summer internship 2017 bifurcation analysis with symmetry

ENS Rennes University of Oxford

Summer Internship 2017

Bifurcation analysis with symmetry groups

Author :Nicolas Boulle

Advisor :Prof. Patrick Farrell

1 n. boulle

Abstract. In this work, we adapt a bifurcation analysis technique, the deflated continuation

algorithm, in order to take into account the symmetry groups of the studied problem. We use

Lie groups theory to find symmetry groups and invariants of partial differential equations. Thus,we compute solutions of the two-dimensional Gross-Pitaevskii equation and the two-component

nonlinear Schrodinger system.

This new technique deflates the whole orbits of a given partial differential equation and canbe applied to many problems with different symmetry groups.

Acknowledgments. Thanks to Alexei Gazca for useful discussions about the definition of theG-invariant transformation and to David Emerson for interesting comments on this report.

Contents

1. Introduction 2

2. Lie groups and partial differential equations 2

2.1. Lie groups and prolongation of group actions 2

2.2. Symmetry groups and invariants of partial differential equations 5

3. Introduction to deflation 10

3.1. Deflation operators on Banach spaces 11

3.2. Find roots of nonlinear problems with deflation 12

3.3. Convergence of Newton’s method with deflation 13

4. Computing orbits of partial differential equations symmetry groups 15

4.1. Computing solutions to the two-dimensional Gross-Pitaevskii equation 15

4.2. Application of Lie groups to deflation 16

4.3. Computation of the circularly symmetric transformation 20

5. Numerical Results 22

5.1. The two-dimensional Gross-Pitaevskii equation 22

5.2. The two-component nonlinear Schrodinger system 23

6. Conclusion and future challenges 26

References 30

bifurcation analysis with symmetry groups 2

1. Introduction

This work can be divided in three parts. We begin by introducing Lie groups theory in section 2.This theory provides many useful tools to analyze a given partial differential equation. Actually, it’spossible to calculate symmetry groups and their invariants of a nonlinear equation, which gives a lotof information of the structure of the solutions. As we will see, symmetry groups and invariants ofa partial differential equation can be computed numerically easily by solving a system of equations,given by the infinitesimal criterion.

Sections 3 and 4 are devoted to the deflation. The idea of deflation to solve nonlinear partialdifferential equations has been introduced in [13] and generalized the work of Wilkinson [21]. Inthese sections, we will use Lie groups theory and the notions of symmetry groups and invariantsto introduce new deflation operators. This will extend the use of deflation to the equations whichhave infinite symmetry groups.

Finally, the last section is dedicated to the numerical simulations. We will apply the deflatedcontinuation algorithm with the new deflation operators to compute solutions of the two-dimensionalGross-Pitaevskii equation and the two-component nonlinear Schrodinger system.

2. Lie groups and partial differential equations

Every definition and proposition of this section can be found in [18]. This section introduces Liegroups and methods to find symmetry groups of partial differential equations.

2.1. Lie groups and prolongation of group actions.

Definition 2.1 (Manifold). An m-dimensional manifold is a set M , together with a countablecollection of subsets Uα ⊂ M , called coordinate charts, and one-to-one functions χα : Uα → Vαonto connected open subsets Vα ⊂ Rm, which satisfy the following properties:

• the coordinate charts cover M : ⋃α

Uα = M,

• on the overlap of any pair of coordinate charts Uα ∩ Uβ the composite map

χβ χ−1α : χα(Uα ∩ Uβ)→ χβ(Uα ∩ Uβ)

is a smooth function,• if x ∈ Uα, x ∈ Uβ are distinct points of M , then there exists open subsets W ⊂ Vα, W ⊂ Vβ

with χα(x) ∈W , χβ(x) ∈ W , satisfying

χ−1α (W ) ∩ χ−1

β (W ) = ∅.

Definition 2.2 (Lie Group). An r-parameter Lie group is a group G which also carries the structureof an r-dimensional smooth manifold in such a way that the group product and inversion maps

m : G×G −→G(g, h) 7−→g · h

i : G −→Gg 7−→g−1

are smooth maps between manifolds.

A Lie group is connected (resp. compact) if its manifold structure is connected (resp. compact).

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Definition 2.3 (Local group of transformations). Let M be a smooth manifold and G a Lie group.A local group of transformations is given by an open subset U such that e ×M ⊂ U ⊂ G ×M ,where e is the identity of G, and a smooth map ψ : U →M with the following properties

• if (h, x) ∈ U , (g, ψ(h, x)) ∈ U and (g · h, x) ∈ U , then ψ(g, ψ(h, x)) = ψ(g · h, x),• for all x ∈M, ψ(e, x) = x,• if (g, x) ∈ U , then (g−1, ψ(g, x)) ∈ U and ψ(g−1, ψ(g, x)) = x.

We will now denote ψ(g, x) by g · x.

A first example of local group of transformations is the group of translations T (1,−2) which actson R2 and whose action is defined by, for all ε ∈ R and for all (x, y) ∈ R2,

ψ(ε, (x, y)) = (x+ ε, y − 2ε).

The group of rotation SO2 is also a local group of transformations as it acts on R2 with its smoothmap ψ : SO2(R)× R2 →M defined by, for all θ ∈ R and (x, y) ∈ R2,

ψ

(θ,

(xy

))=

(cos(θ) − sin(θ)sin(θ) cos(θ)

)·(xy

)=

(x cos(θ)− y sin(θ)x sin(θ) + y cos(θ)

).

Definition 2.4 (Tangent vector). Let M be a smooth manifold of dimension m. A smooth curveC on M parametrized by φ : I → M is given by m smooth functions φ(ε) = (φ1(ε), ..., φm(ε)) inlocal coordinates x = (x1, ..., xm). At each point x = φ(ε) of C, the curve has a tangent vector

φ(ε) = (φ1(ε), ..., φm(ε)), denoted by v|x ≡ φ1(ε) ∂∂x1 + ...+ φm(ε) ∂

∂xm .

The previous notation is used to distinguish between local coordinate expressions for points onthe manifold M and coordinates of the tangent vector. Nevertheless, the link between the notation∂/∂xi and the differential operator will be shown by (2.3).

Definition 2.5 (Vector field). A vector field v on M assigns a tangent vector v|x to each pointx ∈M . In local coordinates, a vector field has the form

v|x ≡ ξ1(x)∂

∂x1+ ...+ ξm(x)

∂

∂xm.

Definition 2.6 (Integral curve). An integral curve of a vector field v is a smooth parametrizedcurve x = φ(ε) whose tangent vector at any points coincides with the value of v at the same point

(2.1) φ(ε) = v|φ(ε)

for all ε.

Definition 2.7 (Maximal integral curve). An integral curve φ : I →M of a vector field v throughthe point (0, φ(0)) is called maximal if for every other integral curve ψ : J →M through this pointit holds that J ⊂ I and ψ = φ|J .

By standards existence and uniqueness theorems for ordinary differential equations, (2.1) admitsan unique maximal solution for each x0 such that

φ(0) = x0.

