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.

Ministry of Science, ICT and Future Planning2020R1F1A1A01072197National Research Foundation of Korea1. 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]:
(1)utx=u−3u2,with initial data
(2)ux,0=fx,where the real valued function u satisfies the periodic boundary u0,t=uL,t 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]:
(3)ux,t=∑n∈Zuntei2πnx/L,where u−n=u¯n for all n∈Z. Integrating (1) on 0,L, they obtained
(4)∫0Lu−3u2x,tdx=0,which implies, combined with (3),
(5)u0=3u02+3∑n∈Z,n≠0un2.

Then, they obtained
(6)u0=161±1−36∑n∈Z,n≠0n2un2,from which a restriction on the initial data is imposed to guarantee ∑n∈Z,n≠0n2un2<1/36. Let us consider a homogeneous solution ux,t=ut. To satisfy (4), possible homogeneous solutions are ut=0 and ut=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:
(7)utx=u−3u2−1L∫0Lu−3u2y,tdy,with initial and boundary conditions
(8)ux,0=fx,u0,t=uL,t=gt,where we assume compatibility condition f0=fL=g0. 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
(9)vtx=0,vx,0=fx,we have solutions vx,t=fx+ht, where h is any C1 function with h0=0. For the solution to (7) satisfying (8), we have a compatibility condition, considering u0,t=∑n∈Zunt,
(10)u0t=gt−∑n∈Z,n≠0unt.

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=vnn∈Z:
(11)H=v=vnn∈Z∣∥v∥2<∞,v−n=v¯n,where the norm is defined by
(12)∥v∥2=v02+∑n∈Z,n≠0n2vn2.

The first result is concerned with the global existence of solution.

Theorem 1.

For data g∈C10,∞ and f∈H satisfying f0=g0, problems (7) and (8) have a unique solution u∈C0,∞,H of the form (3).

Remark 2.

Let u∈C0,T,H 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 t≥0, Fourier series solution (3) converges uniformly in x. Its sum is differentiable in x for almost all x∈0,L. The derivative satisfies the condition ux·,t∈L20,L and uxx,·∈C0,∞. Moreover, ux is differentiable in t, and (7) holds for almost all x∈0,L

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
(13)ux,t=ux+ct.

Note that any constant function u=C is a steady solution of (7). We know, for L periodic function u,
(14)∫0Lux+ct−3u2x+ctdx=∫0Lux−3u2xdx=m,where m is a constant. Substituting the ansatz (13) in (7), we obtain
(15)cd2udξ2=u−3u2−mL,where ξ=x+ct.

Theorem 3.

There are nontrivial traveling wave solutions ux,t=ux+ct to (7) for m<0 or 0<m/L<1/16. In fact, we have solutions of the elliptic function
(16)ux+ct=a−bsn2λx+ct,k.

We refer to Section 3 for precise values of a, b, c, λ, and k.

With the change of variable t=1/2T+X and x=1/2T−X, equation (1) becomes a semilinear wave equation
(17)uTT−uXX=u−3u2,and initial condition (2) becomes data on characteristic line T+X=0. The Cauchy problem on the torus Tn for the semilinear wave equation vtt−Δv+fv=0 with initial data vx,0=v0x, vtx,0=v1x 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 A≲B to denote an estimate of the form A≤CB, where C is a positive constant.

2. Proof of Theorem <xref rid="thm1" ref-type="statement">1</xref>

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

Theorem 4.

If f∈H satisfies
(18)∑n∈Z,n≠0n2fn2<172,∫0Lfx−3f2xdx=0,then problems (1) and (2) have one and only one solution of the form (3). For all t≥0, Fourier series (3) converges uniformly in x. Its sum is differentiable in x for almost all x∈0,L. The derivative satisfies the condition ux·,t∈L20,L and uxx,·∈C0,∞. Moreover, ux is differentiable in t, and (1) holds for almost all x∈0,L.

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
(19)Vt+Vx−Vxxt=3V2x.

Taking change of variables Vx,t=Uξ,τ with ξ=x/ε and τ=εt, equation (19) becomes
(20)εUτ+1εUξ−Uξξτ−3U2ξ=0.

Considering small ε, an asymptotic equation Uξ−Uξξτ−3U2ξ=0 is obtained which can be integrated in ξ to give
(21)Uξτ=U−3U2+Hτ,where Hτ is integration constant with respect to variable ξ. The zero condition Hτ=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:
(22)utx=u−3u2−1L∫0Lu−3u2dx,with conditions
(23)ux,0=fx,(24)u0,t=uL,t=gt,where we assume f∈H, g∈C10,∞ and compatibility condition f0=fL=g0. We can check that ∫0Lutxdx=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
(25)ddtun=−iL2πnun−3∑α+β=nuαuβforn≠0.

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
(26)1L∫0Lu−3u2dx=u0−3u02−3∑n∈Z,n≠0un2.

Then, the right-hand side of (22) becomes for n=0(27)u0−3∑α+β=0uαuβ−1L∫0Lu−3u2dx=u0−3u02−3∑α≠0uαu−α−u0−3u02−3∑n∈Z,n≠0un2=0,where u−α=u¯α is used. Relation (24) implies
(28)u0t=gt−∑n∈Z,n≠0unt.

Therefore, we arrive at the following system of ODEs:
(29)ddtun=−iL2πnun−3∑α+β=nuαuβforn≠0,u0t=gt−∑n∈Z,n≠0unt,un0=fn,where fx=∑n∈Zfnei2πnx/L.

