Mathematics and Physics of Shallow Water Waves

Mathematics for web-like patterns of solitary waves in shallow water

Yuji Kodama The Ohio State University

Colloquium Lecture National Cheng Kung University Tainan, Taiwan, December 20, 2018

Real Shallow Water Waves

Mexican beach (by Mark Ablowitz)

Real Shallow Water Waves

Mexican beach (by Mark Ablowitz)

Real Shallow Water Waves

Washington State beach (by Bernard Deconick)

Real Shallow Water Waves

Estonian lake (by Ira Didenkulova)

Shallow Water Wave Equation

Three dimensional water wave profile:

Here we have







λ0=typical wavelength∼horizontal scale h₀=asymptotic depth∼vertical scale a₀ =typical amplitude∼nonlinear scale We consider an irrotational and incompressible fluid:

~

v =∇φ and ∇ ·~v = ∆φ= 0, whereφis the velocity potential.

Shallow Water Wave Equation

Then the 3-dimensional water wave equation is given by the Euler equation (non-dimensional form)

(a) φzz+β∆⊥φ= 0, for 0<z <1 +αη,

(b) φz= 0, at z = 0,

(c)











φt+1

2α|∇_⊥φ|²+ 1 2

α

βφ²_z+η= 0, η_t+α∇_⊥φ· ∇_⊥η= 1

βφ_z,

at z = 1 +αη,

whereη is the surface elevation,∇_⊥= (_∂x^∂ ,_∂y^∂ ), ∆⊥=∂_x²+∂_y². The parametersα,β are defined by

α:= a₀ h0

, β :=

h₀ λ0

2

Asymptotic perturbation method

We now introduce a small parameter 0<0:

Small amplitude and long waves α:= a₀

h0

∼ β:=

h₀ λ0

2

=O().

Examples:

(a) Tohoku tsunami in northern Japan on March 11, 2011:

α= 10m 1000m= 1

100 ∼β=

1km

10km 2

.

(b) Rogue wave observed off South African coast on July, 1909:

α= 20m 200m = 1

10 ∼β =

200m

600m ²

.

(c) Rogue wave observed in Lake Superior on Nov. 1975:

α= 11m 160m = 7

100 ∼β =

160m

600m ²

.

Asymptotic perturbation method

With the assumptionα∼β ∼(i.e. small and long wave), we employ theasymptotic perturbation method to derive the approximate equation:

We first note that the velocity potentialφsatisfying (a) and (b) in the water wave equation can be written in the expansion form,

φ(x,y,z,t) = cos(zp

β∆⊥)ψ(x,y,t)

=ψ−βz²

2∆⊥ψ+O(²).

Recall thatz = 0 implies the bottom, and z = 1 is the asymptotic surface. That is, we have

ψ(x,y,t) =φ(x,y,0,t).

Asymptotic perturbation method

Then from (c) of the water wave equation, the equation for the velocity potentialψ (the Bernoulli equation) becomes

ψt+η+α

2 |∇ψ|²−β

2∆ψt =O(²) and the equation for the surface elevationη is

ηt+ ∆ψ−α∇ ·(ψt∇ψ)−β

6∆²ψ=O(²).

Here we simply write∇_⊥=∇and ∆⊥= ∆, and note that those equations arerotationally invariant on thexy-plane.

The set of these equations for (ψ, η) is called theBoussinesq-type equation (1872).

The Benny-Luke equation

Eliminatingη from the Boussinesq-type equation, we have the (regularized)Benney-Lukeequation (1964),

1−β

3∆

ψ_tt −∆ψ+α(∇ψ· ∇ψ_t+∇ ·(ψ_t∇ψ)) =O(²).

This can be reduced to theKorteweg-de Vries (KdV) equation (1895) for a far field with a unidirectional approximation: let χ=Xcos Ψ0+Ysin Ψ0 be the coordinate of the propagation direction and usingψ_χ=η+O(), we have the KdV equation for the surface elevationH=h0η in the physical coordinates,

HT +c0Hχ+3c₀ 2h0

HHχ+c₀h²₀

6 Hχχχ= 0.

wherec0 =√

gh0 the velocity of the surface wave (g = 9.8m/sec).

