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.