Zero-dispersion limit of the short-wave–long-wave
interaction equations
Chi-Kun Lin
a,b,∗,1, Yau-Shu Wong
b,2aDepartment of Mathematics, National Cheng Kung University, Tainan 701, Taiwan bDepartment of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Canada
Received 23 May 2005; revised 6 February 2006 Available online 8 June 2006
Abstract
The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ= o(ε), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.
©2006 Elsevier Inc. All rights reserved.
MSC: 35Q40; 35Q53; 76Y07
Keywords: Zero-dispersion limit; Semiclassical limit; Long wave; Short wave; WKB analysis; Dispersive perturbation;
Quasilinear hyperbolic system; Scattering sound wave
1. Introduction
In this paper we study the behavior of solutions to the water wave interaction equations in the limit ε→ 0+, where the parameter ε is analogous to the Planck constant in quantum mechanics. This system has the form
* Corresponding author. Present address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan.
E-mail address: [email protected] (C.-K. Lin).
1 Work partially supported by NSC93-2115-M-008-029, NSC94-2115-M-006-003 and Natural Sciences and Engineering Research Council of Canada.
2 Work partially supported by Natural Sciences and Engineering Research Council of Canada. 0022-0396/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
εi∂tψε+ ε2 2∂xxψ ε−αψε2− 1+ Vεψε= 0, (1.1) ∂tVε= −λ∂xψε 2 , (1.2)
where the complex-valued function ψε and the real-valued function Vε represent the envelope of the short wave and the amplitude of the long wave, respectively. The two real parameters α and λ are assumed to be positive for convenience. The initial values are given by
ψε(x,0)= ψ0ε(x)= A0(x)exp i εS0(x) , (1.3) Vε(x,0)= V0ε(x), (1.4) where S0 is a function of Hs(R) (Sobolev spaces) for s large enough, and A0 is a function,
polynomial in ε, with coefficients of Sobolev regularity in x. More precisely, we are concerned with the behavior of solutions to (1.1), (1.2) as ε tends to zero with the rapid oscillating initial data for the short wave. The small parameter ε represents the space and time scales introduced in (1.1), (1.2), as well as the typical wave length of oscillations of the initial data. In the special case of Schrödinger equation with vanishing Planck’s constant this is precisely the semiclassical
limit.
Under the assumptions of long wave–short wave resonance, Benney [4] proposed several systems of dispersive equations. One of the systems is given by (1.1), (1.2) which has frequently been used to model for interactions between long and short waves in a variety of physical settings [4,6,10,11,30,31,34]. For example, Djordjevic and Redekopp [11] derived (1.1), (1.2) for α= 0 as a model for the interaction between long gravity waves and capillary waves on the surface of shallow water, in the case when the group velocity of capillary wave coincides with the velocity of the long wave. They pointed out that the physical significance of Eqs. (1.1), (1.2) is such that the dispersion of the short wave is balanced by the nonlinear interaction of the long wave with the short wave, while the evolution of the long wave is driven by the self-interaction of the short wave. When α= 0 this model is integrable by the inverse scattering method [26]. Another example arises from the study of resonant ion–acoustic/Langmuir wave interactions in plasma under the assumption that the ion–sound wave is unidirectional. This system has also been employed to substitute for the Davey–Stewartson system due to the effect of resonance, a phenomenon which occurs when the group velocity of the short waves matches the phase velocity of the long waves [11,31]. Note that Eqs. (1.1), (1.2) can be also served as the simplified version of the Zakharov system for the Langmuir turbulence of the plasma physics [1,29,31]. When
Vε= 0, (1.1) uncouples from (1.2) and it becomes a nonlinear Schrödinger equation which has
been studied by Zakharov and Shabat [35]. Depending on the sign of α, this equation has soliton or decaying oscillatory solutions. If λ= 0, Vεdepends on the space variable x only and (1.1) is
known as cubic nonlinear Schrödinger equation with stationary potential Vε= V0ε(x).
The solvability of (1.1), (1.2) is considered under various settings. When ε= 1, applying the smoothing effects of the free Schrödinger operator, it was shown by Tsutsumi and Hatano [32,33] that the initial value problem is locally and globally well posed in H12(R) when α = 0
and in H12+m(R), m = 1, 2, . . . when α = 0. For the largest space, i.e., H1(R) × H 1
2(R) and the
conservation laws hold, it was proved by Ogawa [28] (see also [2,3,19] and references therein). To investigate the singular limit employing the structure of the nonlinear Schrödinger equation we will transform the system (1.1), (1.2) into a single equation and serve (1.2) as a constraint
[2,28,32,33]. Integrating (1.2) in t and eliminating Vεfrom (1.2), we can rewrite (1.1) and (1.2)
as a single equation for ψε
εi∂tψε+ ε2 2∂xxψ ε= −λ t 0 ∂xψε(x, τ ) 2 dτ ψε+V0ε(x)+ αψε2− 1 ψε, (1.5) ψε(x,0)= ψ0ε(x), x∈ R. (1.6) The first nonlocal or memory term induced by the long wave on the right-hand side of (1.5) causes the so-called derivative loss phenomenon which prevent one to apply the well-known Se-gal nonlinear semigroup theory in a simple manner. This difficulty can be overcome by using the smoothing-effects estimates of solutions of linear Schrödinger evolution equations developing by Kenig et al. [17,18] (see also [2,3]).
The semiclassical or small dispersion problem (i.e., small ε) has been the subject of research in the last 20 years. According to the correspondence principle, the classical world should emerge from the quantum world whenever the Planck constant is negligible. But the limit as the Planck constant tends to zero is mathematically singular. This fact complicates the reduction to classical mechanics. Thus the mathematical rigorous analysis of the semiclassical limit for Schrödinger type equations (or more general dispersive equations) is an issue of importance and full of chal-lenge to mathematical analysis. For linear Schrödinger equation or Schrödinger–Poisson, the idea of kinetic formulation to solve it global-in-time is the followings. By applying the Wigner transform, we can obtain a kinetic integro-differential equation the so-called Wigner equation. The investigation of the kinetic structure of the Wigner equation and the application of the mo-ments methods to its solutions, which provide information of macroscopic densities, help us to pass limit as the Planck constant tends to 0 in the Wigner equation and the macroscopic den-sities. We have the Vlasov (Vlasov–Poisson) equation, which is the quantum (hydrodynamic) limiting system of the linear Schrödinger-type equations [12,13]. The analysis of the limiting system gives us the similar macroscopic densities and results to those obtained by the geometric optics approach to the WKB limit of Schrödinger equations and reveals a close relation between the semiclassical limit of quantum fluid equations and the kinetic equations [13].
However, the situation is quite different for nonlinear Schrödinger-type equations because the theory of Wigner transform and the semiclassical (or zero-dispersion) limit are still under investigation for nonlinear Schrödinger-type equations. Most of the rigorous global analysis of the limiting behavior are restricted to the integrable nonlinear wave equations (see [15,25] and references therein). Thus, any analytical or numerical results related to (1.1), (1.2) should be important in the study of short-wave–long-wave interactions. In particular, the zero-dispersion limit, when one could expect creation of shock waves and interesting limiting dynamics of the conserved quantities. Therefore the study of the semiclassical limit of (1.1), (1.2) will signifi-cantly enhance our understanding of the general semiclassical behavior of nonlinear dispersive waves. Our approach to the dispersive limit of (1.1), (1.2) is the WKB analysis. Since the WKB method has the drawback of being local in time, thus we treat the local smooth solutions only [8, 9,14]. This work about the zero-dispersion limit of the short-wave–long-wave interaction equa-tions was motivated by the a natural mathematical question; however, the problem is also of direct importance to water wave and plasma physics.
