非线性波与孤子动力学：KdV、KP、NLS和DNLS方程

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"Solitary Waves" 本资源是一本深入探讨现代非线性物理学和数学领域重要且发展迅速的专著，重点关注一维、二维以及三维孤波（Solitary Waves）及其动力学的研究。这些孤波由Korteweg-de Vries（KdV）、Kadomtsev-Petviashvili（KP）、非线性薛定谔（NLS）以及导数非线性薛定谔（DNLS）等类方程描述。书中特别强调了对这些方程的推广，这些推广涵盖了各种复杂物理介质中的高阶色散修正、耗散效应、不稳定性以及波场的随机波动。书的内容涵盖了固体状态科学系列，由多位该领域的权威研究人员撰写，旨在系统而全面地阐述固态物理的基本原理和新进展。该系列书籍旨在为读者提供理论和实验固态物理学的深度理解。书中提到的136至142号著作分别涉及纳米尺度相分离与巨磁阻现象、量子传输在亚微米器件中的理论介绍、软物质物理中的相分离、纳米结构的光学响应、凝聚态物理中的分形概念、低维半导体中的激子理论与应用，以及二维材料的详细研究。这些主题均与孤波理论紧密相关，因为它们都涉及到非线性动力学、波动现象和材料特性。孤波是一种在非线性系统中保持其形状不变并以恒定速度传播的波动现象。KdV、KP、NLS和DNLS方程是描述这种现象的典型数学工具。KdV方程主要应用于描述浅水波或声波的传播，而KP方程则扩展到二维波动，如大气或流体中的波。NLS方程通常用于量子光学和光纤通信中的光孤波，DNLS方程则在描述量子系统中的一维链状结构时非常有用。书中的“高阶色散修正”指的是更精确地考虑波的传播特性，这在处理具有复杂物理特性的介质时尤为重要。耗散效应是指能量损失，它可能来自介质内部的摩擦或其他过程。不稳定性分析则关注孤波是否会在时间或空间上失去稳定性。最后，“随机波动”讨论了波场的随机变化，这在统计物理和混沌理论中有重要意义。这本书不仅提供了孤波理论的深入理解，还涵盖了相关的物理现象和数学模型，适合于对非线性科学、固态物理学、量子力学以及相关领域有浓厚兴趣的学者和研究人员。通过深入学习，读者将能够更好地理解和预测各种复杂系统中的非线性波动行为。

2 Introduction

∂

u + αu∂

u + β∂

u =0, (0.1)

where u = u(t, x), was derived for the ﬁrst times by Korteweg and de Vries

in 1895 [12]; however, the term “soliton” was introduced in 1965 by Zabusky

and Kruskal [1] who demonstrated that the Korteweg–de Vries equation (KdV

equation) reveals hidden linear properties, allowing a solution in the form of a

nonlinear solitary wave propagating without changing its proﬁle. It appeared

later that the KdV equation describes a wide class of one-dimensional nonlin-

ear physical systems with “real” dispersion when the wave frequency is given

by ω ≈ c

(1 + δ

) (here, c

is the phase velocity of the wave and δ is the

“dispersion length”), and, in addition to hydrodynamics, appears in plasma

physics, magnetohydrodynamics, theory of lattices, etc. [2,3,13,14], i.e., it is

quite universal.

If the considered media are viscous and/or heat conducting, the dispersion

is “imaginary”, i.e., ω ≈ c

(1 − iνk

) (here, ν is the viscosity), and the

approach of Kortweg and de Vries leads to the Burgers equation [15]

∂

u + αu∂

u = ν∂

u. (0.2)

Solutions of the Burgers equation describe such formations as shock waves.

