J.Nonlinear Sci.V ol.12:pp.283–318(2002)
DOI:10.1007/s00332-002-0466-4
©2002Springer-Verlag New York Inc. Boussinesq Equations and Other Systems for
Small-Amplitude Long Waves in Nonlinear Dispersive Media.I:Derivation and Linear Theory
J.L.Bona,1M.Chen,2and J.-C.Saut3
1Department of Mathematics,Statistics,and Computer Science,University of Illinois
at Chicago,Chicago,IL,60607,USA
2Department of Mathematics,Purdue University,West Lafayette,IN47907,USA
e-mail:chen@math.purdue.edu
3UMR de Math´e matiques,Universit´e de Paris-Sud,Bˆa timent425,91405Orsay,France
Received April30,2001;accepted March26,2002
Online publication July1,2002
Communicated by G.Iooss
Summary.Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations.Such equations werefirst derived by Boussinesq to describe the two-way propagation of small-amplitude,long wavelength, gravity waves on the surface of water in a canal.The systems ari also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion,the resulting system may or may not have a linearization about the rest state which is well pod.Even when well pod,the linearized system may exhibit a lack of conrvation of energy that is at odds with its status as an approximation to the Euler equations.In the prent script,we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surfaceflow and formulate criteria to help decide which of the equations one might choo in a given modeling situation.The analysis of the systems according to the criteria is initiated.
Key words.water waves,two-way propagation,Boussinesq systems,local well-pod-ness,global well-podness
1.Introduction
In a continuum approximation,waves on the surface of an idealfluid under the force of gravity are governed by the Euler equations.The are expected to provide a good model of irrotational waves on the surface of water,say,in situations where dissipative
284J.L.Bona,M.Chen,and J.-C.Saut and surface tension effects may be safely ignored.In manyfield and laboratory studies and in engineering applications,the full Euler equations appear more complex than is necessary for the modeling situation at hand,and conquently there have appeared many approximate models applying to restricted physical regimes.
A regime that aris in practical situations is that of waves in a channel of approx-imately constant depth h that are uniform across the channel,and which are of small amplitude and long wavelength,and such that the associated nonlinear and dispersive effects are balanced.Similar waves appear as long-crested disturbances on larger bod-ies of water.If A connotes a typical wave amplitude and a typical wavelength,the conditions just mentioned amount to
α=A
h 1,β=h
2
2
1,S=
α
β
=A
2
h3
≈1.(1.1)
In the1870s,Boussinesq derived some model evolution equations which are appli-cable in principle to describe motions that are nsibly two-dimensional and which have the form of a perturbation of the one-dimensional wave equation.Perhaps the best known is the equation
w tt=w xx+(w2)xx+w xxxx,(1.2) or its regularized version
w tt=w xx+(w2)xx+w xxtt(1.3) (e Boussinesq[29],Keulegan&Patterson[44],Urll[59],Benjamin et al.[9],Kano &Nishida[42],Bona&Sachs[21]).The variables appearing in(1.2)and(1.3)are dimensionless but unscaled,so w itlf is of orderα;w x,w t are of orderαβ12;w xx,w xt,
and w tt are of orderαβ,and so on.Contrary to what one might guess,the equations are derived directly from the Eulerian formulation of the water wave problem using an assumption,among others,that the waves travel only in one direction.(It is worth noting that the assumption of unidirectionality is not needed in the derivation for the Lagrangian formulation,as Craig showed in[36,p.799].This configuration is not under consideration here,however.)In conquence,(1.2)and(1.3)are formally comparable to the well-known model put forth by Korteweg and de Vries[45],but also written down by Boussinesq(e[29]).Boussinesq[28]also derived from the Euler equations a system of two coupled equations,
ηt+w x+(wη)x=0,
w t+ηx+ww x+1
3ηxtt=0,
(1.4)
or its regularization(cf.Whitham[61]),
ηt+w x+(wη)x=0,
w t+ηx+ww x−1
3w xxt=0,
(1.5)
which are free of the presumption of unidirectionality that is the hallmark of(1.2),(1.3), and Korteweg–de Vries–type equations as models of surface-wave propagation.One
Boussinesq Equations and Other Systems 285therefore expects that the Boussinesq systems will have more intrinsic interest than the one-way models (an appellation we append to any model derived under the assumption of unidirectionality of propagation)on account of their considerably wider range of potential applicability.Becau of the very rich mathematical theory that obtains for unidirectional models like the Korteweg–de Vries equation,and which for the most part ems to hav
e no counterpart for systems of equations,much of the existing mathematical discussion has been centered around one-way models.However,it is the Boussinesq systems to which the prent paper is devoted.
As with one-way models,there are potentially many different but formally equivalent Boussinesq systems.As explained by Bona and Smith [26](and e also Section 2of the prent script),the plethora of possibilities is owed in the main to the fact that the lower-order relations can be ud systematically to alter the higher-order terms without disturbing the formal level of approximation,and to the considerable choice of dependent variables available for the description of the motion.Despite their formal equivalence as models for small-amplitude long waves,the systems may have rather different mathematical properties.