The plan of the paper is as follows. In Section 2, we derive the hydrodynamical structure and the local conservation laws of the short-wave–long-wave equations (1.1), (1.2). The formal
dispersive limit is also discussed. We employ the modified Madelung transformation to represent the water wave interaction equations as a perturbation of a quasilinear symmetric hyperbolic system in Section 3. For suitable initial data in the Sobolev space Hs(R) with s sufficiently
large, the classical solutions of (1.1)–(1.4) exist for a time T independent of ε and converge pointwise together with some number of derivatives to a classical solution of the compressible Euler equation with nonlocal potential. In Section 4, we study the zero-dispersion limit of (1.1),
(1.2) directly motivated by the defocussing cubic nonlinear Schrödinger equation [5,25]. In the case when there are no vortices (uniform bounded energy as ε→ 0), we show that the limit of the wave functions solve the wave map equations and the associated phase functions satisfy a linear wave equation which is the same as the defocussing cubic nonlinear Schrödinger equation. This concludes that the long wave plays no role in the zero-dispersion limit.
2. Hydrodynamical structures and conservation laws
The semiclassical limit of Eqs. (1.1), (1.2) is to determine the limiting dynamics of any func-tions of the fields ψε as ε→ 0. However, it is not clear directly from (1.1) what form such
a dynamics might take. Insight into this question can be gained by considering the conserva-tion laws associated with (1.1), (1.2). To this end, we make the geometric optic (semiclassical) ansatz [8,9,14] ψε(x, t )= Aε(x, t )exp i εS ε(x, t ) = ρε(x, t )exp i εS ε(x, t ) . (2.1)
This transformation is usually called the Madelung transformation and was originally introduced in the context of the linear Schrödinger equation for quantum mechanics. The real amplitude Aε, phase function Sεand Vε obey the following equations:
∂tAε+ ∂xAε∂xSε+ 1 2A ε∂ xxSε= 0, (2.2) ∂tSε+ 1 2 ∂xSε 2 +αAε2− 1+ Vε= ε 2 2Aε∂xxA ε, (2.3) ∂tVε+ 2λAε∂xAε= 0, (2.4)
where Eq. (2.3) is the quantum deformation of the Hamilton–Jacobi equation by quantum poten-tial. Consider the new variables
ρε≡Aε2=ψε2, uε≡ ∂xSε, (2.5)
we have the following two local conservation laws
∂tρε+ ∂x ρεuε= 0, (2.6) ∂tuε+ ∂x |uε|2 2 + αρ ε+ Vε =ε2 2 ∂x ∂xx√ρε √ ρε . (2.7)
Equations (2.6), (2.7) comprise a closed system governing ρεand uε, which have the form of a
∂tVε+ λ∂xρε= 0, (2.8) which is equivalent to Vε(x, t )= V0ε(x)− λ t 0 ∂xρε(x, τ ) dτ. (2.8)
This is the local conservation law of mass (or amplitude) of the long wave Vε. From (2.6) and (2.7) we can also derive the equation for the canonical momentum με≡ ρεuε
∂tμε+ ∂x |με|2 ρε + α 2ρ ε2 + ρε∂ xVε= ε2 4 ∂x ρε∂xxlog ρε , (2.9)
which is not conservative because of the coupling. However, employing (2.8), we still have the local conservation law of momentum in the following form
∂t με+ 1 2λV ε2 + ∂x |με|2 ρε + α 2ρ ε2 + ρεVε =ε2 4∂x ρε∂xxlog ρε , (2.10)
where με+2λ1|Vε|2= ρεuε+2λ1|Vε|2is the noncanonical momentum which means that even if the fluid velocity (short wave) vanishes, i.e., uε= 0, the flow still has background momentum caused by the long wave. This implies that solitons of the water wave interaction equation (1.1),
(1.2) have nontrivial static limit. In the field of theoretical language, we can say that the spectrum of excitations has always a gap (like in superfluidity). Besides the principles of conservation of mass and momentum upon which Eqs. (1.1), (1.2) are formulated, the conservation of energy is another principle of great physical and mathematical importance. Define the energy density Eε by Eε= E1ε+ Eε2+ E3ε+ E4ε≡|μ ε|2 2ρε + α 2ρ ε2 + ρεVε+ε2 8 |∂xρε|2 ρε , (2.11)
i.e., the total energy of the short-wave and long-wave interaction equations is constituted by the classical part, Eε1the kinetic energy, Eε2+E3εthe potential energy and the quantum part E4εwhich is of order O(ε2). The crossing term E3ε= ρεVεcomes from the interaction of the short-wave and long-wave. They propagate according to
∂tEε1+ ∂x E1ε· uε+ uε∂x α 2ρ ε2 + ρεuε∂ xVε= ε2 4 u ε∂ x ρε∂x2log ρε, (2.12) ∂tE2ε+ ∂x 2E2ε· uε− uε∂x α 2ρ ε2 = 0, (2.13) ∂tE3ε+ ∂x E3ε· uε− ρεuε∂xVε+ ∂x λ|ρε|2 2 = 0, (2.14) ∂tEε4+ ∂x Eε4· uε+ε 2 4 ∂x ∂xρε∂xμε ρε − με∂xxρε ρε = −ε2 4u ε∂ x ρε∂xxlog ρε . (2.15)
Summing (2.12)–(2.15) yields the energy equation ∂tEε+ ∂x Eε+ E2εμ ε ρε + λ 2ρ ε2 =ε2 4 ∂x με∂xxρε ρε − ∂xρε∂xμε ρε , (2.16)
which is conservative. Therefore we obtain the quantum hydrodynamics equations of (1.1), (1.2) (see [12,22,23] for the similar models). The above hydrodynamical structures imply the local conservation laws of the water wave interaction equations (1.1), (1.2).
