SEMICLASSICAL LIMIT TO THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION∗†

Boling Guo

(Institute of Applied Physics and Computational Math.,China Academy of Engineering Physics,Beijing 100088,PR China)

Guoquan Qin‡

(Graduate School of China Academy of Engineering Physics,Beijing 100088,PR China)

Abstract In this paper,we investigate the semiclassical limit of the generalized nonlinear Schrödinger equation for initial data with Sobolev regularity.Also,we will analyze the structure of the fluid dynamical system with quantum effect corresponding to the semiclassical limit of the generalized nonlinear Schrödinger equation.

Keywords quantum hydrodynamics;dispersive limit;compressible Euler equation

1 Introduction

Hydrodynamics equations with quantum effect describe the hydrodynamical properties and states of some important physical phenomena such as semiconductor,superconductor and super flow.This kind of equations have theoretical significance and practical value.From the semiclassical limit of the nonlinear Schrodinger(NLS)equation with Plank constant h,we can derive various hydrodynamics equations with quantum effect when h→0.

It is well known that the quantum hydrodynamics equations(QHD)can be derived based on the moment method,which is analogous to the derivation of the compressible Euler equation from the Boltzmann equation by taking the zeroth,first and second order velocity moments of the quantum Boltzmann equation and resulting in a hydrodynamical model which then has to be closed in an approximate way,that is,a reasonable macroscopic approximation for the quantum heat flow tensor has to be derived by using additional(quantum)physical properties of the particle ensembles.Moreover,in the case of high electric fields,small mean-free-path asymptotics have been used to derive QHD-models.

When the time and distance scales are large enough relative to the Plank constant h,the system will approximately obey the laws of classical,Newtonian mechanics.That is,quantum mechanics becomes Newtonian mechanics as h→0.The asymptotics of quantum variables as h→0 are known as semiclassical expressing this limiting behavior.

In the semiclassical limit or WKB limit and when ∇xand ∂tscale like ϵ as ϵ→ 0(ϵ is the scaled Planck constant),the quantum-mechanical pressure becomes negligible.The isentropic compressible Euler equation can be formally recovered from the nonlinear Schrödinger equation in this limit.This fact was proven rigorously by Jin,Levermore and McLaughlin[5,6]for the one-dimensional integrable case using the inverse scattering technique and by Grenier[3]for higher dimensions in situations where no vortices are involved.

Very similar model equations have been used for quite a while in other areas of theoretical and computational physics,for instance,in super fluidity[11,12]and in superconductivity[2].

2 Semiclassical Limit to the Nonlinear Schrödinger Equation in Short Time Range

In this section,we consider the following nonlinear Schrödinger(NLS)equation with rapidly oscillating data

where f∈C∞(R+,R),S0(x)∈Hs(Rd)for s large enough.And a0is a function,polynomial in h with coefficients of Sobolev regularity in x.h is the Plank constant and ψhis the wave function.

We will study the semiclassical limit of equation(2.1)-(2.2)and determine the limiting dynamics of any function of the field ψhas h → 0.

Remark 2.1When f(x)=x,equation(2.1)appears in the phenomenological description of super fluidity of an almost ideal Bose gas[10].In this case,the squared modulus of the wave function ψ¯ψ is interpreted as the particle number density in the condensate state,while the gradient of the phase is proportional to the super fluid velocity u= ∇argψ.Moreover,the nonlinear Schrödinger equation is very helpful for the mathematical analysis of the isentropic irrotational QHD-system[4–6,15].

Employing the WKB method,we will look for the solution to(2.1)-(2.2)having the following form

where

and aj(x,t)satisfies certain equations so that we could solve(2.1)locally in time.

Let

then(2.1)is transformed to

The above equation is a perturbation of the following isentropic compressible Euler equation

If f′>0,then equation(2.7)-(2.8)admits a local smooth solution in[0,T∗]for T∗sufficiently small.In fact,we have the following theorems.

Theorem 2.1SupposeHs(Rd)and a0(x,h)be uniformly bounded in Hs(Rd)with respect to x.Then there exists a constant T >0 such that equation(2.1)-(2.2)admits a solution ψh=ah(x,t)exp(iSh(x,t)/h),where ahand Share bounded in L∞([0,T];Hs).

Theorem 2.2Under the assumption of Theorem 2.1,assume further a0(x,h)as h → 0 and equation(2.7)-(2.8)with initial data(ρ(0,x),v(0,x))=admits a solution.Then,equation(2.1)admits a formal solution ψh(x,t)=ah(x,t)exp(iSh(x,t)/h)on[0,T]satisfying the initial condition(2.2)for h small enough,where ahand Share uniformly bounded in L∞([0,T],Hs)with respect to h.