In the physics of plasmas, aero- and hydrodynamics, as well as in many

other areas of physics, there is obvious interest to the wave dynamics of

non-one-dimensional systems with the nonlinearity of the hydrodynamic type

where stable stationary structures in the form of non-one-dimensional solitons

can exist [7]. Many such systems are described by the class of equations given

∂

u + αu∂

u + β∂

u = R, (0.3)

where u = u(t, x, y, z) is the function of the wave ﬁeld and R = R[u]is

some linear functional of u. The particular form of the right-hand side of

(0.3) is deﬁned by the wave properties of the medium and the sign of its

dispersion. For example, sound waves in a plasma with weak dispersion when

the wave numbers of the harmonics in the wave packet are small and satisfy

the inequalities

kδ  1 and k

 k

⊥

, (0.4)

while the dispersion relation in the linear approximation given by

ω ≈ c



⊥

+ δ



, (0.5)

are described by equation of the type (0.3) with R = κ∇

⊥

w and ∂

w = ∇

⊥

∂



∂

u + c

∂

u − c

∂

u + χu∂



= ±(c

/2) ∆

⊥

u. (0.6)

Equation (0.6) for u = u(t, x, y) and ∆

⊥

= ∂

was ﬁrst derived in 1970 by

Kadomtsev and Petviashvili [16]; it is called now the Kadomtsev–Petviashvili

Introduction 3

equation (KP equation). Later, the KP equation was generalized in the three-

dimensional case, with ∆

⊥

= ∂

+ ∂

[7]. For the ion-acoustic waves in a

plasma, when u stands for the ion velocity, c

= c

=(T

)

1/2

is the

ion-acoustic speed (T

is the electron temperature in the energy units and

is the ion mass) and δ

= r

/2=T

/8πn

is the electron Debye

length and n

is the electron number density), we have the positive sign on

the right-hand side of (0.6), corresponding to the “negative” dispersion (for a

number of other cases the dispersion can be “positive”, i.e., the negative sign

appears on the right-hand side of (0.6)). This type of dispersion is typical for

isotropic media, but can also appear in some anisotropic media. If the char-

acteristic frequencies of the ion-acoustic wave packet in a magnetized plasma

signiﬁcantly exceed the ion cyclotron frequency ω

= eB

c (here, B

the external magnetic ﬁeld and c is the light speed), the anisotropy can be

neglected, but in the opposite case ω  ω

this assumption is inadmissible.

In the latter case, an additional term proportional to ω

i ×v (here, i is the

unit vector in the x-direction, and v is hydrodynamic velocity) appears on

the right-hand side of the corresponding magnetohydrodynamic equations,

and the sign of the second term in the dispersion equation (0.5) (and, corre-

spondingly, the sign of the term in the right hand side of (0.6)) changes to

the opposite, i.e., negative. We obtain in this case the equation of the class

(0.3), but with R = κ∆

⊥

∂

u, known as the Zakharov–Kuznetsov equation

[17].

In a strongly magnetized plasma with B

 8πnT (here, nT stands

for the plasma kinetic pressure) in the frequency range ω  ω

, the fast

magnetosonic waves (FMS waves) can propagate [3]. For FMS waves, taking

into account that c

= v

= B

/(4πnm)

1/2

, where v

is the Alfv´en velocity

and nm is the plasma density, the dispersion law is also described by (0.5) and

the equation for the dimensionless amplitude of the wave ﬁeld h = B

∼

is the magnetic ﬁeld of the wave) can be also written in the form (0.6).

In the reference frame moving along the x-axis with the Alfv´en velocity v

the wave equation is given by

∂

h +

sin θh∂

h − v

∂

h = −



−∞

∆

⊥

hdx, (0.7)

where θ is the angle between the magnetic ﬁeld B

and k

and

2ω



cot

θ −



. (0.8)

Here, ω

=(4πn

)

1/2

is the ion plasma frequency and n

is the ion

number density.

For gravity-capillary waves in a shallow water, the equation of the type

(0.3) also holds. We have in this case R = −(c

/2)∇

⊥

w, α =3c

/2H, β =

−c

, where c

=(gH)

1/2

, H is the depth,

4 Introduction



3σ

ρg

− H



, (0.9)

σ stands for the coeﬃcient of the surface tension, and ρ is the density of the

ﬂuid.

