Hirota’s Bilinear Method(The Direct Method)

David GeroskiApplied Physics Program

MATH 651 Final Presentation

Outline

• The Direct Method

• Motivating Example – Korteweg-de Vries (KdV) Equation

• Mechanics of the Direct Method

• Soliton solutions to the Nonlinear Schrödinger Equation (NLS)

• Physical Applications – the Benjamin-Ono Equation

• Soliton solutions to the Benjamin-Ono Equation

The Direct Method, The Hirota Method

• Developed in parallel to the Inverse Scattering Transform

• C. 1970

• Direct Method (Japan), Hirota/Bilinear Method (everywhere else)

• Seeks to directly find the form of solitons solutions admitted by a given nonlinear PDE

• Develops a framework applicable to many nonlinear PDEs

Inverse Scattering vs. Direct Method𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0 (KdV)

𝑢0(𝑥)

𝑟(𝑘; 𝑡 = 0)

𝑢𝑡 + 𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0𝑢(𝑥, 𝑡)

Direct Scattering Transform

Explicit Time Evolution

Inverse Scattering Transform

Overall goal

Bilinear Equations

Soliton Solutions

“Bilinear Ansatz”

𝑟(𝑘; 𝑡)

Example of the Direct Method: Soliton Solution to KdV

• Hirota’s original discovery of the exact solution by bilinearization

𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 𝑢𝑡 + 3 𝑢2 𝑥 + 𝑢𝑥𝑥𝑥 = 0

𝑢 = 2 log 𝑣 𝑥𝑥 = 2𝑣𝑋𝑣 𝑥

2𝑣𝑥𝑣 𝑥𝑡

+ 12𝑣𝑋𝑣

2

𝑥

+ 2𝑣𝑥𝑣 𝑥𝑥𝑥𝑥

= 0

𝑣𝑥𝑡𝑣 − 𝑣𝑥𝑣𝑡 + 𝑣𝑣𝑥𝑥𝑥𝑥 − 4𝑣𝑥𝑥𝑥𝑣𝑥 = 𝐶 = 0𝑣 = Σ𝑛𝜖

𝑛𝑣𝑛; 𝑣0 = 1𝜖1: 𝑣1 𝑥𝑡 + 𝑣1 𝑥𝑥𝑥𝑥 = 0 ⇒ 𝑣1 = exp 𝜂 ; 𝜂 = 𝑃1𝑥 + Ω𝑡 + 𝜂0, 𝑃1

3 + Ω = 0

𝜖2:𝜕

𝜕x

𝜕

𝜕𝑡+

𝜕3

𝜕𝑥32𝑣2 + 𝑣1 ∗ 𝑣1 = 0

⇒ 𝑣 = 1 + 𝜖𝑣1

Soliton Solution to KdV (cont.)

• This method yields the single soliton solution quite simply, despite the relatively simple choice of the perturbation series

• Interactions between solitons can be determined systematically by taking linear superpositions in the first order of the perturbation

• This equation is linear, so everything is still satisfied

• This will require higher-order matching, but everything is still done systematically, based on how many solitons we want to have interacting

𝑣 = 1 + 𝜖𝑣1 = 1 + 𝜖 exp 𝜂𝑢 = 2 log 1 + 𝜖 exp 𝜂

𝑢 =2𝑃1

2 exp 𝜂

1 + exp 𝜂 2

𝑢 =𝑃12

2sech2

𝜂

2

Single Soliton Solution to KdV

Single Soliton Solution to KdV (cont.)