Theorem 2.3Under the assumption of Theorem 2.2,if a0(x,h)admits the following expansion

where N∈N,s−2N−2−d/2>0 and rNsatisfy

Then for interval[0,T]given in Theorem 2.2,one has as h→0

where S and ajare determined by the WKB method and

Proof of Theorem 2.1Suppose

Substituting(2.13)into(2.1)yields

which can be rewritten as

Setting ωh= ∇xSh,we have

where

The matrix A(uh,ξ)can be symmetrized for f′>0 by

with S symmetry.

For the first term,we have

Since

employing the Sobolev embedding and the equation(2.14)yield

For the second term,one easily obtains

Using integration by parts leads to

The second term on the right hand side of(2.15)can be written as

Invoking the symmetry of SAi(uh)yields

therefore,one finds

By the commutator estimates,there holds

Consequently,we find

Proof of Theorem 2.2Assume there exists a solution(ρ,v) ∈ L∞([0,T],Hs+2(Rd))of(2.7)-(2.8)on[0,T]withand the initial condition

We will prove there exists a solution of(2.14)in a interval[0,T)with T independent of h for h small enough and the solution is uniformly bounded in L∞([0,T];Hs).The formal limit of equation(2.14)is

where u=(a1,a2,ω)admits a solution in the maximal interval[0,T′]with T′≥ T.Set vh=uh−u,then we find

Note that

so

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where C(h)→0 and h→0.We thus complete the proof of Theorem 2.2.

Proof of Theorem 2.3We prove the theorem in four steps.

1.Zero order approximation

From Theorem 2.2,we know that ahand ωhare uniformly bounded in L∞([0,T];Hs(Rd)).Therefore, ∂tahand ∂tωhare bounded in L∞([0,T];Hs−2(Rd)). We can extract subsequences of ahand ωhconverging toand,respectively in L∞([0,T];Hs−2(Rd))and the limits are the unique solution of

2.First order approximation

Let vh=uh−u,we can prove the following energy estimates

and the initial condition

The solution to this problem is unique.In fact,there exists a subsequence of evhconverge to

3.Higher order approximation

Assume the N-th approximation to be

4.WKB expansion

We have expanded the formal solution ahand Shto any order.To return to the WKB expansion,one can write the following two formal series

3 Semiclassical Limit to the Derivative Schrödinger equation

Now,let us consider the following derivative Schrödinger equation(GDNLS)

where f∈C∞(R+,R),S0∈Hs(R)for s large enough,is a polynomial in h with coefficients of Sobolev regularity in x.We consider the limit of(3.1)-(3.2)when h→0,for−∞

When f(x)=x in equation(3.1),the resulting equation is used to describe the nonlinear propagation of magnetosonic wave trains parallel to the magnetic field in a hot or collisionless ideal plasma with dispersion due to Hall currents[7,13].

Introducing the new variables ρ =A2=|ψ|2,u=Sx,we have

where

Thus,equation(3.5)-(3.6)can be regarded as a perturbation to the following compressible Euler equation

Multiplying(3.6)by ρ and using(3.5)lead to

where µ = ρu is the momentum and P′(ρ)=2ρf′(ρ)= ρΦ′(ρ).From(3.5),using P(ρ)= ρΦ′(ρ)− Φ(ρ),one finds

or

where

Thus,there holds

Adding(3.10)and(3.12)leads to

As in[1],we denote by M= µ+Φ(ρ)the noncanonical momentum.Using Q(ρ)=equation(3.5)-(3.6)can be written as

and this can be rewritten as the local conservation laws of ρ,M,Φ

Collecting the above arguments,we obtain the following theorem.

Theorem 3.1Equation(3.1)is equivalent to the dispersive perturbation of the quasilinear hyperbolic equation(3.14)-(3.15)or(3.16)-(3.17).The density ρ and the noncanonical momentum M are the conserved quantities of the GDNLS equation.In particular,whenand,(3.16)-(3.17)is equivalent to

In addition,we will prove the following theorem.

Theorem 3.2Let∗be the complex conjugate.The GDNLS equation admits the following conserved quantities

where the fluid dynamical variablescan be represented by the wave function ψ as

ProofObviously,the theorem can be deduced directly by(3.14)-(3.15).Here,we will derive it from(3.1)and thus we can better understand the relation between the classical mechanics and the quantum mechanics.

Note that we have the following equality

Because the third term in(3.26)only depends on ψψ∗=|ψ|2,we can integrate it with respect to x to obtain

This proves(3.20).