The Benney-Luke equation

The KdV equation has one-soliton solution in the form, H=a₀sech²

s3a₀ 4h³₀

χ−c₀

1 + a₀

2h0

T

. An example of interactions of those solitary waves of the Benney-Luke equation:

Here the left figure is the initial wave given by a linear combination of two KdV solitons propagating different directions. The right figure is the solution after some timet>0.

Numerical simulation of the Benney-Luke equation

A simulation of the ion-acoustic wave (similar to the Benney-Luke) equation by Kako and Yajima (1982):

Numerical simulation of the Benney-Luke equation

Notice the change of interaction, and astem wave starts to develop when the interaction angle takes acriticalvalue.

Waves in a pool

Instantgeneration of those waves with stems in a pool:

Can be alaptop experiment for exotic wave patterns.

Asymptotic perturbation method continued

We make a further assumption:

Quasi-two dimensionality:

ζ :=√

γy, γ =O().

That is, we assume a weak dependence in they-direction. Also note that thisbreaks the rotational symmetry. Then we have

ψ_t+η+α

2ψ²_x+β

2η_xx =O(²) ηt+ψxx−α(ψtψx)_x−β

6ψxxxx +γψ_ζζ =O(²).

Asymptotic perturbation method continued

We then consider afar fieldwith the moving coordinate, Unidirectional approximation:

ξ:=x−t, τ :=t.

Then eliminateη and define v :=ψ_ξ, we have 2vτ ξ+ 3α(vvξ)ξ+β

3vξξξξ+γvζζ =O(²) Introduce thenew(scaled) variables (x,y,t,u) defined by

ξ=p

βx, ζ=p

βγy, τ =−3 2p

βt, v = 2 3αu.

The Kadomtsev-Petviashvili (KP) equation (1970) (−4u_t+ 6uux+uxxx)x+ 3uyy = 0

The KP equation

We write the solutionu(x,y,t) in the form with theτ-function u(x,y,t) = 2 ∂²

∂x²lnτ(x,y,t),

Then theτ-function satisfies a bilinear equation, called Hirota bilinear form,

−4(τ τxt −τxτt) + (τ τxxxx −4τxτxxx+ 3τ_xx² ) + 3(τ τyy−τ_y²) = 0.

This can be written in the following nice form,

P(Dx,Dy,Dt)τ◦τ := [Dx(−4D_t+D_x³) + 3D_y²]τ◦τ = 0.

whereD_zⁿ’s are the Hirota (skew) derivatives defined by D_zⁿf ◦g =

∂

∂z − ∂

∂z⁰ n

f(z)g(z⁰) z=z⁰.

The KP equation

Remark: The KP bilinear operator gives thedispersion relation, P(k_x,k_y, ω) =−4k_xω+k_x⁴+ 3k_y³= 0.

This can be parametrized by

k_x =κ_i−κ_j, k_y =κ²_i −κ²_j, ω =κ³_i −κ³_j, whereκ_i’s are arbitrary constants. From this, one can find a solution ofτ in the sum of exponentials,

τ =a₁e^θ1+a₂e^θ2+. . .+a_Me^θM with θi =κix+κ²_iy+κ³_it,

whereκ_i’s are arbitrary constants (i.e. P(D_x,D_y,D_t)e^θi ◦e^θj = 0).

Note here that theτ-function satisfies thelinear equations τy =τxx, τt =τxxx.

The KP equation

In general, we writeτ(x,y,t) in theWronskian form:

τ = Wr(f1,f2, . . . ,f_N) =

f₁⁽⁰⁾ f₁⁽¹⁾ · · · f₁^(N−1) f₂⁽⁰⁾ f₂⁽¹⁾ · · · f₂^(N−1)

... ... . .. ... f_N⁽⁰⁾ f_N⁽¹⁾ · · · f_N^(N−1)

,

wherefi’s are arbitrary smooth functions of the variables{x,y,t}, andf_j⁽ⁿ⁾= ∂ⁿfj

∂xⁿ for n= 0,1, . . . ,N−1.