We say that a function u∈C0,T,H is a solution to problem (22)–(24), if the Fourier coefficients un satisfy (29) for all n. For v∈C0,T,H, we define an operator Φ:C0,T,H→C0,T,H: (30)Φnvt=fn−iL2πn∫0tvn−3∑α∈Zvαvn−αforn≠0,Φ0vt=gt−∑n∈Z,n≠0Φnvt.

We will prove a local existence part of Theorem 1 by applying a standard contraction mapping theorem. Denote by Fg∈C0,T,H the function Fgx,t=fx+gt and consider a space
(31)STM=v∈C0,T,H: ∥v−Fg∥≤M,where T>0 and M>0.

Proposition 5.

Let f∈H and g∈C1. Then, the mapping (30) is a contraction mapping from STM to STM 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 of Φ0v in (30) from (6). For v,ω∈STM, we have
(32)∑n≠0n2Φnv−Φnω2≲∑n≠0∫0t∣vn−ωn∣+∑k=−∞∞vk−ωkvn−k+ωkvn−k−ωn−kds2≲t∑n≠0∫0tvn−ωn2+∑k=−∞∞∣vk−ωk∣∣vn−k∣+∣ωn−k∣2ds≲t∑n≠0∫0tvn−ωn2+v0−ω02vn2+ωn2+∑k≠01k2∑k≠0k2vk−ωk2vn−k2+ωn−k2ds≲t∫0t1+∑n≠0vn2+ωn2ds∥v−ω∥2≲T21+∥v∥2+∥ω∥2∥v−ω∥2.

We also have
(33)Φ0v−Φ0ω2≤∑k≠0Φkv−Φkω2≤∑k≠01k2∑k≠0k2Φkv−Φkω2≤CT21+∥v∥2+∥ω∥2∥v−ω∥2,where (32) is used. Combining (32) and (33) and considering v,ω∈STM, we have
(34)∥Φv−Φω∥2≤CT21+∥Fg∥2+M2∥v−ω∥2,which is contraction mapping for sufficiently small T>0.

Let u∈C0,T,H be a solution of the equation u=Φu. 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.

Proposition 6.

Let u∈C0,T,H be a solution of (22). Then, we have
(35)∑n∈Zn2unt2=∑n∈Zn2fn2.

Proof.

Multiplying ∂xu on both sides of (22) and integrating on 0,L, we have
(36)ddt∫0L12∂xu2dx=∫0L12∂xu2−∂xu3dx−1L∫0Lu−3u2dx∫0L∂xudx=0,which implies
(37)∫0L∂xux,t2dx=∫0L∂xux,02dx.

Moreover, a direct calculation implies that
(38)∫0L∂xux,t2dx=∑n∈Z4π2Ln2unt2,from which we can derive (35).

From Proposition 5, we have a local solution u∈C0,T,H 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.

3. Traveling Waves

Here, we consider a traveling wave solution to (7) of the form
(39)ux,t=ux+ct,where a positive constant c will be determined later. Note that we have, for L periodic function u,
(40)∫0Lux+ct−3u2x+ctdx=∫0Lux−3u2xdx=m,where m is a constant. Substituting the ansatz (39) in (7), we obtain
(41)cd2udξ2=u−3u2−A,where ξ=x+ct and A=m/L. We integrate (41) to obtain
(42)c2dudξ2=−u3+12u2−Au≔hu.

We will consider the cases of 0<A<1/16 or A<0.

For 0<A<1/16, h has three distinct real roots 0<α<β, where

(43)α=1−1−16A4,β=1+1−16A4.

Applying change of variable u=β−β−αz2, we derive an equation for z(44)dzdξ2=λ21−z21−k2z2,where λ2=β/2c and k2=β−α/β. It is well known in [18] that the solution of (44) is given by the elliptic function zξ=snλξ,k. Therefore, we have
(45)uξ=β−β−αsn2λξ,k.

Since the period of sn2x,k is 2∫0π/2dy/1−k2sin2y, we impose the following condition from which the period of (45) becomes L:
(46)2n∫0π/2dy1−k2sin2y=λLforsomen∈N,which can be rewritten as
(47)2n∫0π/21−21−16A1+1−16Asin2y−1/2dy=Lc1+1−16A81/2.

For a given 0<A<1/16, the constant c is determined by (47).

For A<0, h has three distinct real roots α1<0<α2, where

(48)α1=1−1−16A4,α2=1+1−16A4.

Applying change of variable u=α2−α2z2, we have an equation for z(49)dzdξ2=λ21−z21−k2z2,where λ2=α2−α1/2c and k2=α2/α2−α1. Then, we have
(50)uξ=α2−α2sn2λξ,k.

To make the solution (50) L periodic, we impose
(51)2n∫0π/21−1+1−16A21−16Asin2y−1/2dy=L2c1−16A1/4forsomen∈N.

For a given A<0, the constant c is determined by (51).

Remark 7.

For A=0, we have cdu/dξ2=u21−2u which can be integrated as
(52)1−2u−11−2u+1=eξ+a/c,where a is an integration constant. We know that u≤1/2. If uξ0=0 for some ξ0, we have 0=eξ0+a/c which is a contradiction. So, we have 0<u≤1/2 or u<0. For a periodic function u, we have u0=uL. Then, we obtain eL+a/c=ea/c which is a contradiction. The similar argument can be applied for the case of A=1/16 to show that there is not a nontrivial periodic solution u.

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).

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