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 satisfies the periodic boundary and describes a small amplitude wave depending on space variable and time variable .
The solution of the Fourier series was considered in [5]: where for all . Integrating (1) on , they obtained which implies, combined with (3),
Then, they obtained from which a restriction on the initial data is imposed to guarantee . Let us consider a homogeneous solution . To satisfy (4), possible homogeneous solutions are and 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 . 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 [6–11]. Even for the linear equation we have solutions , where is any function with . For the solution to (7) satisfying (8), we have a compatibility condition, considering ,
We refer to Section 2 for more information of (7) and (8).
We consider the solution of Fourier series (3). Let us denote by the space of complex sequences : where the norm is defined by
The first result is concerned with the global existence of solution.
Theorem 1. For data and satisfying , problems (7) and (8) have a unique solution of the form (3).
Remark 2. (1)Let be a solution of equation (7). Then, we can show several regularity properties by applying the same argument as Proposition 2.3 in [5]. In fact, for all , Fourier series solution (3) converges uniformly in . Its sum is differentiable in for almost all . The derivative satisfies the condition and . Moreover, is differentiable in , and (7) holds for almost all (2)It is an interesting problem to consider an initial value problem of (1) on the whole line . We refer to [7, 10, 11] for more informationOur next result is concerned with the existence of the traveling wave solutions of the form Note that any constant function is a steady solution of (7). We know, for periodic function , where is a constant. Substituting the ansatz (13) in (7), we obtain where .
Theorem 3. There are nontrivial traveling wave solutions to (7) for or . In fact, we have solutions of the elliptic function
We refer to Section 3 for precise values of , , , , and .
With the change of variable and , equation (1) becomes a semilinear wave equation and initial condition (2) becomes data on characteristic line . The Cauchy problem on the torus for the semilinear wave equation with initial data , was studied in [12, 13]. Stability of periodic waves of KdV, Schrödinger, Klein-Gordon equations was studied in [14–16]. 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 to denote an estimate of the form , where is a positive constant.
2. Proof of Theorem 1
Let us introduce the following main result of [5].
Theorem 4. If satisfies then problems (1) and (2) have one and only one solution of the form (3). For all , Fourier series (3) converges uniformly in . Its sum is differentiable in for almost all . The derivative satisfies the condition and . Moreover, is differentiable in , and (1) holds for almost all .
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 with and , equation (19) becomes
Considering small , an asymptotic equation is obtained which can be integrated in to give where is integration constant with respect to variable . The zero condition 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 , and compatibility condition . We can check that 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 , 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 where is used. Relation (24) implies
Therefore, we arrive at the following system of ODEs: where .
We say that a function is a solution to problem (22)–(24), if the Fourier coefficients satisfy (29) for all . For , we define an operator
We will prove a local existence part of Theorem 1 by applying a standard contraction mapping theorem. Denote by the function and consider a space where and .
Proposition 5. Let and . Then, the mapping (30) is a contraction mapping from to for a sufficiently small .
Proof. We follow the argument of Proposition 2.2 in [5] with little modifications which come from the different definition of in (30) from (6). For , we have We also have where (32) is used. Combining (32) and (33) and considering , we have which is contraction mapping for sufficiently small .
Let be a solution of the equation . 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 norm.
Proposition 6. Let be a solution of (22). Then, we have
Proof. Multiplying on both sides of (22) and integrating on , we have
which implies
Moreover, a direct calculation implies that
from which we can derive (35).
From Proposition 5, we have a local solution of (22)–(24) for a sufficiently small . By Proposition 6, we can extend a local solution to a global one which completes the proof of Theorem 1.
3. Traveling Waves
Here, we consider a traveling wave solution to (7) of the form where a positive constant will be determined later. Note that we have, for periodic function , where is a constant. Substituting the ansatz (39) in (7), we obtain where and . We integrate (41) to obtain
We will consider the cases of or . (1)For , has three distinct real roots , where
Applying change of variable , we derive an equation for where and . It is well known in [18] that the solution of (44) is given by the elliptic function . Therefore, we have
Since the period of is , we impose the following condition from which the period of (45) becomes : which can be rewritten as
For a given , the constant is determined by (47). (2)For , has three distinct real roots , where
Applying change of variable , we have an equation for where and . Then, we have
To make the solution (50) periodic, we impose
For a given , the constant is determined by (51).
Remark 7. For , we have which can be integrated as where is an integration constant. We know that . If for some , we have which is a contradiction. So, we have or . For a periodic function , we have . Then, we obtain which is a contradiction. The similar argument can be applied for the case of to show that there is not a nontrivial periodic solution .
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01072197).