Generalizing equations (0.6) and (0.7), we write (0.6) in the reference

frame moving along the x-axis with the velocity c

and obtain the KP equa-

tion in the standard form

∂



∂

u + αu∂

u + β∂



= κ∆

⊥

u. (0.10)

Here, the sign of the ratio β/κ deﬁnes the character of dispersion. The KP

equation (0.10) describes many nonlinear physical systems (where the linear

dispersion law is given by (0.5)) and it is thus universal in the same sense as

the KdV equation.

The diﬃculty in the analytical solution of nonlinear problems is in the

proper choice of the eﬀective sequence of methods to construct approximate

solutions of the nonlinear systems, asymptotic in a small parameter. The per-

turbation theory is severely impeded in the non-one-dimensional case because

nonlinear resonances, instabilities, secular eﬀects, etc., can lead to singulari-

ties [18]. The discovery by Gardner, Green, Kruskal, and Miura in 1967 [19]

of the inverse scattering transform method (IST method) for the KdV equa-

tion and subsequent developments of this idea (we would single out here the

classic paper by Zakharov and Faddeev [20] as well as Refs. [21–24] where

non-one-dimensional generalizations of the IST method were considered) has

lead to active development of the theory of nonlinear waves [24–28]. In par-

ticular, the complete integrability of the KP equation (0.10) for ∆

⊥

= ∂

was

proved [29,30], and using the “dressing method” the exact two-dimensional

soliton solution (ﬁrst numerically found by Petviashvili [31]) was obtained

for β/κ > 0. The latter for α = −6, β = −1, and κ = −3 is given by

u(t, x, y)=2∂

ln det B, (0.11)

where

= δ



x − iν

y − ξ

− 3ν



+(1− δ

)

− ν

K+l

= −ν

∗

,ξ

K+l

= ξ

∗

,n,m=1,...,2K, and l =1,...,K

(ν

and ξ

here deﬁne the amplitudes, phases, velocity vectors, and other

soliton parameters). The invariants were also found [24]:



udxdy, J

≡P=



dxdy,

≡H=





β (∂

κw

− u



dxdy, (0.12)

Introduction 5

where ∂

w = ∂

u. The ﬁrst invariant, J

is related to the divergence form of

the KP equation, the second one, J

, is due to the translational invariance

of the KP equation and is its momentum P, and the third one, J

, is due to

the time invariance of the KP equation and is its Hamiltonian H.

Signiﬁcant progress was also achieved for a number of other exactly inte-

grable models. In the context of problems considered in this book, the most

interest is paid to the non-stationary derivative nonlinear Schr¨odinger equa-

tion (DNLS equation) given in the one-dimensional case by [32,33]:

i∂

h +is∂



|h|



+ λ∂

h =0. (0.13)

The DNLS equation describes, in the frequency range ω  ω

, the evolution

of nonlinear ﬁnite amplitude Alfv´en waves [34] propagating in a magnetized

plasma along the external magnetic ﬁeld with the ratio of the kinetic plasma

pressure to the magnetic pressure p =4πnT/B

. Here, the dimensionless

function h(t, x)=(B

+iB

)/2B

|1 − p|

1/2

describes the right circularly-

polarized wave when λ = 1 and s = sgn(1−p). The change h →−sh

∗

with the

change of the dispersion sign on the opposite one (λ = −1) allows description

of the left circular-polarized wave. Note that (0.13) is obtained from the

full set of the one-ﬂuid magnetohydrodynamics equations (MHD equations)

in the dimensionless units t → ω

t/2, x → x/r

, and r

⊥

→ r

⊥

√

2/r

= v

/ω

) in the reference frame moving along the x-axis with the

Alfv´en velocity v

taking into account ∂

= ∂

=0.