Theorem 2.1. Let bar¯ denote the complex conjugate and t ∈ [0, ∞). The following quantities
are conservation integrals of (1.1), (1.2) ∞ −∞ ρε(x, t ) dx= C1, (2.17) ∞ −∞ uε(x, t ) dx= C2, (2.18) ∞ −∞ Vε(x, t ) dx= C3, (2.19) ∞ −∞ με(x, t )+ 1 2λV ε(x, t )2 dx= C4, (2.20) ∞ −∞ Eε(x, t ) dx= C5, (2.21)
where the hydrodynamic variables ρε, uε, με, Vε and Eε are given in terms of the envelope of the short wave function ψεas follows
ρε(x, t )=ψε(x, t )2= ψε(x, t ) ¯ψε(x, t ), (2.22) uε(x, t )=iε 2 ∂x¯ψε(x, t ) ¯ψε(x, t ) − ∂xψε(x, t ) ψε(x, t ) , (2.23) με(x, t )=iε 2 ψε(x, t )∂x ¯ψε(x, t )− ¯ψε(x, t )∂xψε(x, t ) , (2.24) Vε(x, t )= V0ε(x)− λ t 0 ∂xψε(x, τ ) 2 dτ, (2.25) Eε(x, t )=ε 2 2 ∂xψ ε(x, t )2 +α 2ψ ε(x, t )4 + Vε(x, t )ψε(x, t )2 . (2.26)
The conservative quantities of the water wave interaction equations may be recast from the action principle. The Lagrangian formulation allows us to systematically derive conserved quan-tities by means of Noether’s theorem which assigns to each of these symmetries a corresponding conserved quantities by taking the form of the integral of a multinomial in ψε, Vε and their x-derivatives. Equations (1.1), (1.2) are trivially invariant under the phase rotation
ψε(x, t )→ eiθψε(x, t ), θ∈ R, and have C1= R ψε(x, t )2dx= R ρε(x, t ) dx
as the corresponding constant of the motion. The equations have no explicit dependence on x and hence
ψε(x, t )→ ψε(x− x, t), Vε(x, t )→ Vε(x− x, t)
must be symmetry. This space translation invariant gives C4 as its conserved quantity, which
can be thought of as the momentum of the equations. Similarly, the equations have no explicit dependence on t so
ψε(x, t )→ ψε(x, t− t), Vε(x, t )→ Vε(x, t− t)
must also be symmetry. The corresponding conserved quantity is the Hamiltonian or the total energy C5.
Remark. The Madelung formulation relies on the assumption that the amplitude of ψεis not zero
and the phase Sεis not singular, otherwise the transformation is not well-defined and the system (2.6), (2.7) becomes singular even though (1.1), (1.2) is still regular. Therefore, we treat only the regime of smooth phase functions. However, the conservation laws do not rely on the Madelung transformation, we can still derive the same result from (1.1), (1.2) directly. In contrast to all earlier applications of the Madelung transformation, we can avoid making explicit use of the phase function Sεand do not work with (2.6), (2.7). By the three conservation laws, mass (2.17), momentum (2.20) and energy (2.21), Ogawa [28] proves the global well-posedness of the system (1.1)–(1.4) for ε= 1 in the largest class of initial data.
In the formal semiclassical limit ε→ 0, one neglects the contribution from the quantum po-tential ∂xx(√ρε)/√ρε in (2.6)–(2.8), and the limiting densities ρ, μ= ρu and V satisfy the
Euler system ∂tρ+ ∂x(ρu)= 0, (2.27) ∂tμ+ ∂x |μ|2 ρ + α 2ρ 2 + ρ∂xV = 0, (2.28)
with initial conditions inferred from (1.3) given by
where V is formally given by V (x, t )= V0(x)− λ t 0 ∂xρ(x, τ ) dτ. (2.30)
Here V0(x)is the limit of V0ε(x). If λ= 0 then the potential V is stationary, V (x, t) = V0(x).
These are the classical Euler equations of the compressible fluid. This argument is self-consistent only if the solution of (2.22)–(2.25) remains classical (i.e., before the development of the first shock). In that case the limiting energy density will be given by
E=|μ| 2 2ρ + α 2|ρ| 2+ ρV (2.31) and will satisfy
∂tE+ ∂x E+ E2(ρ) μ ρ + λ 2|ρ| 2 = 0, (2.32)
where E2(ρ)=α2ρ2plays the role similar to the pressure.
It is clear from (2.2)–(2.4) that the formal dispersionless limit equations associated with (1.1),
(1.2) are given by ∂tA+ ∂xA∂xS+ 1 2A∂xxS= 0, (2.33) ∂tS+ 1 2∂xS 2+αA2− 1+ V= 0, (2.34) ∂tV + λ∂x A2= 0, (2.35)
where (A, S, V ) is the formal limit of (Aε, Sε, Vε)of (2.2)–(2.4). Note that Eq. (2.34) for the phase S is a classical Hamilton–Jacobi equation for the action of a particle with respect to the potential α(A2− 1) + V . Introducing the new complex wave function
ϕ(x, t )= A(x, t) expiS(x, t ), (2.36) system (2.33)–(2.35) is equivalent to the following modification of the water wave interaction equations (1.1), (1.2): i∂tϕ+ 1 2∂xxϕ− α|ϕ|2− 1+ Vϕ=1 2 ∂xx|ϕ| |ϕ| ϕ, (2.37) ∂tV = −λ∂x |ϕ|2, (2.38) with the quantum potential on the right-hand side of (2.37). The quantum potential contribu-tion to the right-hand side with fixed strength completely compensates U (1) gauge invariant contribution to dispersion on the left-hand side. This potential, the so-called Bohm potential or the internal self-potential was introduced by deBroglie and later explored by Bohm to make a hidden-variable theory, is responsible for producing the quantum behavior, so that all quantum features are related to its special properties. The role of the quantum potential is to change the
dispersion of the Schrödinger equation. If the strength of the quantum potential deviates from the critical value as given in dispersionless equations (2.37), (2.38), then we have the deformed wave equations i∂tϕε+ 1 2∂xxϕ ε−αϕε2− 1+ Vεϕε=1+ ε21 2 ∂xx|ϕε| |ϕε| ϕ ε, (2.39) ∂tVε= −λ∂xϕε2 (2.40) of the dispersionless system (2.37), (2.38), which is determined by the deformation parameter ε (the Planck constant) and the semiclassical ansatz
ϕε(x, t )= A(x, t) expiS(x, t )/ε. (2.41) Moreover, for the classically inaccessible regions simulated by analytical continuation of the Planck constant to a pure imaginary value ε→ iε, instead of (2.39), (2.40), we have
i∂tϕε+ 1 2∂xxϕ ε−αϕε2− 1+ Vεϕε=1− ε21 2 ∂xx|ϕε| |ϕε| ϕ ε, (2.42) ∂tVε= −λ∂xϕε2 . (2.43)
Furthermore, written in terms of the two real-valued functions
Q±(x, t )= A(x, t) exp±S(x, t)/ε= ρ(x, t )exp±S(x, t)/ε, (2.44) the diffusion equations in duality analog of (1.1), (1.2)
ε∂tQ±± ε2 2∂xxQ ±+αQ+Q−− 1+ VQ±= 0, (2.45) ∂tV = −λ∂x Q+Q− (2.46)
are derived. Thus system (1.1), (1.2) is intrinsically in the theory of diffusion processes as an equation in the context of time reversal of diffusion processes, namely, diffusion equations in duality. The origin of the idea of considering diffusion process for quantum mechanics goes back to Schrödinger (1931), in which he formulated Brownian motions in a symmetric form of time reversal. Schrödinger’s time-symmetric theory of diffusion process revealed the deep relation between diffusion theory and quantum theory. We can further represent this system as the decoupled pair of Burgers’ equations by the well known Hopf–Cole transformation which suggests to introduce the pair of velocity fields
u+= ε∂x log Q+= ε∂xQ + Q+ , u −= −ε∂ x log Q−= −ε∂xQ − Q− , (2.47)
such that instead of (2.38), (2.39), we have the coupled system of Burgers’ equations with nega-tive and posinega-tive viscosities
∂tu++ u+∂xu+= − ε 2∂xxu +− ∂ xΩ, (2.48) ∂tu−+ u−∂xu−= ε 2∂xxu −+ ∂ xΩ, (2.49) ∂tV = −λ∂x Q+Q−, (2.50)
with the potential function given by
Ω= αQ+Q−− 1+ V. (2.51) These relative velocities
u±= u ± u∗= ∂xS± ε
2
∂xρ
ρ (2.52)
characterize two motions, the center of mass motion with velocity u= ∂xSand internal
oscilla-tions in the envelope with velocity u∗=2ε∂xρ ρ .
