AMPAdvances in Mathematical Physics1687-91391687-9120Hindawi10.1155/2021/27772652777265Research ArticleExistence of Global Solution and Traveling Wave of the Modified Short-Wave Equationhttps://orcid.org/0000-0001-9844-845XHuhHyungjinMeiMingDepartment of MathematicsChung-Ang UniversitySeoul 06974Republic of Koreacau.ac.kr2021352021202194202126420213520212021Copyright © 2021 Hyungjin Huh.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Korea
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  numerically for periodic and nonperiodic boundary conditions. The following characteristic initial value problem for the periodic short-wave equation was discussed in : (1)utx=u3u2,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 : (3)ux,t=nZuntei2πnx/L,where un=u¯n for all nZ. Integrating (1) on 0,L, they obtained (4)0Lu3u2x,tdx=0,which implies, combined with (3), (5)u0=3u02+3nZ,n0un2.

Then, they obtained (6)u0=161±136nZ,n0n2un2,from which a restriction on the initial data is imposed to guarantee nZ,n0n2un2<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=u3u21L0Lu3u2y,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 . 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=nZunt, (10)u0t=gtnZ,n0unt.

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=vnnZ: (11)H=v=vnnZv2<,vn=v¯n,where the norm is defined by (12)v2=v02+nZ,n0n2vn2.

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

Theorem 1.

For data gC10, and fH satisfying f0=g0, problems (7) and (8) have a unique solution uC0,,H of the form (3).

Remark 2.

Let uC0,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 . In fact, for all t0, Fourier series solution (3) converges uniformly in x. Its sum is differentiable in x for almost all x0,L. The derivative satisfies the condition ux·,tL20,L and uxx,·C0,. Moreover, ux is differentiable in t, and (7) holds for almost all x0,L

It is an interesting problem to consider an initial value problem of (1) on the whole line xR. 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+ct3u2x+ctdx=0Lux3u2xdx=m,where m is a constant. Substituting the ansatz (13) in (7), we obtain (15)cd2udξ2=u3u2mL,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=absn2λx+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=1/2TX, equation (1) becomes a semilinear wave equation (17)uTTuXX=u3u2,and initial condition (2) becomes data on characteristic line T+X=0. The Cauchy problem on the torus Tn for the semilinear wave equation vttΔv+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 . 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 AB to denote an estimate of the form ACB, 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 .

Theorem 4.

If fH satisfies (18)nZ,n0n2fn2<172,0Lfx3f2xdx=0,then problems (1) and (2) have one and only one solution of the form (3). For all t0, Fourier series (3) converges uniformly in x. Its sum is differentiable in x for almost all x0,L. The derivative satisfies the condition ux·,tL20,L and uxx,·C0,. Moreover, ux is differentiable in t, and (1) holds for almost all x0,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+VxVxxt=3V2x.

Taking change of variables Vx,t=Uξ,τ with ξ=x/ε and τ=εt, equation (19) becomes (20)εUτ+1εUξUξξτ3U2ξ=0.

Considering small ε, an asymptotic equation UξUξξτ3U2ξ=0 is obtained which can be integrated in ξ to give (21)Uξτ=U3U2+Hτ,where Hτ is integration constant with respect to variable ξ. The zero condition Hτ=0 was considered in .

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=u3u21L0Lu3u2dx,with conditions (23)ux,0=fx,(24)u0,t=uL,t=gt,where we assume fH, gC10, 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=iL2πnun3α+β=nuαuβforn0.

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)1L0Lu3u2dx=u03u023nZ,n0un2.

Then, the right-hand side of (22) becomes for n=0(27)u03α+β=0uαuβ1L0Lu3u2dx=u03u023α0uαuαu03u023nZ,n0un2=0,where uα=u¯α is used. Relation (24) implies (28)u0t=gtnZ,n0unt.

Therefore, we arrive at the following system of ODEs: (29)ddtun=iL2πnun3α+β=nuαuβforn0,u0t=gtnZ,n0unt,un0=fn,where fx=nZfnei2πnx/L.

We say that a function uC0,T,H is a solution to problem (22)–(24), if the Fourier coefficients un satisfy (29) for all n. For vC0,T,H, we define an operator Φ:C0,T,HC0,T,H: (30)Φnvt=fniL2πn0tvn3αZvαvnαforn0,Φ0vt=gtnZ,n0Φnvt.

We will prove a local existence part of Theorem 1 by applying a standard contraction mapping theorem. Denote by FgC0,T,H the function Fgx,t=fx+gt and consider a space (31)STM=vC0,T,H: vFgM,where T>0 and M>0.

Proposition 5.

Let fH and gC1. 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  with little modifications which come from the different definition of Φ0v in (30) from (6). For v,ωSTM, we have (32)n0n2ΦnvΦnω2n00tvnωn+k=vkωkvnk+ωkvnkωnkds2tn00tvnωn2+k=vkωkvnk+ωnk2dstn00tvnωn2+v0ω02vn2+ωn2+k01k2k0k2vkωk2vnk2+ωnk2dst0t1+n0vn2+ωn2dsvω2T21+v2+ω2vω2.

We also have (33)Φ0vΦ0ω2k0ΦkvΦkω2k01k2k0k2ΦkvΦkω2CT21+v2+ω2vω2,where (32) is used. Combining (32) and (33) and considering v,ωSTM, we have (34)ΦvΦω2CT21+Fg2+M2vω2,which is contraction mapping for sufficiently small T>0.

Let uC0,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 . We skip the proof. We will prove the conservation of H norm.

Proposition 6.

Let uC0,T,H be a solution of (22). Then, we have (35)nZn2unt2=nZn2fn2.

