Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

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3.8 The Schr¨odinger and the Korteweg–de Vries Equations 81

As a simple model of turbulence, c is replaced by the ﬂuid velocity ﬁeld

u (x , t) to obtain the well-known Burgers equation

+ uu

= νu

, (3.7.9)

where ν is the kinematic viscosity.

Thus the Burgers equation is a balance b etween time evolution, non-

linearity, and diﬀusion. This is the simplest nonlinear model equation f or

diﬀusive waves in ﬂuid dynamics. Burgers (1948) ﬁrst developed this equa-

tion primarily to shed light on the study of turbulence described by the

interaction of the two opposite eﬀects of convection and d iﬀusion. However,

turbulence is more complex in the sense that it is both three dimensional

and statistically random in nature. Equat ion (3.7.9) arises in many phys-

ical problems including one-dimensional turbulence (where this equ ation

had its origin), sound waves in a viscous medium, shock waves in a viscous

medium, waves in ﬂuid-ﬁlled viscous elastic tubes, and magnetohydrody-

namic waves in a medium with ﬁnite electrical conductivity. We note that

(3.7.9) is parabolic provided the coeﬃcient of u

is constant, whereas the

resulting (3.7.9) with ν = 0 is hyperbolic. More importantly, the proper-

ties of the solution of the parabolic equation are signiﬁcantly diﬀerent from

those of the hyperbolic equation.

3.8 The Schr¨odinger and t he Korteweg–de Vries

Equations

We consider the following Fourier integral representation of a quasi-mono-

chromatic plane wave solution

u (x , t)=



∞

−∞

F (k)exp[i {kx − ω (k) t}] dk, (3.8.1)

where the spectrum function F (k) is determined from the given ini-

tial or boundary conditions and has the property F (−k)=F

∗

(k), and

ω = ω (k) is the dispersion relation. We assume that the initial wave is

slowly modulated as it propagates in a dispersive medium. For such a quasi-

monochromatic wave, most of the energy is conﬁned in a neighborhood of

a speciﬁed wave number k = k

, so that spectrum function F (k) has a

sharp peak around the point k = k

with a narrow wave number width

k − k

= δk = O (ε), and the dispersion relation ω (k) can be expanded

about k

in the form

ω = ω

+ ω

′

(δk)+

′′

(δk)

′′′

(δk)

+ ··· , (3.8.2)

where ω

= ω (k

), ω

′

= ω

′

), ω

′′

= ω

′′

), and ω

′′′

= ω

′′′

82 3 Mathematical Models

Substituting (3.8.2) into (3.8.1) gives a new form

u (x , t)=A (x, t)exp[i (k

x − ω

t)] + c.c., (3.8.3)

where c.c. stands for the complex conjugate and A (x, t)isthecomplex

wave amplitude given by

A (x, t)=



∞

F (k

+ δk)exp





(x − ω

′

t)(δk) −

′′

(δk)

−

′′′

(δk)



d (δk) , (3.8.4)

where it has been assumed that ω (−k)=−ω (k). Since (3.8.4) depends on

(x − ω

′

t) δk,(δk)

t,(δk)

t where δk = O (ε) is small, the wave amplitude

A (x, t) is a slowly varying function of x

∗

=(x − ω

′

t)andtimet.

We keep only the term with (δk) in (3.8.4) and neglect all terms with

(δk)

, n =2, 3, ···, so that (3.8.4) becomes

A (x, t)=



∞

F (k

+ δk)exp[i {(x − ω

′

t)}(δk)] d (δk) . (3.8.5)

A simple calculation reveals that A (x, t) satisﬁes the evolution equation

∂A

∂t

+ c

∂A

∂x

=0, (3.8.6)

where c

= ω

′

is the group velocity.

In the next step, we retain only terms with (δk)and(δk)

in (3.8.4) to

obtain

A (x, t)=



∞

F (k

+ δk)exp





(x − ω

′

t)(δk) −

′′

(δk)



d (δk) .