3. Modified Madelung transformation
In this section, we will employ the modified Madelung transformation to transform (1.1), (1.2) into a linear dispersive perturbation of a quasilinear symmetric hyperbolic system to which the Lax–Friedrich–Kato theory can be applied [16,27]. Equations (1.1), (1.2) or (2.6)–(2.8) do not have the explicit form of a first-order hyperbolic system for the variables (ρε, uε, Vε). However, it can be overcome by serving Vεas a forcing term given by (2.8). The limit ε→ 0 cannot be made directly in (2.6)–(2.8) since the phase Sε or the quotients 1/√ρε may be undefined. As
suggested by Grenier [14] (see also [8,9,20–22,24]), the modified Madelung transformation can be utilized in the study of the semiclassical limit. The similar idea had also been used earlier by Schochet and Weinstein to study the nonlinear Schrödinger limit of the Zakharov system [29]. Indeed, we will look for solution ψεof the form
ψε= AεexpSε/ε, Aε= aε+ ibε. (3.1) Note here we allow the phase function Sε to depend on the parameter ε. Now inserting (3.1)
into (1.1), we obtain εi∂tAε− Aε∂tSε+ ε2 2 ∂xxA ε+ εi∂ xAε∂xSε− 1 2A ε∂ xSε 2 +εi 2A ε∂ xxSε =αAε2− α + VεAε,
it can then split into
∂tAε+ ∂xAε∂xSε+ 1 2A ε∂ xxSε =ε 2i∂xxA ε and (3.2) ∂tSε+ 1 2 ∂xSε 2 +αAε2− α + Vε= 0. (3.3)
Considering the change of variables vε≡ ∂
xSε, we have the equivalent form of (1.1)–(1.4)
∂taε+ vε∂xaε+ 1 2a ε∂ xvε= − ε 2∂xxb ε, (3.4) ∂tbε+ vε∂xbε+ 1 2b ε∂ xvε= ε 2∂xxa ε, (3.5) ∂tvε+ vε∂xvε+ 2α aε∂xaε+ bε∂xbε + ∂xVε= 0, (3.6) aε(x,0)= a0ε(x), bε(x,0)= bε0(x), vε(x,0)= v0ε(x)= ∂xSε(x,0), (3.7)
where, according to (1.2), the potential Vεis given explicitly by
Vε(x, t )= V0ε(x)− 2λ t 0 aε∂xaε+ bε∂xbε dτ. (3.8)
The system can be rewritten in the vector form:
∂tUε+ A Uε∂xUε+ Gε= ε 2L Uε, (3.9) Uε(x,0)= U0ε(x)=a0ε(x), bε0(x), v0ε(x)t, (3.10) where Uε= (aε, bε, vε)t, Gε= (0, 0, ∂xVε)t, AUε= ⎡ ⎣ v ε 0 aε/2 0 vε bε/2 2αaε 2αbε vε ⎤ ⎦ , L= ⎡ ⎣∂0xx −∂0xx 00 0 0 0 ⎤ ⎦ . (3.11) Obviously the matrix A(Uε)can be symmetrized by
S= ⎡ ⎣4α0 4α0 00 0 0 1 ⎤ ⎦ ,
which is symmetric and positive definite for α > 0. The antisymmetric operator L in (3.11) reflects the dispersive nature of Eqs. (1.1), (1.2). The special structure of (3.9) will be exploited on the classical solutions. The existence of classical solutions proceeds along the lines of the existence proof for the initial value problem for the quasilinear symmetric hyperbolic system with modification. Indeed, applying the theory of the quasilinear symmetric hyperbolic system, we will obtain the existence of smooth solutions (ψε, Vε)of (1.1), (1.2) on a time interval[0, T ) independent of ε. Furthermore, the bounds that we obtained are uniformly bounded in ε on the solution (ψε, Vε)will allow to pass to the limit ε→ 0 in (3.4)–(3.6) and this justify the WKB hierarchy.
In addition to the linear dispersive perturbation of the quasilinear symmetric hyperbolic sys-tem nature, the modified Madelung transformation also give us more information about the phase transportation. Since Aεis complex-valued, we introduce the polar coordinates:
Applying the chain rule, we obtain aε∂xxbε− bε∂xxaε= ∂x ρε∂xθε , (3.13)
then from (3.4)–(3.6) we derive the system
∂tρε+ ∂x ρεvε+ ερε∂xθε = 0, (3.14) ∂tθε+ vε∂xθε+ ε 2∂xθ ε2 =ε 2 ∂xx(√ρε) √ ρε , (3.15) ∂tvε+ vε∂xvε+ ∂x αρε+ Vε= 0, (3.16) where Vεis given by Vε(x, t )= V0ε(x)− λ t 0 ∂xρε(x, τ ) dτ. (3.17)
The quantum effect in this system is of order O(ε) not O(ε2)comparing with (2.7). Note the transport equation for ρεhas an extra term of order O(ε) comparing with the typical equation of continuity. By formally letting ε→ 0, we have
∂tρ+ ∂x(ρv)= 0, (3.18) ∂tθ+ v∂xθ= 0, (3.19) ∂tv+ v∂xv+ ∂x(αρ+ V ) = 0, (3.20) V (x, t )+ λ t 0 ∂xρ(x, τ ) dτ= V0(x). (3.21)
Since (3.19) is the pure transport equation then θ= 0 for the trivial initial data, thus we have the same limit system as (2.22)–(2.24).
We now first establish the existence and uniqueness of the classical solution of the dispersive perturbation of the quasilinear symmetric hyperbolic system (3.8)–(3.11).
Theorem 3.1. Let s >52. Suppose M0 1, M and T are given such that
M0+ cM02+ MT ecM0T 2M 0, (3.22) then
(i) if Gε ∈ L∞([0, T ]; Hs(R)) ∩ C([0, T ]; Hs−2(R)) such that Gε
Hs M and the ini-tial data U0ε= (aε0, bε0, v0ε)∈ Hs(R) × Hs(R) × Hs(R) satisfying Uε
0 Hs(R) M0 are given, then the IVP for the (3.8), (3.9) has a unique solution Uε ∈ C([0, T ]; Hs(R)) ∩ C1([0, T ]; Hs−2(R)) such that Uε
(ii) if Uε= Uε (0), U
ε
(1)are the solutions corresponding with Gε= G
ε (0), G
ε
(1)for the same initial condition U0εsatisfying condition (i), then
U(ε0)− U(ε1)Hs−2 ce
cM0TTGε (0)− G
ε
(1)Hs−2; (3.23)
(iii) if ρ0ε(x)= (aε0)2+ (b0ε)2>0, then ρε(x, t ) >0 for all t 0; if ρ0ε has a compact support, then ρε(·, t) does too for any t ∈ [0, T ] and
Rρε(·, t) Rρ0ε+ (1 + ε)CT , (3.24)
where R{u} ≡ sup{|x|: u(x) = 0}.