This solution is called the maximal integral curve of v.

Definition 2.8 (Flow). Let v be a vector field. We denote the parametrized maximal integral curvepassing through x ∈M by ψ(ε, x), which is called the flow generated by v. The flow generated by avector field is the same as a local group of action of the Lie group R on M called a one parametergroup of transformations. The vector field v is called the infinitesimal generator of the action since


ψ(ε, x) = x + εξ(x) + O(ε2), where ξ = (ξ1, ..., ξm) are the coefficients of v. Then ψ satisfies thefollowing system of equations:

d

dεψ(ε, x) = v|ψ(ε,x),

ψ(0, x) = x.

Conversely, if ψ(ε, x) is a one parameter group of transformations,

(2.2) v|x =d

dε

∣∣∣∣ε=0

ψ(ε, x).

According to (2.2), the vector field of the previous group of translations T (1,−2) is given byv(x,y) = (1,−2). Then, in local coordinates,

v = ∂x − 2∂y.

The vector field of the group of rotations is given by v(x,y) = (−y, x). Then, in local coordinates,

v = −y∂x + x∂y.

Given a smooth real-valued function of p independent variables f(x) = f(x1, ..., xp), there are

pk ≡(p+ k − 1

k

)different possible k-th order partial derivatives of f . We employ the multi-index notation

∂Jf(x) ≡ ∂kf(x)

∂xj1 ...∂xjk

for these derivatives. More generally, if f : X → U is a smooth function from X = Rp to U = Rq,u = (f1(x), ..., fq(x)), there are q · pk numbers

uαJ ≡ ∂Jfα(x), 1 ≤ α ≤ q

needed to represent all the different k-th order derivatives of the components of f at a point x.We let Uk ≡ Rq·pk be the Euclidean space of this dimension and U (n) ≡ U × U1 × ... × Un be theCartesian product space whose coordinates represent all the derivatives of the function u = f(x) ofall orders from 0 to n.

Definition 2.9 (n-th prolongation and n-th order jet space). Let u = f(x) be a smooth function,where f : X → U . There is an induced function u(n) = pr(n)f(x) called the n-th prolongation off , which is defined by the equations

uαJ = ∂Jfα(x),

where 1 ≤ α ≤ q and #J ≤ n. The total space X×U (n), whose coordinates represent the independentvariables, the dependent variables and their derivatives up to order n is called the n-th order jetspace of the space X × U . The n-th order jet space of an open subset M ⊂ X × U is defined by

M (n) ≡M × U1 × ...× Un.

Thus, pr(n)f is a function from X to U (n) and for each x in X, pr(n)f(x) is a vector whose q ·p(n)

entries represent the values of f and all its derivatives up to order n at point x. For instance, if fis a smooth function of class Cn from R to R, its n-th prolongation pr(n)f is a function from R toU (n) = Rn+1 defined by

pr(n)f(x) = (f(x), f ′(x), ..., f (n)(x)).

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Definition 2.10 (Prolongation of group actions). Let G be a local group of transformations actingon an open subset M ⊂ X × U of the space of independent and dependent variables. There is aninduced local action of G on the n-jet space M (n), called the n-th prolongation of the action of Gand denoted pr(n)G.

Definition 2.11 (Total derivative). Let P (x, u(n)) be a smooth function of x, u and derivatives ofu up to order n, defined on an open subset M (n) ⊂ X×U (n). The total derivative of P with respectto xi depends on derivatives of u up to order n+ 1 on M (n+1) and is defined by

DiP ≡∂P

∂xi+

q∑α=1

∑#J≤n

uαJ,i∂P

∂uαJ,

where, for J = (j1, ..., jk),

uαJ,i =∂uαJ∂xi

=∂k+1uα

∂xi∂xj1 ...∂xjk.

Theorem 2.12 (General Prolongation Formula [18, Theorem 2.36]). Let

v =

p∑i=1

ξi(x, u)∂

∂xi+

q∑α=1

φα(x, u)∂

∂uα

be a vector field defined on an open subset M ⊂ X × U . The n-th prolongation of v is the vectorfield

pr(n)v ≡ v +

q∑α=1

∑#J≤n

φJα(x, u(n))∂

∂uαJ

defined on the corresponding jet space M (n) ⊂ X×U (n). The coefficient functions φJα of pr(n)v aregiven by the following formula

φJα(x, u(n)) = DJ

(φα −

p∑i=1

ξiuαi

)+

p∑i=1

ξiuαJ,i,

where uαi = ∂uα

∂xi and uαJ,i =∂uαJ∂xi .

2.2. Symmetry groups and invariants of partial differential equations.

Definition 2.13 (Symmetry group). Let ∆ be a system of differential equations. A symmetrygroup of the system ∆ is a local group of transformations G acting on an open subset M of thespace of independent and dependent variables for the system with the property that whenever (u, x)is a solution of ∆, and whenever g · (u, x) is defined for g ∈ G, then g · (u, x) is also a solution ofthe system.

T (1,−2) is a symmetry group of the differential equation

∂u

∂x+∂u

∂y= 0.

In fact, if (x, y) 7→ u(x, y) is a solution to the above equations, it is also the case for (x, y) 7→u(x− ε, y + 2ε), for all ε ∈ R.

SO2(R) is a symmetry group of the following equation

u(x, y)2 + v(x, y)2 = 1.

Indeed, if (u, v) ∈ H1(R2) ×H1(R2) is a solution to this equation then, for all θ ∈ R, (u cos(θ) −v sin(θ), u sin(θ) + v cos(θ)) is also a solution.


We define the maximal rank condition for smooth manifolds and specify this definition to systemsof differential equations with Definition 2.15.

Definition 2.14 (Maximal rank condition). Let F : M → N be a smooth mapping from an m-dimensional manifold M to an n-dimensional manifold N . The rank of F at a point x ∈M is therank of the n×m Jacobian matrix (∂F i/∂xj) at x. The mapping F is of maximal rank on a subsetS ⊂M if for each x ∈ S the rank of F is as large as possible (i.e. the minimum of m and n).

Definition 2.15 (System of maximal rank). Let

∆k(x, u(n)) = 0, 1 ≤ k ≤ lbe a system of differential equations. The system is said to be of maximal rank if the l× (p+ qp(n))Jacobian matrix

J∆(x, u(n)) =

(∂∆k

∂xi,∂∆k

∂uαJ

)k

=

∂∆1

∂x1 · · · ∂∆1

∂xi · · · ∂∆1

∂ux1· · · ∂∆1

∂uαJ...

. . ....

. . ....

. . ....

∂∆k

∂x1 · · · ∂∆k

∂xi · · · ∂∆k

∂ux1· · · ∂∆k

∂uαJ...

. . ....

. . ....

. . ....

∂∆l

∂x1 · · · ∂∆l

∂xi · · · ∂∆l

∂ux1· · · ∂∆l

∂uαJ

of ∆ with respect to all the variables (x, u(n)) is of rank l whenever ∆(x, u(n)) = 0. Here p is thenumber of independent variables, q the number of dependent variables and n is the order of theequation.