Proof.

Multiplying xu on both sides of (22) and integrating on 0,L, we have (36)ddt0L12xu2dx=0L12xu2xu3dx1L0Lu3u2dx0Lxudx=0,which implies (37)0Lxux,t2dx=0Lxux,02dx.

Moreover, a direct calculation implies that (38)0Lxux,t2dx=nZ4π2Ln2unt2,from which we can derive (35).

From Proposition 5, we have a local solution uC0,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+ct3u2x+ctdx=0Lux3u2xdx=m,where m is a constant. Substituting the ansatz (39) in (7), we obtain (41)cd2udξ2=u3u2A,where ξ=x+ct and A=m/L. We integrate (41) to obtain (42)c2dudξ2=u3+12u2Auhu.

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=1116A4,β=1+116A4.

Applying change of variable u=ββαz2, we derive an equation for z(44)dzdξ2=λ21z21k2z2,where λ2=β/2c and k2=βα/β. It is well known in  that the solution of (44) is given by the elliptic function zξ=snλξ,k. Therefore, we have (45)uξ=ββαsn2λξ,k.

Since the period of sn2x,k is 20π/2dy/1k2sin2y, we impose the following condition from which the period of (45) becomes L: (46)2n0π/2dy1k2sin2y=λLforsomenN,which can be rewritten as (47)2n0π/212116A1+116Asin2y1/2dy=Lc1+116A81/2.

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=1116A4,α2=1+116A4.

Applying change of variable u=α2α2z2, we have an equation for z(49)dzdξ2=λ21z21k2z2,where λ2=α2α1/2c and k2=α2/α2α1. Then, we have (50)uξ=α2α2sn2λξ,k.

To make the solution (50) L periodic, we impose (51)2n0π/211+116A2116Asin2y1/2dy=L2c116A1/4forsomenN.

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

Remark 7.

For A=0, we have cdu/dξ2=u212u which can be integrated as (52)12u112u+1=eξ+a/c,where a is an integration constant. We know that u1/2. If uξ0=0 for some ξ0, we have 0=eξ0+a/c which is a contradiction. So, we have 0<u1/2 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).

GamaS. M.KraenkelR. A.MannaM. A.Short-wave instabilities in the Benjamin-Bona-Mahoney-Peregrine equation: theory and numericsInverse Problems200117486387010.1088/0266-5611/17/4/3182-s2.0-0035419873MannaM. A.MerleV.Asymptotic dynamics of short waves in nonlinear dispersive modelsPhysical Review E19985756206620910.1103/PhysRevE.57.62062-s2.0-0032069680BenjaminT. B.BonaJ. L.MahonyJ. J.Model equations for long waves in nonlinear dispersive systemsPhilosophical Transactions of the Royal Society of London197227212204778PeregrineD. H.Long waves on a beachJournal of Fluid Mechanics196727481582710.1017/S00221120670026052-s2.0-84958426983GamaS. M.SmirnovG.The Cauchy problem for a short-wave equationElectronic Journal of Differential Equations200989AlexiewiczA.OrliczW.Some remarks on the existence and uniqueness of solutions of the hyperbolic equation ²z/xy=fx,y,x,z/x,z/yStudia Mathematica195515220121510.4064/sm-15-2-201-215CabetA.Local existence of a solution of a semi-linear wave equation in a neighborhood of initial characteristic hypersurfacesAnnales de la Faculté des Sciences de Toulouse Mathématiques20031214710210.5802/afst.1044CabetA.Local existence of a solution of a semilinear wave equation with gradient in a neighborhood of initial characteristic hypersurfaces of a Lorentzian manifoldCommunications in Partial Differential Equations200833122105215610.1080/036053008025013352-s2.0-57249084827DossaM.Tagne WafoR.Solutions with a uniform time of existence of a class of characteristic semi-linear wave equations near S+Communications in Partial Differential Equations2019441094098910.1080/03605302.2019.16118452-s2.0-85067031669JokhadzeO.On existence and nonexistence of global solutions of Cauchy-Goursat problem for nonlinear wave equationsJournal of Mathematical Analysis and Applications200834021033104510.1016/j.jmaa.2007.09.0302-s2.0-38049089404JokhadzeO.Cauchy-Goursat problem for one-dimensional semilinear wave equationsCommunications in Partial Differential Equations200934436738210.1080/036053009027689662-s2.0-68749120707DelortJ.-M.On long time existence for small solutions of semi-linear Klein-Gordon equations on the torusJournal d'Analyse Mathématique2009107116119410.1007/s11854-009-0007-22-s2.0-63849187758KapitanskiiL. V.The Cauchy problem for a semilinear wave equation. IJournal of Soviet Mathematics19904951166118610.1007/BF022087132-s2.0-34249956761BronskiJ. C.JohnsonM. A.KapitulaT.An index theorem for the stability of periodic travelling waves of Korteweg–de Vries typeProceedings of the Royal Society of Edinburgh201114161141117310.1017/S03082105100012162-s2.0-84863422044GustafsonS.le CozS.TsaiT. P.Stability of periodic waves of 1D cubic nonlinear Schrödinger equationsApplied Mathematics Research eXpress20172017243148710.1093/amrx/abx0042-s2.0-85031095235NataliF.CardosoE.Jr.Orbital stability of periodic standing waves for the logarithmic Klein-Gordon equationJournal of Mathematical Analysis and Applications2020484212372310.1016/j.jmaa.2019.123723HartmanP.WintnerA.On hyperbolic partial differential equationsAmerican Journal of Mathematics195274483486410.2307/2372229BowmanF.Introduction to Elliptic Functions1953New YorkWiley