Then we have the following Proposition for the solution of the KP equation:

The KP equation

Proposition

The Wronskian form provides a solution of the KP equation, if those{f_i(x,y,t) :i = 1, . . . ,N} satisfy thelinear equations,

∂f_i

∂y = ∂²f_i

∂x², ∂f_i

∂t = ∂³f_i

∂x³.

Then considerfinite dimensional solutions (finite Fourier series):

f_i(x,y,t) =

M

X

j=1

a_ije^θj(x,y,t), i = 1, . . . ,N<M, where θj(x,y,t) :=κjx+κ²_jy+κ³_jt, j = 1, . . . ,M.

This solutionu(x,y,t) of the KP equation is then completely determined by theκ-parameters and theN×M matrixA= (aij).

The KP equation

Mathematical remark: We have a picture of theGrassmannian Gr(N,M), which is defined as follows:

Span_R{E~j = (1, κj, . . . , κ^M−1_j )^Te^θj :j = 1, . . . ,M} ∼=R^M. Span_R{~f_i :i = 1, . . . ,N} forms an N-dimensional subspace in R^M,

f~_i =

M

X

j=1

a_i,jE~_j for i = 1, . . . ,N, where the coefficients {a_i,j}define the N×M matrix

A=







a11 · · · a_1M ... . .. ... ... aN1 · · · aNM





∈MN×M(R), with full rank, i.e. rank(A) =N.

The KP equation

For anyH ∈GL_N(R), theτ-function with the new base (~g1, . . . , ~gN) = (~f1, . . . , ~fN)H gives thesame solution, since Wr(g1, . . . ,g_N) = det(H)Wr(f1, . . . ,f_N). This implies that the τ-function can be identified as a point of the Grassmannian, Gr(N,M), i.e.

Gr(N,M)∼= GLN(R)\M_N×M(R), with dim Gr(N,M) =NM−N²=N(M−N).

H ∈GL_N(R) gives a row reduction of theA-matrix. For example, a genericA can be written in the reduced row echelon form (RREF),

A=







1 · · · 0 ∗ · · · ∗ ... . .. ... ... ... ...

0 · · · 1 ∗ · · · ∗





 Here 1’s are called ”pivot” ones.

One-soliton and Y-soliton

Example 1: N= 1, M = 2. This gives oneline-soliton solution:

WithA= (1a)∈Gr(1,2), the τ-function is τ =e^θ1+ae^θ2 =e^θ1+e^θ˜2 =e^{12(θ1+˜θ2)}

e^{12(θ1−θ˜2)}+e^{−12(θ1−θ˜2)}

= 2e^{12(θ1+˜θ2)}cosh1

2(θ₁−θ˜₂), whereθi’s are given by

θ₁=κ₁x+κ²₁y+κ³₁t, θ˜₂ =κ₂x+κ²₂y+κ³₂t+ lna Then we have

u = 2 ∂²

∂x²lnτ = 1

2(κ1−κ2)²sech²1

2(θ1−θ˜2) Denote this solution by[1,2]-soliton. In general, [i,j]-soliton.

One-soliton and Y-soliton

3D figure of the solutionu_A, and the contour plot. Numbers(i) represent thedominantexponential term e^θj’s withκ1< κ2 .

The line of the wave crest is given byθ₁ = ˜θ₂ =θ₂+ lna, i.e.

x+ (κ1+κ2)y+ (κ²₁+κ²₂+κ1κ2)t− 1

κ₁−κ₂ lna= 0.

One-soliton and Y-soliton

Example 2: N= 1 andM = 3, i.e. A= (1a b)∈Gr(1,3). The τ-function is

τ =e^θ1+ae^θ2+be^θ3, with θ_i =κ_ix+κ²_iy+κ³_it.

There are three distinct regions in thexy-plane, and for a fixedt, in each region one of the exponentiale^θj dominates. Then the solution has a Y-shape pattern (calledY-type solution):

One-soliton and Y-soliton

As we noted that each line-soliton exists as abalance between two dominant exponentials, and crossing the soliton, the index set of the dominant exponential changes. The following diagram, called chord diagram, represents this change:

One-soliton with {i,j}, i.e. [i,j]-soliton:

ki kj π = ^{i j} _{j i}

Y-solitons with {i,j,l}, i.e. [i,j]-, [j,l]- and [i,l]-solitons:

ki kj 0 kl

ki kj 0 kl

These give the permutations π= ^{i j l}_{l i j}

andπ= ^{i j l}_{j l i} .