For instance, consider Laplace’s equation

∆(x, y, u(2)) = uxx + uyy = 0.

This system of one equation is of rank 1 and then of maximal rank since its Jacobian matrix withrespect to all the variables (x, y;u;ux, uy;uxx, uxy, uyy) in X × U (2) is

J∆(x, y, u(2)) = (0, 0; 0; 0, 0; 1, 0, 1),

which is of rank 1 whenever ∆(x, y, u(2)) = 0.

Before studying Lemma 2.16, we will use the chain rule and Definition 2.8, for every x ∈ M tointroduce the following equality, which describes the action of the flow v of a Lie group on functions,

(2.3)d

dεF (ψ(ε, x)) =

m∑i=1

ξi(ψ(ε, x))∂F

∂xi(ψ(ε, x)) ≡ v(F )[ψ(ε, x)].

Lemma 2.16 ([18, Theorem 2.8]). Let G be a connected local Lie group of transformations actingon the m-dimensional manifold M . Let F : M → Rl, l ≤ m, define a system of algebraic equations

Fk(x) = 0, 1 ≤ k ≤ l,and assume that the system is of maximal rank in the sense of Definition 2.15. Then G is asymmetry group of the system if and only if

(2.4) v[Fk(x)] = 0, 1 ≤ k ≤ l, whenever F (x) = 0,

for every infinitesimal generator v of G.

Proof. Suppose G is a symmetry group of the system of algebraic equations and let

v =

m∑i=1

ξi(x)∂

∂xi

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be an infinitesimal generator of G defined on M . Since G is a symmetry group, for all ε ≥ 0

Fk(ψ(ε, x)) = 0, ∀1 ≤ k ≤ l, whenever F (x) = 0.

Thus, after differentiating the right member of the previous equality as in (2.3) and setting ε = 0,we have

v[Fk(x)] = 0, 1 ≤ k ≤ l, whenever F (x) = 0,

which proves the necessity of (2.4).

Let x0 be a solution to the system of equations. By the maximal rank condition and [18,Theorem 1.8], we can change the local coordinates by y = (y1, ..., ym) such that x0 = 0 and F hasthe form F (y) = (y1, ..., ym). Let

v =

m∑i=1

ξi(y)∂

∂yi

be an infinitesimal generator of G expressed in the new coordinates. Equation (2.4) written in thenew coordinates means that

(2.5) v(yk) = ξk(y) = 0, 1 ≤ k ≤ l,

whenever y1 = ... = yl = 0. The flow φ(ε) = ψ(ε, x0) of v through x0 = 0 satisfies the followingsystem of ordinary equations for all 1 ≤ i ≤ m:

d

dεφi(ε) = ξi(φ(ε)),(2.6)

φi(0) = 0.(2.7)

However, using (2.5), the system (2.6)-(2.7) has a unique solution φk(ε) = 0, for 1 ≤ k ≤ l and εsufficiently small. Thus, we have shown that if x0 is a solution to F (x) = 0 and ε sufficiently small,then ψ(ε, x0) is again a solution to this system of equations. Let

ε+ ≡ supε > 0 | ψ(ε, x0) ∈ S∆, ψ(ε, x0) is defined.

Since S∆ = x | F (x) = 0 is closed and ε 7→ ψ(ε, x0) is continuous, ψ(ε+, x0) is a solution toF (x) = 0. This allows us to assume that ψ(δ, ψ(ε+, x0)) is again a solution for δ > 0 sufficientlysmall. Then, by Definition 2.3,

ψ(δ, ψ(ε+, x0)) = ψ(δ + ε+, x0),

which implies that ε+ = supε > 0 | ψ(ε, x0) is defined. The same reasoning with ε− < 0 showsthat g · x0 is a solution to F (x) = 0 for all g = ψ(ε, ·) in the connected one-parameter subgroupof Gx0

= g | g · x0 is defined generated by v. The end of the proof is an application of [18,Proposition 1.24].

The following definition introduces the notion of submanifolds and will be used in the Lemma2.18.

Definition 2.17 (Submanifold). Let M be a smooth manifold. A submanifold of M is a subset

N ⊂ M with a smooth one-to-one map φ : N → N ⊂ M satisfying the maximal rank conditioneverywhere, where the parameter space N is some other manifold and N = φ(N) is the image of φ.

Lemma 2.18 ([18, Theorem 2.27]). Let M be an open subset of X×U and suppose ∆(x, u(n)) = 0is an n-th order system of differential equations defined over M , with corresponding subvarietyS∆ ⊂M (n). Suppose G is a local group of transformations acting on M whose prolongation leavesS∆ invariant, meaning that whenever (x, u(n)) ∈ S∆, we have pr(n)g · (x, u(n)) ∈ S∆ for all g ∈ Gsuch that this is defined. Then G is a symmetry group of the system of differential equations.


Proof. Suppose u = f(x) is a local solution to ∆(x, u(n)) = 0. This means that

Γ(n)f = (x,pr(n)f(x)) ⊂ S∆,

where Γ(n)f is the graph of the prolongation pr(n)f . If g ∈ G is such that g · f is well defined, by the

definition of the n-th prolongation, the graph of its prolongation Γ(n)g·f has the following property

Γ(n)g·f = pr(n)g(Γ

(n)f ).

Since S∆ is invariant under pr(n)g, Γ(n)g·f lies entirely in S∆, which means that the transformed

function g · f is a solution to ∆.

The infinitesimal criterion allows us to find the symmetry groups of a system of differentialequations. It follows directly from Lemmas 2.16 and 2.18.

Theorem 2.19 (Infinitesimal criterion [18, Theorem 2.31]). Suppose

∆k(x, u(n)) = 0, 1 ≤ k ≤ l,is a system of differential equations of maximal rank defined over M ⊂ X ×U. If G is a local groupof transformations acting on M ,

pr(n)v[∆k(x, u(n))] = 0, ∀1 ≤ k ≤ l, whenever ∆(x, u(n)) = 0,

for every infinitesimal generator v of G, then G is a symmetry group of the system.

Theorems 2.12 and 2.19 allow us to find the symmetry groups of a given partial differentialequation as we will see in the following example.

Example 2.20 (Heat equation). Consider the equation that describes the variation of temperaturein a one-dimensional rod

ut = uxx,

where the thermal diffusivity has been normalized to unity. There are two independent variablesx and t and one dependent variable u. The equation is of second order and can be identified withthe subvariety in X × U (2) determined by the vanishing of

(2.8) ∆(x, t, u(2)) = ut − uxx.This system is of maximal rank as its Jacobian matrix

J∆(x, t, u(2)) = (0, 0; 0; 0, 1;−1, 0, 0)

is of rank 1 whenever ∆(x, t, u(2)) = 0. Then, we can apply Theorem 2.19 in order to find symmetrygroups of the heat equation. Let

v = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ φ(x, t, u)

∂

∂ube a vector field on X × U . According to Theorem 2.12, the second prolongation of v is

pr(2)v = v + φx∂

∂ux+ φt

∂

∂ut+ φxx

∂

∂uxx+ φxt

∂

∂uxt+ φtt

∂

∂utt,

whose coefficients can be determined by Definition 2.11. After applying pr(2)v to (2.8), the infini-tesimal criterion becomes

(2.9) φt = φxx

and must be satisfied whenever ut = uxx. However,

φt = φt − ξtux + (φu − τt)ut − ξuuxut − τuu2t ,

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φxx = φxx + (2φxu − ξxx)ux − τxxut + (φuu − 2ξxu)u2x − 2τxuuxut − ξuuu3

x − τuuu2xut

+ (φu − 2ξx)uxx − 2τxuxt − 3ξuuxuxx − τuutuxx − 2τuuxuxt.