• Frequency content of the single soliton

2-Soliton Solution to KdV

• Following the intuition of the single soliton solution, take the same ansatz for KdV

𝑣𝑥𝑡𝑣 − 𝑣𝑥𝑣𝑡 + 𝑣𝑣𝑥𝑥𝑥𝑥 − 4𝑣𝑥𝑥𝑥𝑣𝑥 = 𝐶 = 0𝑣 = Σ𝑛𝜖

𝑛𝑣𝑛; 𝑣0 = 1

𝜖1: 𝑣1 𝑥𝑡 + 𝑣1 𝑥𝑥𝑥𝑥 = 0 ⇒ 𝑣1 = exp 𝜂1 + exp(𝜂2); 𝜂𝑖 = 𝑃𝑖𝑥 + Ω𝑖𝑡 + 𝜂0𝑖 , 𝑃𝑖3 + Ωi = 0

𝑣 = 1 + 𝜖 exp 𝜂1 + exp 𝜂2 + 𝜖2𝑎12 exp 𝜂1 + 𝜂2 ; 𝑎12 =𝑃1 − 𝑃2

2

𝑃1 + 𝑃22

𝑢 =𝑣𝑥𝑥𝑣 − 𝑣𝑥

2

𝑣2

2-Soliton Solution to KdV (cont.)

2-Soliton Solution to KdV (cont.)

N-Soliton Solution to KdV

• Given the 2-soliton solution, we can guess what will happen for 𝑁 ≥ 2

• Essentially, we will have terms like individual solitons, and interaction terms for when solitons are located in the same place in spacetime

• This is Hirota’s original result in 1971

𝑓1 = Σ𝑛 exp 𝜂𝑛𝑓 = Σ𝜇=0,1 exp Σ𝑗𝜇𝑗𝜂𝑗 + Σ𝑘<𝑗𝜇𝑗𝜇𝑘𝐴𝑗𝑘

exp 𝐴𝑗𝑘 =𝑃𝑗 − 𝑃𝑘

2

𝑃𝑗 + 𝑃𝑘2

Mechanics of the Direct Method

• Solving KdV:

• Assume a form which yields a bilinear equation

• Solve the new equation using a proper perturbation series

• First order evolution equation is linear, yielding soliton solutions

• Higher order terms can be used to balance interactions between solitons

• Eventually, high-order terms can be set to 0, giving a convergent series

• Hirota’s Formulation

• Assume a bilinear form of the solution

• Introduce a new operator, defining a class of Hirota equations

• Several Hirota equations have been solved directly

• All other Hirota equations for which we have an answer obtained by transform

Hirota Derivatives• Define the Hirota Derivative by MacLaurin Series:

• Evaluate the first few terms by hand:

L 𝜑, 𝜕𝑡, 𝜕𝑥 𝜑 = 𝑠 𝑥, 𝑡 = 0

𝜑 = 𝑓 𝑥 ∘ 𝑔 𝑥 = 𝑓 𝑥 + 𝑦 ∘ 𝑔 𝑥 − 𝑦 ቚ𝑦=0

𝑓 𝑥 + 𝑦 ∘ 𝑔 𝑥 − 𝑦 = Σ𝑛1

𝑛!𝐷𝑥𝑛 𝑓 ∘ 𝑔 𝑦𝑛

𝜑 = 𝑓 𝑥 + 𝑦 ∘ 𝑔 𝑥 − 𝑦 ቚ𝑦=0

= Σ𝑛1

𝑛!