(3.8.7)

A simple calculation shows that A (x, t) satisﬁes the li near Schr¨odinger

equation



∂A

∂t

+ ω

′

∂A

∂x



′′

∂

∂x

=0. (3.8.8)

Using the slow variables

ξ = ε (x − ω

′

t) ,τ= ε

t, (3.8.9)

the modulated wave amplitude A (ξ, τ) satisﬁes the linear Schr¨odinger equa-

tion

′′

ξξ

=0. (3.8.10)

3.9 Exercises 83

On the other hand, for the frequencies at which the group velocity ω

′

reaches an extremum, ω

′′

= 0. In this case, the cubic term in the dispersion

relation (3.7.2) plays an important role. Consequently, equation (3.8.4) re-

duces to a f orm similar to (3.8.7) with ω

′′

= 0 in the exponential factor.

Once again, a simple calculation from the resulting integral (3.8.4) reveals

that A (x, t) satisﬁes the linearized Korteweg–de Vries (KdV) equation

∂A

∂t

+ ω

′

∂A

∂x

′′′

∂

∂x

=0. (3.8.11)

By transferring to the new variables ξ = x−ω

′

t and τ = t which correspond

to a r eference system moving with the group velocity ω

′

, we obtain the

linearized KdV equation

∂A

∂τ

′′′

∂

∂ξ

=0. (3.8.12)

This describes waves in a dispersive medium with a weak high frequency

dispersion.

One of the remarkable nonlinear model equations is the Korteweg–de

Vries (KdV) equation in the form

+ αu u

+ βu

xxx

=0, −∞ <x<∞,t>0. (3.8.13)

This equation arises in many physical problems including water waves, ion

acoustic waves in a plasma, and longitudinal dispersive waves in elastic rods.

The exact solution of this equation is called the soliton which is remarkably

stable. We shall discuss the soliton solution in Chapter 13.

Another remarkable nonlinear model equation describing solitary waves

is known as the nonlinear Schr¨odinger (NLS) equation written in the stan-

dard form

′′

+ γ |u|

u =0, −∞ <x<∞,t>0. (3.8.14)

This equation admits a solution called the solitary waves and describes the

evolution of the water waves; it arises in many other physical systems that

include nonlinear optics, hydromagnetic and plasma waves, propagation of

heat pulse in a solid, and nonlinear instability problems. The solution of

this equation will be discussed in Chapter 13.

3.9 Exercises

1. Show that the equation of motion of a long string is

= c

− g,

where g is the gravitational acceleration.

84 3 Mathematical Models

2. Derive the damped wave equation of a string

+ au

= c

where the damping force is proportional to the velocity and a is a

constant. Considering a restoring f orce proportional to the displacement

of a string, show that the resulting equation is

+ au

+ bu = c

where b is a constant. This equation is called the telegraph equation.

3. Consider the transverse vibration of a uniform beam. Adopting Euler’s

beam theory, the moment M at a point can be written as

M = −EI u

where EI is called the ﬂexural rigidity, E is the elastic modulus, and

I is the moment of inertia of the cross section of the beam. Show that

the transverse motion of the beam may be described by

+ c

xxxx

=0,

where c

= EI/ρA, ρ is the density, and A is t he cross-sectional area

of the beam.

4. Derive the deﬂection equation of a thi n elastic plate

∇

u = q/D,

where q is the uniform load per unit area, D is the ﬂexural rigidity of

the plate, and

∇

u = u

xxxx

+2u

xxyy

+ u

yyyy

5. Derive the one-dimensional heat equation

= κu

, where κ is a constant.

Assuming that heat is also lost by radioactive exponential decay of the

material in the bar, show that the above equation becomes

= κu

+ he

−αx

where h and α are constants.

6. Starting from Maxwell’s equations in electrodynamics, show that in

a conducting medium electric intensity E, magnetic intensity H,and

current density J satisfy

∇

X = µεX

+ µσX

where X represents E, H,andJ, µ is the magnetic inductive capacity,

ε is the electric inductive capacity, and σ is the electrical conductivity.

3.9 Exercises 85

7. Derive the continuity equation

+div(ρu)=0,

and Euler’s equation of motion

ρ [u

+(u · grad) u]+gradp =0,

in ﬂuid dynamics.

8. In the derivation of the Laplace equation (3.6.11), the potential at Q

which is outside the body is ascertained. Now determine the potential

at Q when it is inside the body, and show that it satisﬁes the Poisson

equation

∇

u = −4πρ,

where ρ is the density of the body.