The above equalities and (2.9) lead us to the following equalities after replacing ut by uxx andidentifying the monomials 1, ux, u2

x, ...

τu = 0(2.10)

τx = 0(2.11)

ξu = −τxu(2.12)

τt = τxx + 2ξx(2.13)

φuu = 2ξxu(2.14)

ξt = −2φxu + ξxx(2.15)

φt = φxx(2.16)

According to (2.10) and (2.11), τ = τ(t) depends only on t. Then by (2.12), ξu = 0, which meansthat ξ doesn’t depend on u, whereas (2.13) requires that τt = 2ξx. This implies that there existsσ, a function of t, such that

ξ(x, t) =1

2τt(t)x+ σ(t).

φ is linear in u owing to (2.14) so there exists two functions α and β such that

φ(x, t, u) = β(x, t)u+ α(x, t).

However, by (2.15), ξt = −2βx and

β(x, t) = −1

8τtt(t)x

2 − 1

2σt(t)x+ ρ(t),

where ρ is a function of t. Finally, according to (2.16), α and β are both solutions to the heatequation. Then,

βt = βxx,

which implies that

τttt = 0, σtt = 0, ρt = −1

4τtt.

Then, the infinitesimal generators of the symmetry groups of the heat equation have coefficientfunctions of the form

ξ = c1 + c4x+ 2c5t+ 4c6xt,

τ = c2 + 2c4t+ 4c6t2,

φ = (c3 − c5x− 2c6t− c6x2)u+ α(x, t),

where c1, ..., c6 are arbitrary constants and α is any solution of the heat equation. Thus, thesymmetries of the heat equation are given by the following vector fields

v1 = ∂x,

v2 = ∂t,

v3 = u∂u,

v4 = x∂x + 2t∂t,

v5 = 2t∂x − xu∂u,v6 = 4tx∂x + 4t2∂t − (x2 + 2t)u∂u,


and

vα = α(x, t)∂u,

where α is any solution of the heat equation. The corresponding symmetry groups can be foundusing Definition 2.8.

Definition 2.21 (Invariant function). Let G be a local group of transformations acting on a man-ifold M . A function F : M → N , where N is another manifold is called G-invariant function if forall x ∈M and all g ∈ G such that g · x is defined,

F (g · x) = F (x).

A real-valued G-invariant function ξ : M → R is simply called an invariant of G.

Proposition 2.22 ([18, Proposition 2.6]). Let G be a connected group of transformations actingon the manifold M . A smooth real-valued ξ : M → R is an invariant function for G if and only if

(2.17) v(ξ) = 0

for every infinitesimal generator v of G.

Proof. The necessity of (2.17) follows from the equality (2.3) by setting ε = 0.

Conversely, if

v(ξ) = 0 for all x ∈M,

then by (2.3)

d

dεξ(ψ(ε, x)) = 0 for all x ∈M.

Thus, ξ(g · x) = ξ(x) for every g in the connected one-parameter subgroup Gvx , generated by v,

which is the orbit of the group action on x:

Gx ≡ g ∈ G, g · x is defined.

However, by [18, Proposition 1.24], every element of Gx can be written as a finite product

gv1x · ... · gvnx ,

where gvkx ∈ Gvkx , which proves that ξ is constant on Gx.

An invariant function of T (1,−2) is ξ : (x, y) 7→ 2x+ y, indeed

∂ξ

∂x− 2

∂ξ

∂y= 0.

ξ : (u, v) 7→ u2 + v2 is an invariant function of SO2(R).

3. Introduction to deflation

The aim of this section is to present a computational technique for finding distinct solutions ofa system of partial differential equations, which is called deflation. This algorithm is an extensionof Newton’s method in infinite Banach spaces.

11 n. boulle

3.1. Deflation operators on Banach spaces. Consider a polynomial p(x) of R[X] and supposea root x0 of this polynomial has already been computed by Newton’s method. Then, another rootcan be computed by applying the same algorithm to the deflated function

q(x) =p(x)

x− x0.

The other roots of p can be computed as well by iterating this process, historically introduced byWilkinson [21].

In [4], Brown and Gearhart introduced the concept of a deflation matrix in order to computesolutions of a system of n real nonlinear equations in n real unknowns. However, this technique hasbeen generalized in infinite Banach spaces by Farrell, Birkisson and Funke [13] to find solutions ofa system of partial differential equations.

Definition 3.1 (Deflation operator on a Banach space [3]). Let V , W and Z be Banach spaces, andU be an open subset of V . Let F : U ⊂ V →W be a Frechet differentiable operator with derivativeF ′. For each r ∈ U , u ∈ U\r, let M(u; r) : W → Z be an invertible linear operator. We say thatM is a deflation operator if for any F such that F(r) = 0 and F ′(r; ·) is non singular, we have

(3.1) lim infn→+∞

‖M(un; r)F(un)‖Z > 0

for any sequence un converging to r, un ∈ U\r.

The following lemma, introduced by Brown and Gearhart in its original form [4, Lemma 2.1], isuseful to determine whether an operator can serve as a deflation operator.

Lemma 3.2 (Sufficient condition for identifying deflation operators [13, Lemma 2.2]). Let F : U ⊂V →W be a Frechet differentiable operator. Suppose that the linear operator M(u; r) : W → Z has

the property that for each r ∈ U , and any sequence uiU→ r, ui ∈ Ur = U\r, if

(3.2) ‖ui − r‖M(ui; r)wiZ−→ 0 =⇒ wi

W−→ 0

for any sequence wi, wi ∈W , then M is a deflation operator.

Proof. We proceed via a proof by contradiction. If (3.2) holds and M is not a deflation operatoras defined by Definition 3.1, then there exists a Frechet differentiable operator F : U ⊂ V → Wand an r ∈ U such that F(r) = 0, F ′(r) nonsingular and

lim infi→+∞

‖M(ui; r)F(ui)‖Z = 0

for some sequence ui converging to r, ui ∈ Ur. Then, there exists a subsequence vi such that

limi→+∞

‖M(vi; r)F(vi)‖Z = 0.

Let wi ∈W be the sequence defined by

wi =F(vi)

‖vi − r‖U.

Thus,

‖vi − r‖UM(vi; r)wiZ−→ 0.

By (3.2), this implies that wiW→ 0 i.e.

(3.3)F(vi)

‖vi − r‖UW−→ 0.