Classification Theorem

Now weclassify the soliton solutions generated by the Wronskian form,

τ = Wr(f₁,f₂, . . . ,f_N).

Usingf_i =P_M

j=1a_i,je^θj, this can be expressed as,

τ = Wr(f1,f2, . . . ,f_N) =

f₁⁽⁰⁾ f₁⁽¹⁾ · · · f₁^(N−1) f₂⁽⁰⁾ f₂⁽¹⁾ · · · f₂^(N−1)

... ... . .. ... f_N⁽⁰⁾ f_N⁽¹⁾ · · · f_N^(N−1)

=







a_1,1 a_1,2 · · · a_1,M ... ... . .. ... a_N,1 a_N,2 · · · a_N,M

















e^θ1 κ1e^θ1 · · · κ^N−1₁ e^θ1 e^θ2 κ₂e^θ2 · · · κ^N−1₂ e^θ2

... . .. ...

e^θM κMe^θM · · · κ^N−1_M e^θM











Classification Theorem

Then using theBinet-Cauchy lemma, we have:

Lemma (the expansion formula of theτ-function) Theτ-function can be expanded as

τ(x,y,t) = X

I={i1<···<i_N}

∆I(A)EI(x,y,t),

where∆_I(A)is the N ×N minor of the A-matrix with the set of columns I ={i₁, . . . ,iN} ⊂ {1,2, . . . ,M}, and

E_I = Wr(e^θi1. . . ,e^θiN) =



 Y

1≤j<l≤N

(κ_ij −κ_il)



e^{θi1+···+θiN} >0, where we have assumed the orderκ1 < κ2<· · ·< κ_M.

Classification Theorem

Remark: We are interested in the case where the matrixA has

∆_I(A)≥0 for all I ={i₁<i₂ <· · ·<i_N}. Such matrix is called totally non-negativematrix. Then the set of all such matrices defines the totally non-negative Grassmannian:

Totally nonnegative Grassmannian

Gr⁺(N,M):={A∈Gr(N,M) : ∆_I(A)≥0,∀I ={i₁< . . . <i_N}}

Then we have the following theorem:

Theorem (K. -Williams, 2013) The solution u(x,y,t) = 2 ∂²

∂x²lnτ is non-singularif and only if A∈Gr⁺(N,M).

Classification Theorem

IrreducibleAmatrix

We say that theA-matrix isirreducible, if

in each column, there is at least one nonzero element, in each raw, there is at least one more nonzero element in addition to the pivot.

Example: ForN = 2 andM = 4, there are only two types of irreducibleA-matrices in RREF:

1 0 ∗ ∗ 0 1 ∗ ∗

,

1 ∗ 0 ∗ 0 0 1 ∗

.

Note that other cases can be expressed by a smaller matrix of N⁰×M⁰ with either N⁰<N or M⁰ <M.

Classification Theorem

Example: ForN = 2 andM = 4, there are totally seven types of theA-matrices in RREF which are bothirreducible andtotally non-negative:

1 0 −c −d

0 1 a b

1 0 −b −c

0 1 a 0

1 0 0 −c

0 1 a b

1 0 0 −b

0 1 a 0

1 a 0 −c

0 0 1 b

1 a 0 0 0 0 1 b

Herea,b,c and d are positive numbers, and for the first one, eitherad −cb>0 or = 0. The total number of nonzero minors

∆i,j(A) is at least four, and the maximal number is six. Note that there aresevenderangements (permutations without fixed points) of the symmetric groupS₄ with two excedances.

Classification Theorem

LetAbe an N×M irreducibleand totally non-negativematrix, and let{e₁, . . . ,eN} be the pivot indices and {g₁, . . . ,gM−N} be thenon-pivotindices. Then we have:

Theorem (Chakravarty-K. 2008)

The soliton solution associated with the A-matrix has (a) for y 0,∃ N solitons of [e_n,j_n]-type for some j_n, (b) for y 0,∃ M−N solitons of[im,gm]-type for some im.