9. Setting U = e

ikt

u in the wave equation U

= ∇

U and setting U =

−k

u in the heat equation U

= ∇

U, show that u (x, y, z) satisﬁes

the Helmholtz equation

∇

u + k

u =0.

10. The Maxwell equations in vacuum are

∇×E = −

∂B

∂t

, ∇×B = µε

∂E

∂t

∇·E =0, ∇·B =0,

where µ and ε are universal constants. Show that the magnetic ﬁeld

B =(0,B

(x, t) , 0) and the electric ﬁeld E =(0, 0,E

(x, t)) satisfy

the wave equation

∂

∂t

= c

∂

∂x

where u = B

or E

and c =(µε)

−

is the speed of light.

11. The equations of gas dynamics are linearized for small perturbations

about a constant state u =0,ρ = ρ

,andp

= p (ρ

)withc

= p

′

(ρ

In terms of velocity potential φ deﬁned by u = ∇φ, the perturbation

equations are

+ ρ

div u =0,

p − p

= −ρ

= c

(ρ − ρ

) ,

ρ − ρ

= −

86 3 Mathematical Models

Show that f and u satisfy the three dimensional wave equations

= c

∇

f, and u

= c

∇

where f = p, ρ,orφ and

∇

≡

∂

∂x

∂

∂y

∂

∂z

12. Consider a slender body moving in a gas with arbitrary constant ve-

locity U, and suppose (x

) represents the frame of reference in

which the motion of the gas is small and described by the equations of

problem 11. The body moves in the negative x

direction, and (x, y, z)

denotes the coordinates ﬁxed with respect to the body so that the co-

ordinate transformation is (x, y, z)=(x

+ Ut,x

). Show that the

wave equation φ

= c

∇

φ reduces to the form



− 1



= Φ

+ Φ

where M ≡ U/c

is the Mach number and Φ is the potential in the new

frame of reference (x, y, z).

13. Consider the motion of a gas in a taper tube of cross section A (x).

Show that the equ ation of continuity and the equation of motion are

ρ = ρ



1 −

∂ξ

∂x

−

∂A

∂x



= ρ



1 −

∂

∂x

(Aξ)



and

∂

∂t

= −

∂p

∂x

where x is the distance along the length of the tube, ξ (x) is the dis-

placement function, p = p (ρ) is the pressure-density relation, ρ

is the

average density, and ρ is the local density of the gas.

Hence derive the equation of motion

= c

∂

∂x



∂

∂x

(Aξ)



∂p

∂ρ

Find the equation of motion when A is constant. If A (x)=a

exp (2αx)

where a

and α are constants, show that the above equation takes the

form

= c

(ξ

+2αξ

) .

14. Consider the current I (x, t) and the potential V (x, t)atapointx

and time t of a uniform electric transmission line with resistance R,

3.9 Exercises 87

inductance L, capacity C, and leakage conductance G per unit length.

(a) Show that both I and V satisfy the system of equati ons

+ RI = −V

+ GV = −I

Derive the telegraph equati on

− c

−2

− au

− bu =0, for u = I or V,

where c

=(LC)

−1

, a = RC + LG and b = RG.

(b) Show that the telegraph equation can be written in the form

− c

+(p + q) u

+ pq u =0,

where p =

and q =

u = v exp



−

(p + q) t



to transform the above equation into the form

− c

(p − q)

(d) When p = q, show that th ere exists an undisturbed wave solution

in the form

u (x , t)=e

−pt

f (x + ct) ,

which propagates in either direction, where f is an arbitrary twice dif-

ferentiable function of its argument.

If u (x, t)=A exp [i (kx − ωt)] is a solution of the telegraph equation

− c

− αu

− βu =0,α= p + q, β = pq,

show that the di spersion relation holds

+ iαω −



+ β



=0.

Solve the dispersion relation to show that

u (x , t)=exp



−



exp





kx −



+(4q − p

)



When p

=4q, show that the solution represents attenuated nondisper-

sive waves.

88 3 Mathematical Models

(e) Find the equations for I and V in the following cases:

(i) Lossless tr ansmission line (R = G = 0),

(ii) Ideal submarine cable (L = G = 0),

(iii) Heaviside’s distortionless line (R/L = G/C = constant = k).