Since F is Frechet differentiable, we can expand it in a Taylor series around r to give

F(vi) = F(r) + F ′(r; vi − r) + o(‖vi − r‖U )

= F ′(r; vi − r) + o(‖vi − r‖U )

as F(r) = 0. Then, we have

F(vi)

‖vi − r‖U=

1

‖vi − r‖U[F ′(r; vi − r) + o(‖vi − r‖U )]

≈ F ′(r; vi),where

vi =vi − r‖vi − r‖

∈ Ur

is a function with unit norm for all vi ∈ Ur. But then, (3.3) lead to a contradiction of thenonsingularity of F ′(r).

We now introduce the class of shifted deflation operators.

Definition 3.3 (Shifted deflation [13]). Shifted deflation specifies

(3.4) Mp,α(u; r) =

(1

‖u− r‖pU+ α

)IW ,

where α ≥ 0 is the shift, p ≥ 1 the power and IW is the identity on W .

According to Lemma 3.2, shifted deflation operators satisfy (3.1) and are deflation operators.

3.2. Find roots of nonlinear problems with deflation. In [12], Farrell, Beentjes and Birkissonproposed the Algorithm 1, called deflated continuation for computing bifurcation diagrams.

This algorithm is capable of computing solutions to

(3.5) F (u, λ) = 0,

where F : U ×R→ Y is the C1 problem residual, U and Y are isomorphic Banach spaces, u ∈ U isthe solution and λ ∈ R the parameter. The associated bifurcation diagram shows how the solutionschange as λ varies over some interval [λmin, λmax].

The heart of the algorithm is deflation. Indeed, if λ is fixed in (3.5), this becomes a nonlinearproblem

F (u) = 0.

Suppose Newton’s method is applied to F from an initial guess u0 and gives a solution u∗1, whoseFrechet derivative F ′(u∗1) is nonsingular. Then, we apply a deflation operator M(u, u∗1) to theresidual F in order to construct a new problem

G(u) =M(u, u∗1)F (u).

G preserves the solutions of F as for all u 6= u∗1, G(u) = 0 if and only if F (u) = 0 and Newton’smethod applied to G will not discover u∗1 as

lim infu→u∗1

‖G(u)‖ > 0.

Indeed, if a sequence un converges to u∗1 then the deflated residual does not converge to zero.Thus, if Newton’s method is applied to G, it will converge to a distinct solution u∗2 6= u∗1 and we canrepeat this process until no more solutions are found in a specified number of Newton iterations.In their article, the authors used the shifted deflation operator (3.4) and obtained their numericalresults by taking p = 2 and σ = 1.

13 n. boulle

Algorithm 1 Deflated continuation

Input: Initial parameter value λmin.Input: Final parameter value λmax > λmin.Input: Step size ∆λ > 0.Input: Nonlinear residual f(u, λ).Input: Deflation operator M(u;u∗).Input: Initial solutions S(λmin) to f(·, λmin).

1: λ← λmin

2: while λ < λmax do3: F (·)← f(·, λ+ ∆λ) . Fix the value of λ to solve for.4: S(λ+ ∆λ)← ∅5: for u0 ∈ S(λ) do . Continue known branches.6: Apply Newton’s method to F from initial guess u0.7: if solution u∗ found then8: S(λ+ ∆λ)← S(λ+ ∆λ) ∪ u∗ . Record solution.9: F (·)←M(·, u∗)F (·) . Deflate solution.

10: for u0 ∈ S(λ) do . Seek new branches.11: success ← true12: while success do13: Apply Newton’s method to F from initial guess u0.14: if solution u∗ found then . New branch found.15: S(λ+ ∆λ)← S(λ+ ∆λ) ∪ u∗ . Record solution.16: F (·)←M(·, u∗)F (·) . Deflate solution.17: else18: success ← false19: λ← λ+ ∆λ

20: return S

3.3. Convergence of Newton’s method with deflation. The aim of this section is to givesufficient conditions in order to ensure convergence to two solutions with Newton’s method anddeflation. The following results have been proved in [12] by Farrell, Beentjes and Birkisson. Westart with Theorem 3.4, which proves the convergence of Newton’s method to a solution of a givenproblem under certain conditions.

Theorem 3.4 (Affine-covariant Rall-Rheinboldt [19, 20]). Let F : D → Y be a continuously Frechetdifferentiable function on the open convex D ⊆ U . Suppose that there exists a u∗ ∈ D such thatF (u∗) = 0, and suppose further that

i) F ′(u∗)−1 exists,ii) ‖F ′(u∗)−1(F ′(u)− F ′(v))‖ ≤ w∗‖u− v‖ for all u, v ∈ D.

Then any ρ∗ ≤ 23w∗ such that B = B(u∗, ρ∗) ⊂ D has the property that starting at u0 ∈ B, the

Newton sequence is well-defined and remains within B. The Newton sequence converges to u∗ ∈ B.Furthermore, if we define ρ+ = 1

w∗ , then u∗ is unique within D ∩B(u∗, ρ+).

Lemma 3.5 (Product of Lipschitz continuous functions [12, Lemma 3.3]). Let X, Y and Z beBanach spaces and let L(Y, Z) be the vector space of bounded linear operators from Y to Z withinduced operator norm. Let F : X → Y and G : X → L(Y,Z) be Lipschitz continuous functions on


the open subset D ⊆ X with Lipschitz constants ωF and ωG respectively. Assume further that Fand G are bounded on D, i.e. there exists NF , NG ∈ R such that ‖F (x)‖ < NF and ‖G(x)‖ < NGfor all x ∈ D. Then, the product GF : X → Z is bounded and Lipschitz continuous on D withLipschitz constant (NFωG +NGωF ).

The above lemma will be useful for the proof of Theorem 3.6. Under sufficient conditions, thistheorem guarantees convergence to another solution u∗2 after the deflation of a known solution u∗1.

Theorem 3.6 ([12, Theorem 3.4]). Let F : D → Y be a continuously Frechet differentiable functionon the open convex subset D ⊆ U . Suppose there exists u∗2 ∈ D such that F (u∗2) = 0. Further assumethere exists u∗1 ∈ D, u∗1 6= u∗2, such that F (u∗1) = 0. This solution is deflated with a deflation operatorM(·;u∗1) : D\u∗1 → GL(Y, Y ). Suppose there exists an open bounded convex subset E ⊆ D\u∗1with u∗2 ∈ E such that the following conditions hold:

i) F ′(u∗2)−1 exists,ii) ‖F ′(u∗2)−1(F ′(u)− F ′(v))‖ ≤ ω∗‖u− v‖ for all u, v ∈ E,

iii) M(u;u∗1) is continuously Frechet differentiable for all u ∈ E,iv) ‖M′(u;u∗1)−M′(v;u∗1)‖ ≤ ωM′‖u− v‖ for all u, v ∈ E.

Then there exists a ρ > 0 such that the Newton sequence from u0 ∈ B = B(u∗2, ρ) on the deflatedfunction M(u;u∗1)F (u) is well-defined, remains in B and converges to u∗2 ∈ B.