Interaction Region (e ,e ,...,e )1 2 N+

[e , j ]N+ N+

x = ∞ x = - ∞

y = ∞

y = - ∞

[e , j ] j j [e , j ] k k

[e , j ]1 1

[ i , g ]N- N-

[ i , g ]1 1

[ i , g ]s s

Classification Theorem

Theorem (Chakravarty-K. 2008)

The set of those solitons[e_n,j_n]and [i_m,g_m]are expressed by a uniquechord diagram which corresponds to aderangementof the symmetric groupS_M with N excedances, i.e.

e₁ · · · e_N g₁ · · · gM−N

j₁ · · · j_N i₁ · · · i_M−N

∈ S_M

Theorem (K.-Williams, 2014)

Conversely, for each chord diagram associated with the derangement, one can reconstruct an A-matrix, and the correspondingτ-function gives the solution of the KP equation having line-solitons expressed by the chord diagram. The entries of the A-matrix give thescattering data, i.e. the locations of those line-solitons and their interaction properties.

Classification Theorem

Example: N= 2,M = 4. We have sevendifferent types of

(2,2)-soliton solution, which are parametrized by the derangements ofS₄:

(3412)

(4321)

(2143)

(4312) (3421)

(3142) (2413)

The 4-tuples of the diagrams represent the derangements, π=

1 2 3 4

π(1) π(2) π(3) π(4)

= (π(1), . . . , π(4)).

List of (2, 2)-solitons

The caseπ= (3412): The A-matrix is given by

A=

1 0 −c −d

0 1 a b

The solution contains all possible line-solitons, that is, for any 1≤i <j ≤4, there aresix different solitons of [i,j]-types. In particular, the asymptotic line-solitons are [1,3]- and [2,4]-types.

List of (2, 2)-solitons

The caseπ= (4312): The A-matrix is given by

A=

1 0 −c −d

0 1 a 0

_{1 2 3} π = (4312)

4

The asymptotic line-solitons are (a) for y 0, [1,4]- and [2,3]-types, (b) for y 0, [1,3]- and [2,4]-types.

List of (2, 2)-solitons

The caseπ= (2413): The A-matrix is given by

A=

1 0 −c −d

0 1 a b

₁

2 3

π = (2413) 4

wheread−bc = 0.

The asymptotic line-solitons are (a) for y 0, [1,2]- and [2,4]-types, (b) for y 0, [1,3]- and [3,4]-types.

List of (2, 2)-solitons

The caseπ= (4321): The A-matrix is given by

A=

1 0 0 −d

0 1 a 0

There is no resonant interaction, and this solution is called P-type.

The asymptotic line-solitons are of [1,4]- and [2,4]-types.

List of (2, 2)-solitons

The caseπ= (2143): The A-matrix is given by A=

1 a 0 0 0 0 1 b

There is no resonant interaction, and this solution is called O-type.

The asymptotic line-solitons are of [1,2]- and [3,4]-types.

List of (2, 2)-solitons

Gr(3,6) Example: (9-dimensional solution).

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

[2,5]

[3,6]

[1,4]

1 0 0 g h k A = 0 1 0 -d -e -f 0 0 1 a b c

1 2 3 4 5 6

t = -10 t = 0 t = 10

π = (456123) = [14][25][36]

List of (2, 2)-solitons

Other example inGr(3,6): (7-dimensional solution).

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

-100 -50 0 50 100

[5,6]

[2,5]

[1,4]

[3,6]

[2,4]

[1,3]

1 0 -a -b 0 c A = 0 1 d e 0 -f 0 0 0 0 1 g

1 2 3 4

5

6

t = -30 t = 0 t = 30

π = (451263)

Mach reflection phenomena in shallow water

Incident line-soliton propagating into an inclined wall. The right panel is an equivalent system, and illustrates a typical wave pattern after colliding the inclined wall. It turns out that this can be described by one of the exact solutions of the KP equation.

Mach reflection phenomena in shallow water

Harry Yeh’s experiment of the Mach reflection with Ψ₀<Ψ_c:

The wave is similar to a (3142)-type soliton solution. Notice in particular that the stem length is increasing, i.e. the Mach stem.