15. The Fermi–Pasta–Ulam model is used to describe waves in an anhar-

monic lattice of length l consisting of a row of n identical masses m,

each connected to the next by nonlinear springs of constant κ.The

masses are at a distance h = l/n apart, and the springs when extended

or compressed by an amount d exert a force F = κ



d + αd



where α

measures the strength of nonlinearity. The equation of motion of the

ith mass is

m¨y

= κ

i+1

− y

) − (y

− y

i−1

)+α

(

i+1

− y

)

− (y

− y

i−1

)

where i =1, 2, 3...n, y

is the displacement of the ith mass from its

equilibrium position, and κ, α are constants with y

= y

=0.

Assume a continuum approximation of this discrete system so that the

Taylor expansions

i+1

− y

= hy

xxx

xxxx

+ o





− y

i−1

= hy

−

xxx

−

xxxx

+ o





can be used to derive the nonlin ear diﬀerential equation

= c

[1 + 2αhy

] y

+ o





= c

[1 + 2αhy

] y

xxxx

+ o





where

κh

Using a change of variables ξ = x − ct, τ = cαht, show that u = y

satisﬁes the Korteweg–de Vries (KdV) equation

+ uu

+ βu

ξξξ

= o





,ε= αh, β =

24α

16. The one-dimensional isentropic ﬂuid ﬂow is obtained from Euler’s equa-

tions (3.1.14) in the form

+ uu

= −

,ρ

+(ρu)

=0,p= p (ρ) .

3.9 Exercises 89

(a) Show that u and ρ satisfy the one-dimensional wave equation





− c





=0,

where c

dρ

is the velocity of sound.

(b) For a compressible adiabatic gas, the equation of state is p = Aρ

where A and γ are constants; show that

γp

17. (a) Obtain the two-dimensional u nsteady ﬂuid ﬂow equations from

(3.1.14).

(b) Find the two-dimensional steady ﬂuid ﬂow equations from (3.1.14) .

Hence or otherwise, show that



− u



− uv(u

+ v



− v



=0,

where

= p

′

(ρ) .

tion reduces to the quasi-linear partial diﬀerential equations



− φ



− 2φ



− φ



=0.

(d) Show that the slope of the characteristic C satisﬁes the quadratic

equation



− u







+2uv







− v



=0.

Hence or otherwise derive



− v







− 2uv







− u



=0.

18. For an inviscid incompressible ﬂuid ﬂow under the body force, F =

−∇Φ, the Euler equations are

∂u

∂t

+ u ·∇u = −∇Φ −

∇p, divu =0.

(a) Show that th e vorticity ω

ω = ∇×u satisﬁes the vorticity equation

Dω

∂ω

∂t

+ u ·∇ω

ω = ω

ω ·∇u.

(b) Give the interpretation of this vorticity equation.

Dω

= 0 (conservation of vorticity).

90 3 Mathematical Models

19. The evolution of the probability distribution function u (x, t) in nonequi-

librium statistical mechanics is described by the Fokker–Planck equa-

tion (See Reif (1965))

∂u

∂t

∂

∂x



∂u

∂t

+ x



(a) Use the change of variables

ξ = xe

and v = ue

−t

to show that the Fokker–Planck equation assumes the form with

u (x , t)=e

v (ξ,τ )

= e

ξξ

(b) Make a suitable change of variable t to τ (t), and transform the

above equation into the standard diﬀusion equation

= v

ξξ

20. The electric ﬁeld E (x) and the electromagnetic ﬁeld H (x)infree

space (a vacuum) satisfy the Maxwell equations E

= c curl H, H

−c curl H,divE =0=divH,wherec is the constant speed of light in a

vacuum. Show that both E and H the three-dimensional wave equations

= c

∇

E and H

= c

∇

where x =(x, y, z)and∇

is the three-dimensional Laplacian.

21. Consider longitudinal vibrations of a free elastic rod with a variable

cross section A (x)withx measured along the axis of the rod from

the origin. Assuming that the material of th e rod satisﬁes Hooke’s law,

show that the displacement function u (x, t) satisﬁes the generalized

wave equation

= c

A (x)





where c

=(λ/ρ), λ is a constant that describes the elastic nature of the

material, and ρ is the line density of the r od. W hen A (x) is constant,

the above equation reduces to one-dimensional wave equation.