Proof. F andM(·, u∗1) are Lipschitz continuous because they are continuously Frechet differentiableon E, which is bounded. This implies that F (u), F ′(u), M(u;u∗1) and M′(u;u∗1) are bounded onE. Thus, by triangular inequality and Lemma 3.5, the Frechet derivative of the deflated operator

(M(u;u∗1)F (u))′ =M(u;u∗1)F ′(u) +M′(u;u∗1)F (u)

is Lipschitz continuous with Lipschitz constant L.

Since u∗2 is a root of F ,

(M(u;u∗1)F (u))′|u∗2 =M(u∗2;u∗1)F ′(u∗2).

Moreover, for all u ∈ D, the deflated operator M(u;u∗1) ∈ GL(Y, Y ) and is invertible. Thus, theFrechet derivative of the deflated residual is invertible at u∗2 and

([M(u;u∗1)F (u)]′)−1∣∣∣u∗2

= F ′(u∗2)−1M(u∗2;u∗1)−1.

Then, there exists an (affine covariant)

ω2 ≡ ([M(u∗2;u∗1)F (u∗2)]′)−1L > 0

such that for all u, v ∈ E,

(3.6)∥∥∥([M(u∗2;u∗1)F (u∗2)]′)

−1[(M(u;u∗1)F (u))′ − (M(v;u∗1)F (v))′]

∥∥∥ ≤ ω2‖u− v‖.

Then, by (3.6), the conditions of Theorem 3.4 are satisfied for both F (u) andM(u;u∗1)F (u), whichconcludes the proof.

The following theorem is a corollary of Theorems 3.4 and 3.6 and gives sufficient conditions forconvergence to two solutions with deflation and Newton’s method. This result can be generalizedfor convergence to more solutions.

15 n. boulle

Theorem 3.7 (Deflated Rall-Rheinboldt [2, Theorem 15]). Let F : D → Y be a continuouslyFrechet differentiable function on the open convex subset D ⊆ U . Suppose there exists u∗1, u

∗2 ∈ D

such that F (u∗1) = F (u∗2) = 0, u∗1 6= u∗2. Let E1 be an open bounded convex subset such thatE1 ⊂ D\u∗2 and u∗1 ∈ E1. Furthermore let E2 be an open bounded convex subset such thatE2 ⊂ D\u∗1 and u∗2 ∈ E2. Let M(·;u∗1) : D\u∗1 → GL(Y, Y ) be a deflation operator such thatthe following conditions hold:

i) F ′(u∗1)−1 and F ′(u∗2)−1 exist,ii) ‖F ′(u∗1)−1(F ′(u)− F ′(v))‖ ≤ ω∗1‖u− v‖ for all u, v ∈ E1,

iii) ‖F ′(u∗2)−1(F ′(u)− F ′(v))‖ ≤ ω∗2‖u− v‖ for all u, v ∈ E2,iv) M(u, u∗1) is continuously Frechet differentiable for all u ∈ E2,v) ‖M′(u;u∗1)−M′(v;u∗1)‖ ≤ ωM′‖u− v‖ for all u, v ∈ E2.

Then there exists an ω2 > 0 such that for all u, v ∈ E2, there holds∥∥([M(u∗2;u∗1)F (u∗2)]′)−1[(M(u;u∗1)F (u))′ − (M(v;u∗1)F (v))′]∥∥ ≤ ω2‖u− v‖.

If ‖u∗1 − u∗2‖ < ρ1 + ρ2 for some ρ1 ≤ 23ω∗1

and ρ2 ≤ 23ω2

such that we have B1 = B(u∗1, ρ1) ⊂ E1

and B2 = B(u∗2, ρ2) ⊂ E2, then the intersection B1 ∩B2 is nonempty. Starting from u0 ∈ B1 ∩B2,Newton’s method will first converge to u∗1 ∈ E1 and then after deflation with M(·;u∗1) will convergeto u∗2 ∈ E2.

The following corollary shows that, under sufficient conditions, Algorithm 1 will discover con-nected branches for sufficiently small ∆λ.

Corollary 3.8 (Connected roots [12, Corollary 3.6]). Let f : D × R→ Y and suppose there existsa λc ∈ R such that f(·, λ) : D → Y is a continuously Frechet differentiable function on the opensubset D ⊆ U for λ > λc. Furthermore assume that there exists branches u∗1(λ), u∗2(λ) ∈ D suchthat f(u∗1(λ), λ) = f(u∗2(λ), λ) = 0 and u∗1(λ) 6= u∗2(λ) for λ > λc and u∗1(λc) = u∗2(λc). Assumethat for a fixed λ > λc all conditions from Theorem (3.7) hold for the function f(·, λ) : D → Y sothat ρ1(λ), ρ2(λ) ∈ R as in Theorem (3.7) are well-defined. If

limλ↓λc

‖u∗1(λ)− u∗2(λ)‖ρ1(λ) + ρ2(λ)

< 1,

then an initial guess u0 ∈ D exists which converges to both u∗1 and u∗2 using Newton’s method anddeflation for λ sufficiently close to λc.

According to Farrell, Beentjes and Birkisson, a similar result holds for the case of branchesmeeting as λ ↑ λc, and for more than two roots.

4. Computing orbits of partial differential equations symmetry groups

In this section, we change the deflation operator defined by (3.4) in order to compute solutionsto the two-dimensional Gross-Pitaevskii (GP) equation.

4.1. Computing solutions to the two-dimensional Gross-Pitaevskii equation. This equa-tion has been studied in [6] by Charalampidis, Kevrekidis and Farrell. The problem is definedby

(4.1) F (φ;µ) ≡ −1

2∇2φ+ |φ|2φ+ V (r)φ− µφ = 0,


where ∇2 stands for the standard Laplace operator in 2D and the external potential V (r) assumesthe standard harmonic form of V (r) = 1

2Ω2|r|2, with r2 = x2 +y2 and the normalized trap strength

Ω. The problem is posed on D = (−10, 10)2 with homogeneous Dirichlet boundary conditions.

The two-dimensional Gross-Pitaevskii equation has an infinite symmetry group as SO2(R) actson the space of solutions to (4.1). This means that

F (φ;µ) = 0 =⇒ F (eiθφ;µ) = 0,

for all θ ∈ R. In this case, the algorithm of Farrell, Beentjes and Birkisson does not work becauseit has to deflate the infinite orbit SO2(R)φ1

= eiθφ1 | θ ∈ R, whenever φ1 is a solution to (3.5).The idea of the authors was to change the deflated problem to

(4.2) G(φ) =

(1

‖|φ|2 − |φ1|2‖2U+ 1

)F (φ).

As the amplitude is invariant under the group action, this modified operator eliminates the entiregroup orbit SO2(R)φ1

= eiθφ1 | θ ∈ R, ensuring nonconvergence to any solution related to aknown solution.

(a) (b) (c)

(d) (e) (f)

Figure 1. Density profiles |φ(x, y)|2 of solutions to the Gross-Pitaevskii equationat µ = 0.6.