Mach reflection phenomena in shallow water

Appearance of Mach-stem is an “everyday” occurrence.

KP solitons in Mach reflection

We perform anumerical simulation with V-shape initial wave profile: First recall that one-line soliton of [i,j]-type,

u(x,y,t) =A_[i,j]sech² pA_[i,j]

2 (x+ytan Ψ_[i,j]+C_[i,j]t), where the amplitudeA_[i,j] and the slope tan Ψ_[i,j] are given by

A_[i,j]= 1

2(κ_i −κ_j)², tan Ψ_[i,j]=κ_i+κ_j. We consider asymmetric V-shape initial data as shown in

0 0

−Ψ⁰

Ψ⁰

Solitons fory → ±∞have A₀ = ¹₂(k_i^±−k_j^±)² tan Ψ₀ =∓(k_i^±+k_j^±)

KP solitons in Mach reflection

Numerical result for the case (a), Ψ₀ >Ψ_c (regular reflection):

0 2 4 6

0 0.1 0.2 0.3 0.4

t

E(t)

KP solitons in Mach reflection

3-D figure of the (2143)- (i.e. O-)type completion: The solution consists of [1,2]-soliton for y >0, and [3,4]-soliton for y <0.

Notice that the maximum amplitude is achieved rather quickly, and the interaction point becomes steady to give aconstant phase shift. Also the bow-shape wakes start to leave the main part of the interaction point, and this can be considered to be aseparation of thesoliton from the radiations .

KP solitons in Mach reflection

Numerical result for the case (b), 0<Ψ₀<Ψ_c:

[1,3]

[2,4]

[3,4]

[1,2]

[1,4]

0 5 10

0 0.02 0.04 0.06

t

E(t)

KP solitons in Mach reflection

3-D figure of the (3142)-type completion.

Notice that the amplitude is slowly growing to the maximum, and the interaction part generates a stem wave,Mach stem. Again the bow-shape wakes start to leave the main part of the interaction point, and this can be considered to be a separation of thesoliton and theradiations .

Other examples

Example 1: Waves in an Estonian beach (the photo (video) courtesy by Ira Didenkulova):

Here we estimate the angles,

Ψ₁ = Ψ_[3,4] = 37^◦, Ψ₂= Ψ_[2,4]= 14^◦, Ψ₃ = Ψ_[2,3]= 0^◦, Ψ4 = Ψ_[1,3] =−20^◦, Ψ5 = Ψ_[1,2]=−42^◦.

which gives

(κ1, κ2, κ3, κ4) = (−0.632,−0.268,0.268,0.517)

Other examples

Then we get the KP solution which approximates the video:

The video on the third line is another seen of the same kind.

Other examples

Example 2: The KP exact solution is of π= (2,4,1,5,3).

The photo is taken by Ablowitz at an Mexican beach.

References (mathematical physics and engineering)

1. Y. K., Young diagrams and N-soliton solutions of the KP equation, J. Phys. A: Math. Gen. 37(2004) 11169-11190.

2. S. Chakravarty and Y. K., Classification of the line-solitons of KPII, J. Phys. A: Math. Theor. 41 (2008) 275209 (33pp).

3. S. Chakravarty and Y. K., Soliton solutions of the KP equation and application to shallow water waves, Stud. in Appl. Math. 123 (2009) 83-151.

4. Y. K, KP solitons in shallow water, J. Phys. A: Math. Theor.

43 (2010) 434004 (54pp).

5. W. Li, H. Yeh and Y. K., On the Mach reflection of a solitary wave: revisited, J. Fluid Mech. 672(2011) 326-357.

6. H. Yeh, W. Li and Y.K.,Mach reflection and KP solitons in shallow water, Eur. Phys. J. Special Topics 185 (2010) 97-111.

References (mathematics)

7. Y. K. and L. Williams,KP solitons, total positivity, and cluster algebras, PNAS,108 (2011) 8984-8989 (arXiv:1105.4170).

8. Y. K. and L. Williams, KP solitons and total positivity for the Grassmannian, Invent. Math. 195(2014) (arXiv:1106.0023).

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244 (2013) 979-1032 (arXiv:1204.6446).

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