Proof. The existence of a solution in a sufficiently short time interval is guaranteed by
[16, Theorem II]. It suffices only to find the explicit estimates stated in the theorem. The fol-lowing procedure is attributed to [16,27] (see also [8,9,14,22]).
(i) For further reference, we ignore the superscripts ε. Given
U∈ L∞[0, T ]; Hs(R)∩ C[0, T ]; Hs−2(R),
such that U Hs 2M0, the linear problem ∂tU+ A(U)∂xU+ G =
ε
2L(U ), U (x, 0)= U
ε
0(x) (3.25)
has a unique solution U∈ C([0, T ]; Hs(R)) ∩ C1([0, T ]; Hs−2(R)). Multiplying (3.25) by the
matrix S then taking the inner product with Uand integrating overR yields
d dt U E= R Ax(U )U , U dx+ R SGU , U dx + ε R SL(U )U , Udx, where U E ≡
R S U , U dx is the canonical energy associated with (3.25). The term
R SL(U )U , U dx = 0 contributes nothing to the estimate, by the antisymmetry of L. This
means that the singular perturbation does not create energy. We assume that the matrix A together with its derivatives of any desired order are continuous and bounded uniformly in[0, T ] × R. Since ∂xA(U ) ∞ ∂xA(U ) L2 C1M0and S ∞ c2≡ max{1, α}, we have
max 0tT U (t )L2 U0εL2+ T 0 SG dt eC1M0T (M 0+ c2MT )eC1M0T.
The estimates of the higher derivatives of U can be obtained in the same manner. We write U(ν)= ∂xνU, ν 1. Then U(ν)∈ C[0, T ]; L2(R)∩ C1[0, T ]; H−2(R), satisfies ∂tU(ν)+ A(U)∂xU(ν)+ G(ν)= ε 2LU (ν), (3.26)
where G(ν)= ∂xνG−∂xνA(U )∂xU −A(U )∂xν∂xU = ∂ν xG− ∂xν, A(U ) ∂xU . (3.27)
The commutator[∂xν, A(U )]∂xUconsists of terms of the form ∂αAx· ∂β∂xU , α + β ν − 1 s− 2. Since ∂xA, ∂xU∈ Hs(R), we can apply the Moser-type calculus inequality [8,16,27] to
estimate the commutator terms:
∂α∂xA· ∂β∂xUL2 C ∂xA Hs ∂xU Hs CM02, (3.28)
provided that U Hs 2M0.(Note that A(U) Hs C U Hs.) Thus we have
G(ν)L2∂xνGL2+ C2M02, (3.29)
as long as U Hs 2M0. This implies
U Hs M0+C3M2 0+ M
TeC3M0T 2M
0, (3.30)
for 0 t T provided that T is sufficiently small that the last inequality holds. Moreover, from Eq. (3.9) we have the estimate of the time derivative ∂tU
max 0tT ∂t U Hs−2=−A(U)∂xU+ ε 2L(U ) Hs−2 C4 M02+ M = L. (3.31)
Note that unlike solutions to the quasilinear hyperbolic system considered in [7,16,27], ∂tUε
is only Hs−2and not Hs−1due to the presence of the higher order term L(Uε)in (3.25). It is interesting to mention that the function space C([0, T ]; Hs(R))∩C1([0, T ]; Hs−2(R)) is natural
from the point of view of dimensional analysis. Both function spaces have the same dimension
2
∞+n2− s = 2
∞− 2 +n2− (s − 2). Here we use the fact that the time dimension is 2 other than 1
comparing with the pure quasilinear symmetric hyperbolic system. Now we consider the fundamental set
X=U∈ L∞[0, T ]; Hs(R)∩ C[0, T ]; Hs−2(R): U Hs 2M0,
U (t1)− U(t0)Hs−2 L|t1− t0|
,
and the mapping F : U→ U. Thus we will show the contraction in the lower norm. This is the seminal ideal of Lax and Kato. We have already shown that F maps X into X itself. To apply the fixed point theorem, we make X into a complete metric space with the metric
d(U(1), U(0))= U(1)− U(0) ≡ sup 0tT
U(1)(t )− U(0)(t )L2. (3.32)
We are going to show that F is a contraction if T is sufficiently small. The perturbation δ U=
U(1)− U(0)solves ∂t+ A(U(0))∂x δ U= f +ε 2L(δ U ), δ U (x,0)= 0, (3.33)
where
f=A(U(1))− A(U(0))
∂xU(1). (3.34)
Keeping in mind that A(U ) is linear in U , we see that
f L2 C5 δU U(1) 2C5M0 δU .
Thus | δU | 2C5M0 δU T eC3M0T. Therefore, if 2C5M0T eC3M0T <1, then F is a
con-traction, so has a fixed point in X, which belongs to the required space C([0, T ]; Hs(R)) ∩ C1([0, T ]; Hs−2(R)) and solves (3.8), (3.9). This complete the proof of (i).
(ii) The equation satisfied by the perturbation δU= U(1)− U(0)is
∂t+ A(U(0))∂x+ B(∂xU(1)) δU+ δG = ε 2L(δU ), δU (x,0)= 0, (3.35) where δG= G(1)− G(0)and B(∂xU(1))= A(U(1))− A(U(0)) U(1)− U(0) ∂xU(1)= A (1− θ)U(1)+ θU(0) ∂xU(1). (3.36)
Keeping in mind that A(U(0))∈ C([0, T ]; Hs(R)) with A(U(0)) Hs C6M0, and B(∂xU(1))∈ C([0, T ]; Hs−1(R)) with B(∂xU(1)) Hs−1 C6M0. Let ν s. Then δU(ν)= ∂xνδUsolves
(∂t+ A∂x+ B)δU(ν)= h(ν)+ ε 2L δU(ν), (3.37) where h(ν)= ∂xνδG−∂xν, A ∂x(δU )− ∂xν, B δU. (3.38) Here the bracket[·,·] denotes the commutator. For ν 1, [∂xν, A]∂x(δU )consists of terms of the
form ∂α
xAx· ∂xβ∂x(δU ), α+ β s − 1 with
∂xαAx· ∂xβ∂x(δU )L2 C Ax Hs δU Hs CM0 δU Hs, (3.39)
and[∂ν
x, B]δU consists of terms of the form ∂xαBx· ∂xβδU, α+ β s − 1 with
∂xαBx· ∂xβδUL2 C A Hs δU Hs CM0 δU Hs. (3.40)
Thus h(ν)L2∂ ν xδG +C7M0 δU Hs. (3.41) This implies δU Hs C8 δG HsT eC8M0T, (3.42)
which completes the proof of (ii).