However, after computation, the algorithm gave many copies of the same solution, suggestingthe existence of another symmetry group. Actually, Figure 1 shows the results of the deflationalgorithm at µ = 0.6. Solutions (a), (b) and (c) lie in three different orbits, while (d), (e) and (f)are just rotations of the solution (c). In the next section, we will adapt the deflation operator inorder to compute only solutions which lie in different orbits.

4.2. Application of Lie groups to deflation. Thanks to section 2, given a system of partialdifferential equations, we are able to find the symmetry groups and their relative invariants. Weused Maple and the package PDEtools, introduced in [10], to compute the symmetry groups andthe invariants of the Gross-Pitaevskii equation (4.1).

17 n. boulle

Then, we found two different symmetry groups of (4.1)

(4.3) G1 ≡ SO2(R) acts on U by g · ((x, y), φ) = ((x, y), eiθφ),

generated by v1 = v∂u − u∂v, where u and v are respectively the real and the imaginary part of φ.

(4.4) G2 ≡ SO2(R) acts on X = R2 by g ·((

xy

), φ

)=

((cos(θ) − sin(θ)sin(θ) cos(θ)

)·(xy

), φ

),

generated by v2 = y∂x−x∂y. In the last case, the solution space is extended into R2 by zero outsidethe domain D.

Thus, the invariants are respectively

ξ1(x, y, u, v) = u2 + v2 ξ2(x, y, u, v) = x2 + y2,

which explains why the deflated problem (4.2) works and the fact that the algorithm still computesmany copies of the same solution because (4.2) is not invariant under the action of G2.

We are now able to generalize this problem to a group G acting on U ×X. The aim is to findan operator Au : u ×X → R, for all u ∈ U , such that

(4.5) Au(g · (u, x)) = Au(u, x), for all g ∈ G,and then define A : U → (X → U) by u 7→ Au. If this operator exists, we would be able to buildanother deflation operator from (3.4), invariant under the action of G by

(4.6) MA(u;u∗1) =

(1

‖A(u)−A(u∗1)‖p+ σ

)I.

If G acts on U ×X, then every invariant ξ of the group satisfies (4.5). Thus, if we denote ξu : x 7→ξ(x, u), we can take

A(u) = ξu.

However, this works only if A is not a constant. Indeed, if u 7→ A(u) is constant, then the operatorMA defined by (4.6) is not defined. This leads us to exclude the case of a group G acting only onX. For instance, if G is SO2 acting on R2, then an invariant is

ξ : R2 −→R(x, y) 7−→x2 + y2

and for u ∈ U , A(u) : (x, y) 7→ x2 + y2, which does not depend on u. Then, for every u∗1 6= u,A(u) = A(u∗1) and the deflation operator (4.6) is not defined.

We will now suppose that G acts on the coordinate space X and construct an operator A whichallows (4.6) to be well defined.

Definition 4.1 (G-invariant transformation). Let G be a compact group acting on X, then for allx ∈ X and u ∈ U , let

(4.7) u(x) ≡∫g

u(g · x)dµ(g),

where µ is the Haar measure [14].

If (x, y) is in R2 and G = SO2(R) acting on R2, then

(4.8) u(x, y) =1

2π

∫ 2π

0

u(x cos(θ)− y sin(θ), x sin(θ) + y cos(θ))dθ

is the average of u along the circle of center (0, 0) and radius r =√x2 + y2. The following

propositions summarize the main properties of the G-invariant transformation.


Proposition 4.2. Let G be a compact group acting on X, u ∈ U and x ∈ X.

i) The G-invariant transformation is linear, if v ∈ U and λ ∈ R,

u+ λv = u+ λv.

ii) u is invariant under the action of G as for all g ∈ G,

u(g · x) = u(x).

iii) If u ∈ U is invariant under the action of G then u = u.

Proposition 4.3. Let G be the compact group SO2(R) acting on R2.

i) If u ∈ L2(R2), then u ∈ L2(R2) and

‖u‖L2 ≤ 1√2π‖u‖L2 .

ii) If ∇u ∈ L2(R2), then ∇u ∈ L2(R2) and

‖∇u‖L2 ≤ 1√2π‖∇u‖L2 .

iii) If u ∈ H1(R2), then u ∈ H1(R2) and

‖u‖H1 ≤ 1√2π‖u‖H1 .

Proof. We denote the action of SO2(R) on R2 by, for all θ ∈ R,

ψθ : (x, y) 7−→ (x cos(θ)− y sin(θ), x sin(θ) + y cos(θ)).

Let u ∈ L2(R2), ∫R2

u(x, y)2dxdy =

∫R2

(1

2π

∫ 2π

0

u(ψθ(x, y))dθ

)2

dxdy

≤ 1

4π2

∫ 2π

0

∫R2

u(ψθ(x, y))2dxdydθ,

by Jensen’s inequality and Fubini-Tonelli theorem. Then,

‖u‖L2 ≤ 1

2π

√∫ 2π

0

∫R2

u(x, y)2|det(J−1ψθ

)|dxdydθ ≤ 1√2π‖u‖L2 ,

because |det(J−1ψθ

)| =

∣∣∣∣∣det


)−1)∣∣∣∣∣ = 1 for all θ ∈ [0, 2π].

Let u ∈ L2(R2) with ∇u ∈ L2(R2) and φ ∈ D(R2), according to distribution theory,

< ∂xu, φ > = −∫R2

u∂xφdxdy

= − 1

2π

∫ 2π

0

∫R2

u(ψθ(x, y))∂φ

∂xdxdydθ,(4.9)

by Fubini’s theorem. If we integrate (4.9) by parts using Green’s theorem, we find

< ∂xu, φ > =1

2π

∫ 2π

0

∫R2

[cos(θ)

∂u

∂x(ψθ(x, y)) + sin(θ)

∂u

∂y(ψθ(x, y))

]φdxdydθ

19 n. boulle

=<1

2π

∫ 2π

0

cos(θ)∂u


∂u

∂y(ψθ(x, y))dθ, φ >,

by Fubini’s theorem. This implies that

(4.10) ∂xu(x, y) =1

2π

∫ 2π

0

cos(θ)∂u


∂u

∂y(ψθ(x, y))dθ.

By the same way,

< ∂yu, φ >=<1

2π

∫ 2π

0

− sin(θ)∂u

∂x(ψθ(x, y)) + cos(θ)

∂u

∂y(ψθ(x, y))dθ, φ >

and

(4.11) ∂yu(x, y) =1

2π

∫ 2π

0

− sin(θ)∂u

∂x(ψθ(x, y)) + cos(θ)

∂u

∂y(ψθ(x, y))dθ.