𝜕𝑛

𝜕𝑦𝑛𝑓 ∘ 𝑔 𝑦𝑛

𝑓 ∘ 𝑔 ቚ𝑦=0

= 𝑓 𝑥 𝑔 𝑥 +𝜕𝑓

𝜕𝑦𝑔 − 𝑓

𝜕𝑔

𝜕𝑦𝑦 +

1

2

𝜕2𝑓

𝜕𝑦2𝑔 − 2

𝜕𝑓

𝜕𝑦

𝜕𝑔

𝜕𝑦+ 𝑓

𝜕2𝑔

𝜕𝑦2𝑦2 + …

𝑓 ∘ 𝑔 ቚ𝑦=0

= 𝑓 𝑥 𝑔 𝑥 +𝜕𝑓

𝜕𝑥𝑔 − 𝑓

𝜕𝑔

𝜕𝑥𝑦 +

1

2

𝜕2𝑓

𝜕𝑥2𝑔 − 2

𝜕𝑓

𝜕𝑥

𝜕𝑔

𝜕𝑥+ 𝑓

𝜕2𝑔

𝜕𝑥2𝑦2 + …

𝐷𝑥1 𝑓 ∘ 𝑔 = 𝐷𝑥 𝑓 ∘ 𝑔 =

𝜕𝑓

𝜕𝑥𝑔 − 𝑓

𝜕𝑔

𝜕𝑥, 𝐷𝑥

2 =𝜕2𝑓

𝜕𝑥2𝑔 − 2

𝜕𝑓

𝜕𝑥

𝜕𝑔

𝜕𝑥+ 𝑓

𝜕2𝑔

𝜕𝑥2

𝐷𝑥𝑛 𝑓 ∘ 𝑔 ≡

𝜕

𝜕𝑥−

𝜕

𝜕𝑥′

𝑛

𝑓 𝑥 ∘ 𝑔(𝑥′)

Hirota Equations

• Define a Hirota Equation in the following way:

• Key Properties

• Hirota Equations are explicitly bilinear (e.g. quadratic in function)

• Hirota Equations which do not depend on a constant (𝐷𝑞0) admit constants as solutions

• Hirota Equations which do not depend on a constant 𝐷𝑞0 explicitly admit pseudo-soliton (or better) solutions

• Several famous nonlinear equations can be represented as a Hirota equations

• KdV

• Nonlinear Schrödinger Equations (both Focusing and Defocusing)

• Toda Equations

• Benjamin-Ono

𝑃 𝐷𝑞𝑖 𝑓 ∘ 𝑓 = 0

Pseudo-Soliton Solutions

• Assume a Hirota Equation with no constant derivative

• The first order of the equation is satisfied by a soliton solution

• Represents a sum over j solitons, traveling in a sum over i dimensions

• For a series that eventually converges, these solitons are usually conserved

𝑃 Σ𝑚>0,𝑛𝐷𝑥𝑛𝑚 𝑓 ∘ 𝑓 = 0

𝑓 = 1 + Σ𝑛≥1𝜖𝑛𝑓𝑛

𝜖0: 𝑃 1 ∘ 1 = 0

𝜖1: 𝑃 Σ𝑚>0,𝑛𝜕𝑥𝑛𝑚 𝑓1 1 = 0; 𝜕𝑥𝑛

𝑚 ≔𝜕𝑚

𝜕𝑥𝑛𝑚

𝑓1 = Σ𝑗 exp Σ𝑖𝑘𝑖𝑗𝑥𝑖

Σ𝑖𝑘𝑖𝑗= 0

KdV by Hirota’s Method

• Making the substitution before, KdV yields a Hirota equation

• Given the structure of the Hirota equation, this equation clearly admits soliton solutions

𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0

𝑢 = 2 log 𝑣 𝑥𝑥 =𝑣𝑋𝑣 𝑥

𝑣𝑥𝑡𝑣 − 𝑣𝑥𝑣𝑡 + 𝑣𝑣𝑥𝑥𝑥𝑥 − 4𝑣𝑥𝑥𝑥𝑣𝑥 = 𝐶 = 0𝐷𝑥𝐷𝑡 + 𝐷𝑥

4 𝑣 ∘ 𝑣 = 0

Nonlinear PDE’s Expressed as Hirota Equations

• Defocusing and Focusing NLS

• Toda Lattice

• Benjamin-Ono

• Note: H is a Hilbert transform

𝑖Ψ𝑡 +Ψ𝑥𝑥 + 2𝑐 Ψ 2Ψ = 0; 𝑐 = ±1𝑖𝐷𝑡 + 𝐷𝑥

2 (𝐺 ∘ 𝐹) = 𝜆(𝐺 ∘ 𝐹) (1)𝐷𝑥2 𝐹 ∘ 𝐹 − 2𝑐 𝐺 2 = 𝜆𝐹2

𝜕2

𝜕𝑡2log 1 + 𝑉𝑛 𝑡 = 𝑉𝑛+1 𝑡 − 2𝑉𝑛 𝑡 + 𝑉𝑛−1 𝑡

𝑉𝑛 𝑡 =𝜕2

𝜕𝑡2log 𝑓𝑛 𝑡 =

𝑓𝑛+1 𝑡 𝑓𝑛−1 𝑡

𝑓𝑛 𝑡 2 − 1

𝐷𝑡2 − 4 sinh2

𝐷𝑛2

𝑓𝑛 ∘ 𝑓𝑛 = 0

𝑢𝑡 + 4𝑢𝑢𝑥 + 𝐻𝑢𝑥𝑥 = 0𝑖𝐷𝑡 − 𝐷𝑥

2 𝑓′ ∘ 𝑓 = 0

Nonlinear Schrödinger Equation: Another Application

• NLS can be bilinearized using the following transformation

• This assumption yields what is called an envelope soliton

• The structure of the soliton depends on the choice of parameter 𝜆• Bright soliton: 𝜆 = 0 (FNLSE)

• Dark soliton: 𝜆 = 1 (DNLSE)

𝑖Ψ𝑡 +Ψ𝑥𝑥 + 2𝑐 Ψ 2Ψ = 0; 𝑐 = ±1

Ψ =𝐺

𝐹𝑖𝐷𝑡 + 𝐷𝑥

2 (𝐺 ∘ 𝐹) = 𝜆(𝐺 ∘ 𝐹) (1)𝐷𝑥2 𝐹 ∘ 𝐹 − 2𝑐 𝐺 2 = 𝜆𝐹2

Soliton Solutions to FNLSE: “Bright Solitons”

• Choosing Focusing form on NLS yields Bright Solitons

Ψ =𝐴1 sech 𝜉1 exp(𝑖𝜁1)(cos 𝜙1 + 𝑖𝑠𝑖𝑛 𝜙1 tanh 𝜉2 + 𝐴2 sech 𝜉2 exp 𝑖𝜁2 cos 𝜙2 + 𝑖𝑠𝑖𝑛 𝜙2 tanh 𝜉1

cosh 𝑎 + sinh(𝑎)(tanh 𝜉1 tanh 𝜉2 − sech 𝜉1 sech 𝜉2 cos 𝜁1 − 𝜁2

𝑎 = log𝑃1 − 𝑃2|𝑃1 + 𝑃2

∗|

𝜙1 = arg𝑃1 − 𝑃2𝑃1 + 𝑃2

∗ , 𝜙2 = arg𝑃2 − 𝑃1𝑃2 + 𝑃1

∗

𝐴𝑖 =1

2𝑃𝑖 + 𝑃𝑖

∗ , 𝜉𝑖 = 𝑅𝑒 𝑃𝑖𝑥 − Ω𝑖𝑡 + 𝐶𝑖 , 𝜁𝑖 = 𝐼𝑚 𝑃𝑖𝑥 − Ω𝑖𝑡 + 𝐶𝑖

Ω𝑖 = −𝑖

2𝑃𝑖2

Soliton Solution to FNLSE

Soliton Solution to FNLSE

2-Soliton Solution to FNLSE (Collision)

2-Soliton Solutions to FNLSE

Advantages of solving FNLSE using Direct Method

• Soliton solutions can be found over the course of a day

• Uses a very common ansatz in Direct Method to bilinearize FNLSE

• Bilinearized equations can be solved perturbatively or by making a soliton ansatz

• Parameters in the Soliton solution are clear

• Amplitude(s)

• Speed(s)

• Location of Soliton center(s)