The DNLS equation is completely integrable, has inﬁnite number of the

conservation laws, and can be solved by the IST method [35]. The evolution

of Alfv´en waves in the model (0.13) was studied in terms of the sign of the

DNLS integral of motion, H/2, where the Hamiltonian is given by

H =

∞



−∞



|h|

+ s|h|

∂



dx, ϕ = arg(h).

It was established that the wave evolution can lead either to its spreading or

to formation of the one-dimensional Alfv´en soliton, depending on the sign of

H. From the physical point of view, this means that there are two types of

the nonlinear wave dynamics [33]: the modulationally-stable case when H > 0

and an initial pulse is spreading, loosing its structure; and the modulationally-

unstable case when evolution of the modulational instability [36] ends by the

formation of the one-dimensional soliton [32]

h =





1/2

−Ax

+ie

cosh

(2Ax)

−iA

, (0.14)

where A is the amplitude of the soliton.

Equation (0.13), similar to the KdV equation, can be generalized in the

non-one-dimensional situation. Thus, three-dimensional generalization of the

6 Introduction

DNLS equation was obtained [7,37,38] in the form of the set of coupled equa-

tions

∂



∂

+ ∂



|h|



± i∂



= −∇

⊥



+ h

−



, (0.15)

where h

= h

± ih

. In the case of the negative dispersion, solutions in the

form of the one-dimensional solitons were numerically found [7,37,38]; and no

numerical study of non-one-dimensional solutions was done. The exact two-

dimensional solitary solutions of the set (0.15) can in principle be obtained

by the IST method, similar to the case of the KP equation; it is necessary to

construct the corresponding Lax

L–

A pair [24].

Successes in the theoretical investigation of nonlinear physical systems

having soliton solutions has stimulated experimental studies in various ar-

eas of the physics of nonlinear waves. Here, we would like to mention the

laboratory and space experiments on excitation, evolution, and interaction

dynamics of the ion-acoustic and Alfv´en solitons as well as the experimental

studies of the structure of plasma shock waves [39–50], the modeling of the

solitons’ dynamics in the electric circuits [51,52], the experiments on surface

waves and internal waves in rotating vessels and hydro-trays [53,54], etc. This

in turn has raised new questions and demonstrated the urgency of the theo-

retical investigation of such problems as the stability of non-one-dimensional

solitons, the dynamics of their interaction, the nonlinear resonances and for-

mation of the bound states, the eﬀects of small corrections in equations of

the class (0.3), the inﬂuence of dissipation on the structure and evolution

of the non-one-dimensional and nonlinear waves and solitons, the wave self-

inﬂuence eﬀects (the wave collapse and the wave self-focusing), etc.

It was shown, in Refs. [16,55-57] for some particular cases (concrete values

of the coeﬃcients) of the equation (0.10) with ∆

⊥

= ∂

, and in Ref. [58] for

arbitrary α, β, and κ, that for the negative dispersion (β/κ < 0) the one-

dimensional solitons are stable whereas for the positive dispersion (β/κ > 0)

they are unstable with respect to growth of inﬁnitesimal perturbations. The

case of very small k was studied [16] by the Krylov–Bogolyubov method;in

Ref. [55], the term ∂

yyxx

u was added on the right-hand side of the equation

and the problem was solved by the method of the variation of the action with

the Lagrangian integrated over x and the test functions in the solitary-like

form; in Ref. [56], the IST method was used, and in Ref. [57], the Lyapunov’s

functional method was used. In Ref. [58], the numerical investigation was

done. It was demonstrated that for β/κ < 0 the perturbation can easily

transfer from the soliton into the medium and spread in all directions while

for β/κ > 0 it is “locked” in the soliton. In the region of localization of the

perturbation, the soliton’s velocity diﬀers from the velocity of an unperturbed

soliton. This, taking into account the spreading, leads to the growth of the

perturbation.

For two-dimensional and three-dimensional solitons of the KP equa-

tion, the stability analysis is highly non-trivial. The most illustrative study

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