(iii) To show that ρε(x, t )= (aε(x, t ))2+ (bε(x, t ))2>0 for all 0 t < ∞, we employ the
Since vε+ εθε
x ∈ C1(R × [0, T ]), the well-known theorem for ordinary differential equations
guarantees that the problem
dx dt = v
ε(x, t )+ ε∂
xθε(x, t ), x|t=τ= η, (3.43)
has a unique solution x= Ψ (t) ∈ C1([0, T ]; R). The continuity equation implies d dtρ εΨ (t ), t= −∂ x vε+ ε∂xθε ρεΨ (t ), t.
Integrating over[0, τ], we have
ρε(η, τ )= ρεΨ (0), 0exp − τ 0 ∂x vεΨ (t ), t+ ε∂xθε Ψ (t ), tdt .
Thus ρε(η, τ ) 0, if ρε(Ψ (0), 0)= ρ0ε(Ψ (0)) 0. If ρε(η, τ )= 0, then ρ0ε(Ψ (0))= 0 so that |Ψ (0)| R{ρε 0}, and |η| =Ψ (τ ) = Ψ (0)+ τ 0 vεΨ (t ), t+ ε∇θεΨ (t ), tdt Ψ (0) + τ 0 vε∞+ ε∇θε∞dt Rρ0ε+ (1 + ε)C2τ.
In order to complete the proof of the theorem, we only need to show that Gε∈ L∞([0, T ]; Hs(R)) ∩ C([0, T ]; Hs−2(R)) such that Gε Hs M, which is equivalent to show Vε∈ L∞([0, T ]; Hs+1(R)) ∩ C([0, T ]; Hs−1(R)). Indeed, it follows immediately from the
con-servation laws (2.19), (2.20) and the assumption λ > 0 that Vε∈ L∞([0, T ]; L1∩ L2(R)) if V0ε(x)∈ L1∩ L2(R). Similarly for higher derivative we have Vε∈ L∞([0, T ]; Ws,1∩ Hs(R)) if V0ε(x)∈ Ws,1∩ Hs(R). However, the assumption λ > 0 can be overcome by employing the
ex-plicit representation (2.8)or (3.8) of Vε. Indeed, by Cauchy–Schwarz and Minkowski’s integral inequalities and the imbedding H1→ L2we have
Vε(t )L2V0εL2+ 2|λ| R t 0 aε∂xaε+ bε∂xbε dτ 2 dx 1/2 V0εL2+ 2|λ| t 0 R aε∂xaε+ bε∂xbε 2 dx 1/2 dτ V0ε L2+ 2|λ|Taε 2 H1+bε 2 H1 .
Similarly, for higher derivatives Vε(t )Hs V ε 0Hs+ 2|λ|Ta ε2 Hs+1+b ε2 Hs+1 .
The same computation also shows that Vεsatisfies for any 0 t1< t2 T ,
Vε(t2)− Vε(t1)L2 |λ|C t2 t1 aε(τ )2H1+b ε(τ )2 H1dτ and Vε(t2)− Vε(t1)Hs−1 |λ| t2 t1 aε(τ )2 Hs +aε(τ ) 2 Hsdτ.
Since ρε∈ C([0, T ]; Hs(R))∩C1([0, T ]; Hs−2(R)), thus the above inequality implies that Vε∈
Lip([0, T ]; Hs−1(R)). Indeed, we can prove that Vε∈ C1([0, T ]; Hs−1(R)). 2
Theorem 3.2. Assume Aε0, S0 and V0ε in Hs(R), s > 5/2 then solutions (ψε, Vε) of the
(1.1)–(1.4) exist on a small time interval [0, T ], T independent of ε. Moreover, ψε(x, t )= Aε(x, t )eiSε(x,t )/ε, with Aε, Sεand Vεin L∞([0, T ]; Hs) uniformly in ε, and (ρε, Sxε, Vε), with
√
ρε= Aε, converges to the solution (ρ, u, V ) of (2.22)–(2.25).
Proof. Since Aε= aε+ ibεand vε= ∂xSε, it follows from the theorem that
Aε∈ C[0, T ]; Hs(R)∩ C1[0, T ]; Hs−2(R), Sε∈ C[0, T ]; Hs+1(R)∩ C1[0, T ]; Hs(R),
and thus
Aε∈ C1[0, T ] × R∩ C1[0, T ]; C2(R), Sε∈ C1[0, T ]; C2(R).
For classical solutions, (1.1)–(1.4) is equivalent to the dispersive quasilinear hyperbolic sys-tem (3.4)–(3.8). Applying this equivalent relation, the theorem follows immediately by Theo-rem 3.1. 2
The limiting system of (3.4)–(3.8) or (3.9), (3.10) is the quasilinear symmetric hyperbolic system (formally letting ε→ 0)
∂tU+ A(U)∂xU+ G = 0, U (x,0)= U0(x), (3.44) V (x, t )+ 2λ t 0 (a∂xa+ b∂xb) dτ= V0(x), (3.45)
which is equivalent to (2.27)–(2.30) as long as the solutions are smooth. As an immediate conse-quence, we also prove the existence and uniqueness of the local smooth solutions to the system (2.27)–(2.30).
Theorem 3.3. Assume the hypothesis of Theorem 3.1. Given initial datum U0ε, U0∈ Hs(R) and U0ε(x) converges to U0(x) in Hs(R) as ε → 0. Let [0, T ] be the fixed interval determined in Theorem 3.1. Then as ε→ 0, there exists U ∈ L∞([0, T ]; Hs(R)) and V ∈ C1([0, T ]; Hs−1(R)) such that for all σ > 0
Uε→ U in C[0, T ]; Hs−σ(R), (3.46)
Vε→ V in C1[0, T ]; Hs−σ −1(R). (3.47)
The function U (x, t) belongs to C([0, T ]; Hs(R)) ∩ C1([0, T ]; Hs−1(R)) and is a classical so-lution of (2.27)–(2.30).
Proof. Since Uεis bounded in C([0, T ]; Hs(R)) ∩ C1([0, T ]; Hs−2(R)), by the Arzela–Ascoli
theorem (applied in the time variable), the Rellich compactness theorem (applied in the space variable) and interpolation, we have that for every sequence of ε’s tending to 0,{Uε}ε has a
subsequence that converges in C([0, T ]; Hlocs−σ(R)) for σ > 0, to U. Furthermore, from (3.25) the
convergence takes place as well in C1([0, T ]; Hlocs−2−σ(R)). Since U0ε(x)converges strongly to
U0(x)in Hs(R), this limiting solution has initial data U0(x). Also L(Uε)is uniformly bounded
in Hs(R) therefore the perturbation ε2L(Uε)tends to zero as ε→ 0. Thus the sequence {Uε} converges to a solution of the quasilinear hyperbolic system (3.44). Also, after extraction of a subsequence, the above limit converges weakly in Hs(R). Therefore, by the identity of weak
and strong limits, Uε∈ L∞([0, T ]; Hs(R)) ∩ AC([0, T ]; Hs−2(R)). Since the system admits
a unique solution, it then follows that the convergence to U takes place without passing to the subsequence. 2
To ensure the strong convergence of (ψε, Vε)to a classical solution of the (3.44), (3.45) (or equivalently (2.27)–(2.30)), we require the hypothesis that the solution sequences are near the system (2.27)–(2.30) initially. It means that the regularity of solutions of (2.27)–(2.30) controls that of solutions to (1.1)–(1.4).