Limitations of Direct Method in FNLSE

• Does not solve a general problem for a given nonlinear PDE

• Soliton solutions say nothing about the initial conditions

• Soliton solutions do not necessarily form a complete basis for evolving an initial condition

• Using language learned in class based on Inverse Scattering Transform

• These solutions come from the reflectionless initial condition

• Riemann-Hilbert Problem is solved by placing simple complex poles based on parameters in solution

• Solitary wave solution is not guaranteed to say anything about higher-order poles in the complex plane or an initial condition that is not reflectionless

The Benjamin-Ono Equation

• Another integrable equation:

• H is the Hilbert Transform

• Describes the motion of internal waves in the deep ocean

• Provides environmental mismatch for sonars

• Serves to toss around submersible ships

• Provides some pitch and toss, shaping the deep ocean ecosystem

• Describes the propagation of Rossby waves in a rotating fluid

𝑢𝑡 + 4𝑢𝑢𝑥 + 𝐻𝑢𝑥𝑥 = 0

Bilinearization of Benjamin-Ono Equations

• Assume the following form, for 𝑓 is a linear equation in space

𝑢𝑡 + 4𝑢𝑢𝑥 + 𝐻𝑢𝑥𝑥 = 𝑢𝑡 + 2 𝑢2 𝑥 +𝐻𝑢𝑥𝑥0

𝑢 =𝑖

2

𝜕

𝜕𝑥log

𝑓∗

𝑓𝐻 ≔ 𝑃 න

𝑅

𝑑𝜏𝑢 𝜏

𝑡 − 𝜏

𝐻𝑢 =𝑖

2𝐻

𝜕

𝜕𝑥log

𝑓∗

𝑓=𝑖

2𝐻

𝑓∗′

𝑓∗−𝑓′

𝑓

𝐻𝑢 =𝑖

2𝐻

1

𝑓∗−1

𝑓=𝑖

2

1

𝑓∗+1

𝑓= −

1

2

𝜕

𝜕𝑥log(𝑓∗𝑓)

𝑖

2log

𝑓∗

𝑓𝑥𝑡

+ 2𝑖

2log

𝑓∗

𝑓𝑥

− log 𝑓∗𝑓 𝑥𝑥𝑥 = 0

𝑖 𝑓𝑡∗𝑓 − 𝑓∗𝑓𝑡 − 𝑓𝑥𝑥

∗ 𝑓 − 2𝑓𝑥∗𝑓𝑥 + 𝑓𝑥𝑥𝑓 = 0

𝑖𝐷𝑡 − 𝐷𝑥2 𝑓∗ ∘ 𝑓 = 0

Soliton Solution of the Benjamin-Ono Equation

• 1st Class of soliton solutions: Normalizable solitons

𝑖𝐷𝑡 − 𝐷𝑥2 𝑓∗ ∘ 𝑓 = 0

𝑓 = Σ𝑛𝜖𝑛𝑓𝑛

𝜖1: 𝑖𝐷𝑡 − 𝐷𝑥2 𝑓1

∗ ∘ 𝑓0 + 𝑓0 ∘ 𝑓1 = 0

𝑖𝜕𝑓1

∗

𝜕𝑡−𝜕𝑓1𝜕𝑥

−𝜕2𝑓1

∗

𝜕𝑥2+𝜕2𝑓1𝜕𝑥2

= 0

𝑓1 = 𝑖 𝑥 − 𝑎𝑡 − 𝑥0 +1

𝑎= 𝑖𝜃 𝑥, 𝑡 +

1

𝑎𝜖2: 𝑖𝐷𝑡 − 𝐷𝑥

2 𝑓2∗ ∘ 𝑓0 + 𝑓0 ∘ 𝑓2 + 𝑓1

∗ ∘ 𝑓1 = 0𝑖𝐷𝑡 − 𝐷𝑥

2 𝑓1∗ ∘ 𝑓1 = 0 ⇒ 𝑓2 = 0

𝑓 = 𝜖𝑓1 + 𝑓0𝑓0 = 0

𝑢 =𝑖

2log

𝑓∗

𝑓𝑥

⇒ 𝑢 =𝑎

1 + 𝑎2𝜃 𝑥, 𝑡 2

Behavior of the Soliton Solution of Benjamin-Ono

Frequency Content of the Soliton

2-Soliton Solution of the Benjamin-Ono Equation

• 2-Soliton solution can be found by making the same assumption as in the 1-soliton case, the structure is the same as finding the 2-soliton solution to KdV