Also,one easily deduces

Subtracting(3.28)from(3.29),we obtain

Next,note that the following equality

is equivalent to(3.5)-(3.6)with

Integrating(3.31)yields

Multiplying(3.28)by Φ′(ρ)leads to

Adding(3.30)and(3.34)and using

we obtain

which leads to

This complete the proof.

For the GDNLS equation

with the initial condition

where the amplitude A0(x)is nonnegative,the phase S0(x)is real-valued and smooth and is independent of h.One can take

We can prove that ψhis a dispersive perturbation with O(h2)error as h → 0 to the following deformed Euler equation

with the initial condition

Consider

Then(3.18)-(3.19)can be transformed to

From

we have

On the other hand

with

Consequently,one obtains the energy equation

Assume f(ρ)= ρ,then the Euler equation derived by the semiclassical limit is

with initial data

Energy equation(3.52)is then

One can rewritten(3.54)-(3.55)as

with

The eigenvalues of B are the roots of

or

the corresponding right and left eigenvector are respectively

The Riemann invariants is

The eigenvalues λ+,λ−can be represented by the Riemann invariants as

The equation can be rewritten by the Riemann invariants as

with initial data

Theorem 3.3The blowup time tbcan be estimated as

with

where

and x±(t)satisfying

ProofThe break-time tbcan be estimated by Laxs recipe[8,14].

From(3.65),we obtain

Similarly,

Differentiating(3.68)with respect to x and setting Z+=∂xR+,we obtain

(3.69)leads to

Therefore

Substituting(3.72)into(3.70)yields

one can derive the standard Raccati equation

The solution to(3.75)is

where

The integration is along the characteristic of λ+and the sign of q0K(t)impact its singularity essentially.If the initial data satisfiesnamely,q0<0,then q+(x,t)will tend to infinity in finite time,which implies that q+(x,t)certainly will blowup and 1+q0K(t)=0.Therefore,the blowup time tbcan be estimated as follows.

Let t+,bsatisfy

with

The particle trajectory x=x+(t)satisfies

Similarly,to estimate t−,b,we consider the characteristic of.When f(ρ)= −ρ,we have

and the energy equation is

(3.77),(3.78)can be write as a matrix form as follows

where

of which the eigenvalues are the roots of

that is,

then the corresponding eigenvector are

The Riemann invariants are

The equation can be written as

with the initial data

This completes the proof.

Similarly,we can derive the estimate of the blowup time

Theorem 3.4The blowup timecan be estimated as

where

and

4 Semiclassical Limit to the Generalized NLS:Subsonic,Supersonic,Transonic

Consider the generalized NLS

Setting

one obtains

where

When ϵ=0,the MNLS is

where we define the sonic

We consider the inverse scattering method to MNLS.

To do so,we first consider the initial condition of(4.6),

Assume ρ0(x)andare real-valued Schwartz functions for x ∈ R and

In particular,choose u0(x)and S0(x)as

where S0(0),δ and µ are real-valued parameters andTherefore,

Without loss of generality,let ν=1,S0(0)=0.

When ν =1,Q in the definition of(4.7)becomes

In the transonic domain,Q(x)will necessarily change sign.From(4.12),it is easy to see that Q(x)is a quadratic equation of T(x)and has two roots.For simplicity,suppose T has only one root for T ∈ (−1,1).For this,assume

and

According to inequalities(4.13)-(4.15),(4.12)has a unique solution in(−1,1),

Consequently,when t=0,x0,and the domain is of hyperbolic,x>xc=arctanh(Tc).Next we present our main result.

Theorem 4.1Let ϕϵ(x,t)be a solution of Cauchy problem to the MNLS equation(4.1)with the initial data(4.3),where A0(x)and S0(x)are given by(4.10).Then there exists a smooth curve x=xc(t),t≥ 0,xc(0)=xc,such that for all t≥ 0 and x

ProofWe use Riemann-Hilbert method to solve this problem.Assume the unknown matrix M(k;x,t)admits a discontinuous matrix in the complex plane,

where

are the Pauli spin matrices.

where r(k)= −r(−k)is the reflection coefficient,and

Then the solution to the MNLS equation can be derived from M(k;x,t)

We thus complete the proof.

Remark 4.1Here,the error term holds uniformly in any compact subset of(x,t).A(x,t)and S(x,t)are smooth real-valued functions,independent of ϵ and satisfy A(x,0)=A0(x)and S(x,0)=S0(x).When x0,

satisfy the MNLS equation(4.6).