Theorem 3.4. Let T > 0 be arbitrary and (ρ0, μ0, V0) be such that the IVP of (2.27)–(2.30) has
a classical solution (ρ, μ, V ). Then, there is a critical value of ε, εcand a constant C > 0 such that under the hypotheses:
(1) Aε0and V0ε converge strongly respectively to A0and V0in Hs(R) and Hs−1(R) as ε tends to 0,
(2) ρ0 Hs < C, μ0 Hs < C and V0 Hs−1< C, s >3,
(3) 0 < ε < εc,
the IVP for (1.1)–(1.4) has a unique classical solution (ψε, Vε) on [0, T ] with ψε(x, t )= Aε(x, t )exp(iεSε(x, t )). Moreover, Aε and Sxε are bounded in L∞([0, T ]; Hs(R)) and Vε is bounded in C1([0, T ]; Hs−1(R)) uniformly in ε.
The proof is standard by considering the difference of the two systems and then applying the energy estimate [8,27,29]. We therefore omit the details.
4. Weak solutions and scattering sound
In this section, we consider the semiclassical (WKB) limit of (1.1) directly. For the cubic nonlinear Schrödinger equation, it has been studied by Colin and Soyeur [5] for the case when there are no vortices, and by Lin and Xin [25] when there are vortices in two space dimensions. For the derivative nonlinear Schrödinger equation, we will refer to [9,24].
We multiply (1.1) by ¯ψε and its complex conjugate by ψε, and subtract the latter from the former to obtain the conservation of mass in terms of the wave function:
∂tψε 2 + ∂x iε 2 ψε∂x¯ψε− ¯ψε∂xψε = 0, (4.1)
which is the same as (2.6). We rewrite (4.1) as
∂t |ψε|2− 1 ε + ∂xW ψε= 0, (4.2)
where the linear momentum W is defined by
Wψε= Wψε, ∂xψε ≡ i 2 ψε∂x¯ψε− ¯ψε∂xψε . (4.3)
In the sequel, we assume λ > 0 satisfying λ= o(ε) and the initial data are taken in such way that V0ε ε ∈ L 1(R) ∩ L2(R), Wψε 0 ∈ L1(R), (4.4) ψ0ε∈ H1(R), |ψ ε 0|2− 1 ε ∈ L 2(R), (4.5) then the conservation laws (2.17)–(2.21) imply
∞ −∞ Vε ε dx C0, ∞ −∞ Wψε+ 1 2λεV ε2dx C 1, (4.6) ∞ −∞ 1 2∂xψ ε2 +α 2 |ψε|2− 1 ε 2 + 1 ε2ψ ε2 Vεdx C2, (4.7)
where C0, C1and C2are constants independent of ε. The above bounds imply
ψε is bounded in L∞[0, T ]; H1(R), (4.8) ∂tψε is bounded in L∞ [0, T ]; H−1(R), (4.9) |ψε|2− 1 ε is bounded in L ∞[0, T ]; L2(R), (4.10) Wψε is bounded in L∞[0, T ]; L1(R), (4.11)
for the short wave part and similarly for the long wave part we have Vε ε is bounded in L ∞[0, T ]; L2(R) ∩ L1(R), (4.12) ∂t Vε ε is bounded in L∞[0, T ]; H−1(R) + W−1,∞(R), (4.13)
and thus by interpolation
Vε ε is bounded in L ∞[0, T ]; Lp(R), (4.14) ∂t Vε ε is bounded in L∞[0, T ]; W−1,q(R), (4.15)
for 1 p 2,p1+1q= 1. Although it holds for all 1 p 2, but p = 2 is good enough for our discussion. Moreover, from (4.10)
ψε2→ 1 strongly in L2(R) and a.e. (4.16) It follows from these bounds that{ψε}ε is strongly compact in C([0, T ]; L2(R)) and weakly
compact in L∞([0, T ]; H1(R)), and from the classical compactness arguments there exists a
subsequence still denoted{ψε}ε and a function ψ∈ L∞([0, T ]; H1(R)) such that
ψε→ ψ strongly in C[0, T ]; L2(R), (4.17)
ψε ψ weakly∗in L∞[0, T ]; H1(R). (4.18) Similarly, there exists V∈ L∞([0, T ]; L2(R)) such that
Vε ε → V strongly in C [0, T ]; H−η(R), 0 < η < 1, (4.19) Vε ε V weakly ∗in L∞[0, T ]; L2(R). (4.20) We claim that ψε Vεε converges to ψV in D((0, T ) × R). It is easily seen that ψε ∈
C([0, T ]; L2(R)) implies that ψε∈ L2([0, T ] × (−M, M)) for any M > 0. Indeed, if g is in D((0, T )× R) with compact support Ω, ψε converges strongly in L2(Ω)and since Vε
ε
con-verges weakly to V in L2([0, T ]; L2(Ω)), then lim ε→0 Ω ψε(x, t )V ε(x, t ) ε g(x, t ) dx dt= Ω ψ (x, t )V (x, t )g(x, t ) dx dt, (4.21)
which proves the claim. Also from (4.2), (4.5) and (4.17), we have |ψε(x, t )|2− 1 ε = |ψε 0(x)|2− 1 ε − t 0 ∂xW ψεdτ − t 0 ∂xW (ψ ) dτ, (4.22)
in the sense of distributions. We rewrite (1.2) as ∂t Vε ε +λ ε∂xψ ε2= 0, (4.23)
then using the fact that λ= o(ε) and |ψε|2→ 1 a.e., we can conclude that ∂tV = 0 in the sense
of distribution thus V = lim ε→0 Vε(x, t ) ε = limε→0 V0ε(x) ε = V0(x). (4.24)
Therefore, using the above compactness results, λ= o(ε) and the fact that |ψε|2→ 1 a.e. again, as ε tends to zero in (1.1), we have
i∂tψ+ αψ t
0
∂xW (ψ ) dτ− V0(x)ψ= 0, (4.25)
in the sense of distribution. Since|ψ| = 1, ¯ψ∂xψ+ ψ∂x¯ψ = 0 and W (ψ )= i 2(ψ ∂x¯ψ − ¯ψ∂xψ )= −i ∂xψ ψ = −i∂x(log ψ), (4.25) becomes i∂tψ− iα t 0 ∂xx(log ψ) dτ+ V0(x) ψ= 0. (4.26)
Differentiating (4.26) with respect to t once, we can then derive the wave map equation
∂t tψ− α∂xxψ=
α|∂xψ|2− |∂tψ|2
ψ, |ψ| = 1 a.e. (4.27) with initial data
ψ (x,0)= ψ0(x), i∂tψ (x,0)= V0(x)ψ0(x). (4.28)
Using the fact|ψ| = 1 again, writing ψ = eiθ shows
∂t tθ− α∂xxθ= 0, D
[0, T ] × R, (4.29) i.e., θ is a distribution solution of the linear wave equation. Moreover, θ (x, t)∈ H1(R) implies
that θ is the unique weak solution of (4.28) with finite energy. Thus we have proved the following theorem.