It is our principal purpo to examine some of the properties of a family of Boussinesq systems of the form
ηt +w x +(wη)x +a w xxx −b ηxxt =0,
w t +ηx +ww x +c ηxxx −d w xxt =0,(1.6)
which are all first-order approximations (in the small parameters αand βintroduced in (1.1))to the Euler equations.We also aim to derive Boussinesq systems of the form ηt −b ηxxt +b 1ηxxxxt =−w x −(ηw)x −a w xxx +b (ηw)xxx − a +b −13
(ηw xx )x −a 1w xxxxx ,w t −d w xxt +d 1w xxxxt =−ηx −c ηxxx −ww x −c (ww x )xx −(ηηxx )x
+(c +d −1)w x w xx +(c +d )ww xxx −c 1ηxxxxx ,
(1.7)
which are cond-order approximations.The derivation of the model systems from the full Euler equations is addresd in Section 2.
The parameters a ,b ,c ,...appearing in (1.6)and (1.7)are not independently speci-fied.As will appear in Section 2,the constants in (1.6)obey the relations
a +
b =12 θ2−13
,c +d =12
(1−θ2)≥0,(1.8)a +b +c +d =13,the last of which follows from the first two,where θ∈[0,1]specifies which horizontal velocity variables w reprents (w =w θis the nondimensional horizontal velocity in the flow corresponding to the physical velocity at height θh where h ,as above,is the
286J.L.Bona,M.Chen,and J.-C.Saut undisturbed depth of the liquid).As will appear prently,the constants in(1.6)ari naturally in the form
a=1
2
θ2−1
3
λ,b=1
2
θ2−1
3
(1−λ),
c=1
2
(1−θ2)µ,d=1
2
(1−θ2)(1−µ),
(1.9)
whereλ,µ∈R are modeling parameters(parameters that do not posss a direct physical interpretation as doesθ).Of cour(1.8)follows from(1.9),but the significance ofλandµwill become apparent in Section2.Similar but more elaborate restrictions apply to the parameters in(1.7).
A few specializations of(1.6)have appeared already in the literature.In addition to the regularized ver
sion(1.5)of the classical Boussinesq system,Kaup[43],Bona and Smith[26],and Bona and Chen[14]have put forward models that have attracted further attention(,[46],[53],[35]for the Kaup system;[62],[57],[58]for the Bona-Smith system,and[33]for the system studied in[14]).The and some other interesting specializations are listed now to give concrete form to the discussion:•Classical Boussinesq system(θ2=1
3
,λarbitrary,µ=0)
ηt+w x+(wη)x=0,
w t+ηx+ww x−1
3w xxt=0;
(1.10)
•Kaup system(θ2=1,λ=1,µarbitrary)
ηt+w x+(wη)x+1
3w xxx=0,
w t+ηx+ww x=0;
(1.11)•Bona–Smith system(θ2=(4−µ)/(2−µ),λ=0,µ<0arbitrary)
ηt+w x+(wη)x−bηxxt=0,
w t+ηx+ww x+cηxxx−b w xxt=0,
(1.12) where,in the notation of(1.6),
b=d=
1−µ
3(2−µ)
>0and c=
µ
3(2−µ)
<0;
•Coupled BBM-system(θ2=2
3
,λ=0,µ=0)
ηt+w x+(wη)x−1
6
ηxxt=0,
w t+ηx+ww x−1
6w xxt=0;
(1.13)
Boussinesq Equations and Other Systems287•Coupled K-dV system(θ2=2,λ=1,µ=1)
ηt+w x+(wη)x+1
6
w xxx=0,
w t+ηx+ww x+1
6ηxxx=0;
(1.14)
•Coupled K-dV–BBM system(θ2=2
3
,λ=1,µ=0)
ηt+w x+(wη)x+1
6
w xxx=0,
w t+ηx+ww x−1
6w xxt=0;
(1.15)
•Coupled BBM–K-dV system(θ2=2
3
,λ=0,µ=1)
ηt+w x+(wη)x−1
6
ηxxt=0,
w t+ηx+ww x+1
6ηxxx=0.
(1.16)
Some preliminary commentary is warranted concerning some of the preceding mod-els.Among other things,the Kaup system was featured in Craig’s comparison[36]with the full Euler equations for two-dimensional water waves.On the other hand,as will appear in Section3,Kaup’s version of Boussinesq’s system is linearly ill-pod for the initial-value problem
η(x,0)=ϕ(x),w(x,0)=ψ(x),(1.17) for x∈R.The system(1.14),while referred to as a coupled K-dV system,is not at all the same as a pair of K-dV equations for the two dependent variablesηand w coupled through mixed nonlinear effects.The operator∂t±∂3x which is characteristic of the K-dV
equation does not appear in its pure form.Nevertheless,becau of the appearance of third-order spatial derivatives in the dispersive term,this appellation ems uful.A further change of dependent variables does render(1.14)into the form of a pair of K-dV equations coupled through nonlinearity.System(1.13),on the other hand,is exactly a pair of nonlinear BBM equations or regulariz
ed long-wave equations(e Benjamin et al.[9])coupled through nonlinear effects.The special ca of the system(1.12)in which the parameters take their limiting valuesθ2=1,b=d=13and c=−13,as µ→−∞was considered by Bona and Smith[26].Notice that if reference is made to the definition of c and d in(1.9),the valueθ=1would imply c=d=0.However,
if one takes the limit asµ→−∞,the combination−1
3(ηxxx+w xxt)remains in the
formal limit.When referred to the size of dependent variablesηand w,this quantity has relative size of orderβ2,as will be apparent in Section2.Conquently,its appearance plays no role at a formal level,though Bona and Smith appended it to gain a uful mathematical advantage.It will be en later that their analysis of(1.12)withµ=−∞may be adapted to the system(1.12)for any value ofµ≤0.Of cour,whenµ=0, the coupled BBM-system(1.13)is recovered.
