Evolution of initial discontinuity for the

defocusing complex modified KdV equation

Deng-Shan Wang

Beijing Information Science and Technology University

20 December 2019

Workshop held by IMS National University of Singapore

Emergent Phenomena-from Kinetic Models

to Social Hydrodynamics: 16-20 Dec. 2019

1. Introduction

2. cmKdV-Whitham equations

3. Basic structures

4. Classification of step-like initial condition

5. Further work

Outline

1. Introduction: Discovery of Solitons

▲ 1834: Russell observed nonlinear

wave motions in water waves:

Solitons/Solitary waves

John Scott Russell (1808-1882)

was a Scottish civil engineer, naval architect and

shipbuilder.

▲ 1844: Russell reported on the Waves

▲ 1895: Korteweg-de Vries equation

▲ 1965: Zabusky, Kruskal sovled KdV

numerically and found Solitons again.

Scott Russell Aqueduct

• Long: 89.3m

• Wide: 4.13m

• Deep: 1.52m

• On the union

Canal Near

Edinburgh

Nature, 376 (1995) 373

In the year 1995, the hydrodynamic soliton effect

was reproduced near the place where Russell

observed hydrodynamic solitons in 1834.

Kinds of nonlinear waves in Nature

Line Soliton

Vortex

Rogue Waves

Dispersive shock waves

A brief history of dispersive shock waves

▲ 1954-56: Benjamin & Lighthill; Sagdeev

(dispersive-dissipative --qualitative theory)

▲ 1965: G. B. Whitham

(Whitham modulation theory)

▲ 1973: Gurevich and Pitaevskii

(purely dispersive--general analytical framework

by the Whitham (1965) modulation theory)

Dispersive hydrodynamics is the domain concerned with

fluid motion in which dissipation, e.g., viscosity, is ignored

relative to wave dispersion. There are dispersive shock

waves (DSW) in dispersive hydrodynamics.

▲ 1982-85: Lax, Levermore & Venakides

(rigorous theory of the KdV DSWs using the IST)

▲ 1985-1989: Novikov, Dubrovin, Tsarev, Krichever

(hydrodynamics of integrable soliton lattices)

▲ Others: EL, Ablowitz, Kodama, Tian, Biondini

(Whitham modulation theory to integrable systems)

▲ 2004: E.A.Cornell et al.(Nobel Prize in Physics)

(experimental observation of conservative DSWs

in Bose-Einstein condensates)

The small dispersion KdV equation:

Wave breaking under Zero Dispersion Limit

Dispersive shock wave (DSW) of the KdV Eq.

Structure of a dispersive shock wave

Soliton front

Oscillatory

front

Here cn(x,m) is Jacobi elliptic function

Many other Dispersive shock waves

England: River Severn

DSW on the shallow water

Australia: Morning Glory

in the Gulf of Carpentaria

DSW in the atmosphere

Many other Dispersive shock waves

DSW in optical media

Nat. Phys. 3 (1) (2007)

46–51.

DSW in ultracold atoms

Phys. Rev. A 74 (2006)

023623.

An example of Dispersive shock waves

in the NLS equation.

The famous defocusing NLS equation

with step-like initial data

Defocusing NLS equation

G.A. El, et al. Physica D 87 (1995) 186-192: Whitham theory

R. Jenkins, Nonlinearity 28 (2015) 2131–2180 Deift-Zhou method

Transform defocusing NLS equation into

Madelung variables

One-phase solution

Whitham equations

The self-similar

evolution of the

Riemann invariants.

Result of Whitham theory

The self-similar solution

by Whitham theory.

Result of Riemann–Hilbert Problem

The classification of asymptotic solution.

R. Jenkins, Nonlinearity 28 (2015) 2131–2180 by Deift-Zhou method

The famous KdV equation

is transferred into the modied KdV (mKdV) equation

2. cmKdV-Whitham equations

under the Miura transformation

The complexification of mKdV is the complex

mKdV (cmKdV) equation

The defocusing semi-classical cmKdV equation

has Lax pair

The cmKdV-Whitham equations

Yuji Kodama, SIAM J. Math. Anal. 41 (2008) 26-58.

taking the Madelung transformation

The Whitham Theory

For the defocusing semi-classical cmKdV equation

=

( , ) (x,t)v x t S

xequivalently, if writing then

=(x,t)

(x,t; ) (x,t)iS

q e

The genus-0 cmKdV–Whitham equations

Taking the dispersionless limit

We have the cmKdV equation in the dispersive-hydrodynamic form

we have

which can be represented as cmKdV–Whitham equations in

the diagonal form

The genus-0 cmKdV–Whitham equations

with the Riemann invariants

and the characteristic velocities

The genus-1 cmKdV-Whitham equations

For the Madelung transformation

cmKdV has periodic solution for genus-1 region

The method: finite-gap integration by Lax pair

Where are determined implicitly by genus-1

cmKdV-Whitham equation

where

where

A.M. Kamchatnov, Physics Reports. 286 (1997) 199-270.

The defocusing mKdV equationwith a step-like initial data

Step-like initial data

We transform the initial value problem from physical

variables to the Riemann invariants form

L. Kong, L. Wang, D. Wang, C. Dai, X. Wen, L. Xu, Nonlinear

Dyn. (2019) 98:691–702.

Consider the transformation

3.Basic structures: Rarefaction wave (RW)

RW-1:

RW-2:

RW-3:

For variable the self-similar solution satisfies

Condition of rarefaction: vertex

Soliton front: where m→1, and we have

Basic structures (continuous): DSW

One can see from the expression

Harmonic front: where m→0, and we have

or

Symmetric: DSW-I and DSW-II

DSW-III and DSW-IV

DSW-V and DSW-VI

For DSW-I:

For DSW-III:

Basic structures (continuous): DSW

For DSW-V:

Basic structures (continuous): DSW

4.Classification of step-like initial condition

The initial condition of the cmKdV equation

A:

B:

C:

D:

E:

F:

with

Self-similar solutions under condition (A.1)

I: Plateau, II: RW-Ⅲ, Ⅲ: RW- I, Ⅳ: DSW-III, Ⅴ: Plateau

Self-similar solutions under condition (A.1)

Self-similar solutions under condition (A.1) (continuous):

Outside the boundaries of DSW (genus-1) region are controlled

by rarefaction waves.

Self-similar solutions under condition (A.2)

Indeed, the results of Riemann distribution under A.2 and A.1 are

symmetric with respect to x-axis, and the

density are exactly the same.

Self-similar solutions under condition (A.3)

Condition A.3 gives three possible Riemann distributions:

Example of

Self-similar solutions under condition (B.2)

Self-similar solutions under condition C

Condition C gives three possible Riemann distributions:

Example of (C.3). The boundaries of genus-2 regions are

Self-similar solutions under condition D

Genus-2 regions also appears in condition D, the situation for (C)

and (D) are the same to some extend except the type of DSWs for

collision are different. The boundaries of genus-2 regions are:

A:

B:

C:

D:

E:

F:

The solutions under E and F are the same as those

which are under D and B, respectively.

The cases E and F

5. Further work: Genus-2 cmKdV-Whitham

⚫Derivation of genus-2 cmKdV-Whitham equations:

Expressed by Riemann-Theta functions.

⚫The analysis under general initial condition

where

⚫Other physically interesting initial conditions

Thanks for your

attention!