Theorem 4.1. Let the positive parameter λ be small order of ε, λ= o(ε) and (4.4), (4.5)
be satisfied uniformly in ε. Assume ψ0ε → ψ0 in L2(R), |ψ0| = 1 a.e. and V0ε/ε→ V0 in L2(R), then denoting by (ψε, Vε) a weak solution of (1.1)–(1.4); we have ψε → ψ strongly
in C([0, T ]; L2(R)), Vε/ε V ≡ V
0 weakly∗ in L∞([0, T ]; L2(R)), ψε ψ weakly in L∞([0, T ]; H1(R)) and Vε/ε→ V ≡ V0strongly in C([0, T ]; H−η(R)), then ψ satisfies
∂t tψ− α∂xxψ= α|∂xψ|2− |∂tψ|2 ψ, |ψ| = 1 a.e., ψ (x,0)= ψ0(x), ∂tψ (x,0)= −iV0(x)ψ0(x),
or equivalently ψ= eiθ with the phase function θ satisfying the wave equation ∂t tθ− α∂xxθ= 0.
Remarks. (1) The potential V0disappears in (4.27) because it is stationary, V0= V0(x).
How-ever, from (4.25) or (4.26) the initial data need to satisfy the compatibility condition i∂tψ (x,0)= V0(x)ψ0(x)which shows the long wave effect. For cubic defocussing NLS equation it vanishes, ∂tψ (x,0)= 0 due to the lack of extra potential (see [5,25]).
(2) It is straightforward to pass to the limit from (1.1) because we have the compactness of the long wave{Vε/ε}ε. However we can also consider the limit procedure from (1.5). Since
Vε(·, t)/εεis weakly compact in L2(R) for t ∈ [0, T ],
which according to (1.2) is equivalent to t 0 ∂xψε(x, τ ) 2 /εdτ ε
is weakly compact in L2(R) for t ∈ [0, T ].
We also have
ψε(·, t)
εis strongly compact in L
2(R) for t ∈ [0, T ].
We deduce from the above two statements that t 0 ∂xψε(x, τ ) 2 /εdτ ψε(·, t) ε
is weakly compact in L1(R) for t ∈ [0, T ].
Therefore{[0t∂x(|ψε(x, τ )|2/ε) dτ]ψε}εis uniformly bound in L∞([0, T ]; L1(R)), hence
uni-formly bounded in L1loc(R+; L1(R)). Since λ = o(ε) we can conclude that the nonlocal term
−λ[t
0∂x(|ψ
Acknowledgments
The authors are grateful to the referee for valuable suggestions and comments which have helped to improve the manuscript. C.-K. Lin would like to thank the Department of Mathematical & Statistical Sciences, University of Alberta for the hospitality and support during his visit. He also thanks the Banff International Research Station for providing a stimulating and fruitful research environment. Part of this paper was finished when he participated the Focused Research Group 2004 (the kinetic models for multiscale problems from August 21 to September 4, 2004).
References
[1] H. Added, S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: Smoothness and ap-proximation, J. Funct. Anal. 79 (1988) 183–210.
[2] D. Bekiranov, T. Ogawa, G. Ponce, On the well-posedness of Benney’s interaction equation of short and long waves, Adv. Differential Equations 1 (1996) 919–937.
[3] D. Bekiranov, T. Ogawa, G. Ponce, Interaction equation for short and long dispersive waves, J. Funct. Anal. 158 (1998) 357–388.
[4] D.J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math. 56 (1977) 81–94. [5] T. Colin, A. Soyeur, Some singular limits for evolutionary Ginzburg–Landau equations, Asymptot. Anal. 13 (1996)
361–372.
[6] T. Colin, D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics, Duke Math. J. 107 (2001) 351–419.
[7] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss., vol. 325, Springer, New York, 2000.
[8] B. Desjardins, C.K. Lin, T.C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation, Math. Models Methods Appl. Sci. 10 (2000) 261–285.
[9] B. Desjardins, C.K. Lin, On the semiclassical limit of the general modified NLS equation, J. Math. Anal. Appl. 260 (2001) 546–571.
[10] F. Dias, C. Kharif, Nonlinear gravity and capillary–gravity waves, Annu. Rev. Fluid Mech. 31 (1999) 301–346. [11] V.D. Djordjevic, L.G. Redekopp, On two-dimensional packets of capillary–gravity waves, J. Fluid Mech. 79 (1977)
703–714.
[12] I. Gasser, C.K. Lin, P. Markowich, A review of dispersive limit of the (non)linear Schrödinger-type equation, Tai-wanese J. Math. 4 (2000) 501–529.
[13] I. Gasser, P. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14 (1997) 97–116.
[14] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc. 126 (1998) 523–530.
[15] S. Jin, C.D. Levermore, D.W. McLaughlin, The semiclassical limit of the defocussing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999) 613–654.
[16] T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975) 181–205.
[17] C. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991) 33–67.
[18] C. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 255–288.
[19] P. Laurencot, On a nonlinear Schrödinger equation arising in the theory of water wave, Nonlinear Anal. 24 (1995) 509–527.
[20] J.H. Lee, C.K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit, Chaos Solitons Fractals 13 (2002) 1475–1492.
[21] J.H. Lee, C.K. Lin, O.K. Pashaev, Shock waves, chiral solitons and semiclassical limit of one-dimensional anions, Chaos Solitons Fractals 19 (2004) 109–128.
[22] H.L. Li, C.K. Lin, Semiclassical limit and well-posedness of Schrödinger–Poisson and quantum hydrodynamics, Electron. J. Differential Equations 2003 (2003) 1–17.
[23] H.L. Li, C.K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys. 256 (2005) 195–212.
[24] C.K. Lin, Singular limit of the modified nonlinear Schrödinger equation, in: I.M. Sigal, C. Sulem (Eds.), Nonlinear Dynamics and Renormalization Group, in: CRM Proc. Lecture Notes, vol. 27, Amer. Math. Soc., Providence, RI, 2001, pp. 97–109.
[25] F.-H. Lin, J.X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation, Comm. Math. Phys. 200 (1999) 249–274.
[26] Y.C. Ma, The complete solution of the long wave–short wave resonance equations, Stud. Appl. Math. 59 (1978) 201–221.
[27] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci., vol. 53, Springer, New York, 1984.
[28] T. Ogawa, Global well-posedness and conservation laws for the water wave interaction equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 368–384.
[29] S. Schochet, M. Weinstein, The nonlinear Schrödinger limit of Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986) 569–580.
[30] N. Sepulveda, Solitary waves in the resonant phenomenon between a surface gravity wave packet and an internal gravity wave, Phys. Fluids 30 (1987) 1984–1992.
[31] C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation, Appl. Math. Sci., vol. 139, Springer, New York, 1999. [32] M. Tsutsumi, S. Hatano, Well-posedness of the Cauchy problem for the long wave–short wave resonance equations,
Nonlinear Anal. 24 (1994) 155–171.
[33] M. Tsutsumi, S. Hatano, Well-posedness of the Cauchy problem for Benney first equations of long wave–short wave interaction, Funkcial. Ekvac. 37 (1994) 289–316.
[34] T. Yoshinaga, M. Wakamiya, T. Kakutani, Recurrence and chaotic behavior resulting from nonlinear interaction between long and short waves, Phys. Fluids A 3 (1991) 83–89.
[35] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 34 (1972) 62–69.