𝑓2 = 𝑐3𝜃1𝜃2 + 𝑐2𝜃2 + 𝑐1𝜃1 + 𝑐0

𝑓2 = −𝜃1𝜃2 + 𝑖𝜃1𝑎2

+𝜃2𝑎1

+1

𝑎1𝑎2

𝑎1 + 𝑎2𝑎1 − 𝑎2

2

𝑢 =𝑖

2log

𝑓2∗

𝑓2 𝑥

𝑢 =

𝑎12𝑎2𝜃1 + 𝑎2

2𝑎1𝜃2 𝜃2 + 𝜃1 +𝑎1 + 𝑎2𝑎1 − 𝑎2

2

− 𝑎1𝑎2𝜃1𝜃2 𝑎1 + 𝑎2

𝑎1 + 𝑎2𝑎1 − 𝑎2

2

− 𝑎1𝑎2𝜃1𝜃2

2

+ 𝑎1𝜃1 + 𝑎2𝜃22

2 Soliton Solution

N-Soliton Solutions of the Benjamin-Ono Equation

• As a curiosity, the 2-Soliton solution can be expressed as the following determinant:

• Similarly, the N-Soliton solution (presented here without proof) can be written:

𝑓2 = det

𝑖𝜃1 +1

𝑎1

2

𝑎1 − 𝑎2

−2

𝑎1 − 𝑎2𝑖𝜃2 +

1

𝑎2

𝑓𝑁 = 𝑑𝑒𝑡𝑀𝑁

𝑀𝑁 𝑗𝑘 =

𝑖𝜃𝑗 +1

𝑎𝑗, 𝑗 = 𝑘

2

𝑎𝑗 − 𝑎𝑘, 𝑗 ≠ 𝑘

Review of Topics Covered

• One-, Two-, and N-Soliton solutions of several integrable systems

• Shallow Water Waves (KdV)

• Wave Bullet/Focusing Media (Focusing Nonlinear Schrödinger Equation)

• Deep Ocean Internal Wave Travel (Benjamin-Ono Equation)

• Bilinearization of wide class of nonlinear PDE’s

• Three possible bilinear ansatzes to make (most popular choices)

• Ways to derive soliton expressions

• General form of multi-soliton solutions, once a single soliton has been found

• General pros/cons of the Direct Method

Conclusion

• The Direct Method (also known as the Hirota Method) is a viable way of finding solitary wave solutions to a wide class of nonlinear, integrable partial differential equations

• Allows direct exploration of the exact form of solitary waves

• Assumptions made in solving differential equations tend to involve less rigorous assumptions than inverse scattering

• The Direct Method does not guarantee a solution to the general problem

• Solutions are obtained making an assumption

• Solitary wave solutions usually do not form a complete basis in which to evolve an initial condition

References

• R. Hirota. The Direct Method in Soliton Theory. UK. Cambridge University Press, 2004.

• T. Miwa, Jimbo, and Date. Solitons – Differential Equations, Symmetries and Infinite Dimensional Algebras. UK. Cambridge University Press, 1993.

• Y. Matsuno. Bilinear Transformation Method. Orlanda, FL. Academic Press. 1984.

• R. Hirota. “Exact Envelope-Soliton Solutions of a Nonlinear Wave Equation.” J. Math. Phys. vol. 14, no. 7, 805-809, 1973.