Then by (4.10) and (4.11),

∇u =1

2π

∫ 2π

0

(cos(θ) sin(θ)− sin(θ) cos(θ)

)· ∇u(ψθ)dθ.

ii is a consequence of the above equalities as∫R2

(∂xu(x, y))2 + (∂yu(x, y))2dxdy =

∫R2

(1

2π

∫ 2π

0

cos(θ)∂u

∂x(ψθ) + sin(θ)

∂u

∂y(ψθ)dθ

)2

dxdy

+

∫R2

(1

2π

∫ 2π

0

− sin(θ)∂u

∂x(ψθ) + cos(θ)

∂u

∂y(ψθ)dθ

)2

dxdy

≤ 1

4π2

∫ 2π

0

∫R2

(∂u

∂x(ψθ)

)2

+

(∂u

∂y(ψθ)

)2

dxdydθ,

by Jensen’s inequality and Fubini-Tonelli theorem. Then,

‖∇u‖L2 ≤ 1

2π

√√√√∫ 2π

0

∫R2

[(∂

∂xu(x, y)

)2

+

(∂

∂yu(x, y)

)2]|det(J−1

ψθ)|dxdydθ

≤ 1√2π

√‖∂xu‖2L2 + ‖∂yu‖2L2 =

1√2π‖∇u‖L2 ,

which proves ii. iii follows immediately from i and ii.

According to Propositions 4.2 and 4.3, an SO2-invariant transformation, where SO2 acts on thecoordinate space, of a function u is a function u, which keeps the potential regularity H1 of u and isinvariant under the action of SO2. Then, we can use this transformation to define another deflationoperator, which will be invariant under the action of the two symmetry groups (4.3) and (4.4) of theGross-Pitaevskii equation. We will now denote the SO2-invariant transformation as the circularlysymmetric transformation.

Definition 4.4 (Symmetry-invariant deflation operator). Let G1 and G2 be the two symmetrygroups of the Gross-Pitaevskii equation defined by (4.3) and (4.4). For each r ∈ U , for eachu ∈ U\r, define

(4.12) M(u; r) ≡

1∥∥∥|u|2 − |r|2∥∥∥pU

+ σ

IW .


4.3. Computation of the circularly symmetric transformation. The aim of this section isto provide an efficient method to compute the circularly symmetric transformation defined by (4.7)and (4.8). Indeed, we need to compute the transformation of roots of the Gross-Pitaevskii equationat each step of the deflation algorithm [12]. As we use a finite element discretization, we precalculatethe matrix of the transformation with respect to the finite element basis. In fact, if V is a finite-dimensional subspace of H1

0 with a basis (e1, ..., en), the coefficients (aij)i,j of the matrix A of thetransformation are given for each 1 ≤ j ≤ n by

(4.13) ej ≈ Aej =

n∑k=1

akjek,

where (a1j , ..., anj) is the approximation of the decomposition of ej in the basis (e1, ..., en). In fact,ej does not necessary lie in V . Then, we have to interpolate ej on the finite-dimensional space Vbefore decomposing it in the basis (e1, ..., en).

Let Ω = x ∈ R2 | ‖x‖ ≤ 1 be the domain and assume that there exists a mesh on this domainwith vertices (X1, ..., Xn). The finite-dimensional subspace V of H1

0 (Ω) is the space of piecewiselinear functions generated by the vectors (ei)1≤i≤n, where ei is the unique function of V whose valueis 1 at Xi and zero at every Xj , j 6= k as shown in Figure 2, also known as first-order LagrangeFinite Elements.

(a) (b)

Figure 2. Representation of an element of the basis of V .

Let ei be an element of the basis of V . According to (4.7) and thanks to an integration bysubstitution, for each (x, y) ∈ Ω, we need to compute the following integral

(4.14) ei(x, y) =1

2π

∫ 2π

0

ei(r cos(θ), r sin(θ))dθ,

where r =√x2 + y2 is the radius. We denote (xi, yi) = Xi, Dcell the maximum diameter of a cell

in the discretization of Ω and

Rmin = max

√x2i + y2

i −Dcell, 0

, Rmax = min

√x2i + y2

i +Dcell, 1

.

Then, in order to compute (4.14) and to reduce the time cost of the algorithm, we define a one-dimensional function

f : [0, 1] −→R

r 7−→

12π

∫ 2π

0ei(r cos(θ), r sin(θ))dθ if r ∈ [Rmin, Rmax]

0 otherwise,

21 n. boulle

which is the circularly symmetric transformation of ei along the line [0, 1]×0 ⊂ Ω. Actually, dueto the circular symmetry, we only need to compute the value along one radial. Then, the integral is

computed only if r lies in the interval [Rmin, Rmax] as ei(x, y) = 0 if r =√x2 + y2 is less than Rmin

or greater than Rmax thanks to the definition of ei (Figure 3). However, because ei, is circularlysymmetric, for each (x, y) ∈ Ω,

ei(x, y) = f(√x2 + y2).

This operation reduces significantly the time cost of the algorithm because it allows us to computethe transformation of ei only on the line [0, 1]× 0 instead of the whole domain Ω.

(a) (b)

Figure 3. Computation of the circularly symmetric transformation of the basisof V . (a) represents the intersection between the integration circle of radius r andthe mesh of the domain while (b) shows that r lies in the interval [Rmin, Rmax].

The algorithm which computes the matrix A has been implemented in FEniCS [16], which is aPython finite element library. The integral is traditionally computed in FEniCS with the edges ofthe mesh. However, as shown in Figure 3 (a), if we define a domain of integration (the boundary ofthe red circle) with a mesh, the edges of the two domains do not coincide and FEniCS is not ableto compute the integral around the interface of the two meshes. Then, we used MultiMesh [17], apowerful library recently implemented in FEniCS, which is able to compute the integral over theinterface of overlapping meshes.

(a) (b)

Figure 4. (a) is an element of the basis of V while (b) represents its circularlysymmetric transformation.


Figure 4 shows the application of our algorithm on an element of the basis ei. The result is notperfectly circularly symmetric owing to the geometry of the mesh of the domain.

After calculating the matrix of the transformation in the basis (ei)1≤i≤n, we are now able tocompute the G-invariant transformation of any function efficiently. For instance, if we want toobtain the circularly symmetric transformation of the function f : Ω→ R defined by

(x, y) 7−→ x3 − y2,

and shown in Figure 5, we just have to approximate f by a linear combination of the basis of V

f ≈n∑i=0

λiei =

λ1

...λn

,

in the matrix notation. Thus, if the matrix of the transformation is denoted by A and defined by(4.13), the transformation f of f can be approximated by

f ≈ A ·

λ1

...λn

=

a11 . . . a1n

.... . .

...an1 . . . ann

·λ1

...λn

.

Figure 5 represents the circularly symmetric transformation f of the function f : (x, y) 7→ x3−y2.

The value of f at a point (x0, y0) is the average of f along the circle of center (0, 0) and radius

r0 =√x2

0 + y20 , which is respected in the computation of f as shown in Figure 5.

(a) (b)

Figure 5. Representation of the function (x, y) 7→ x3 − y2 (a) and its circularlysymmetric transformation (b).

The complexity of the algorithm is quadratic as shown in Figure 6. In fact, in order to computethe matrix of transformation, we have to loop over every element of the basis and compute thecircularly symmetric transformation of this element. Then, the transformation has to be expressedin the basis by an interpolation, which is linear in the size of the basis.

5. Numerical Results

5.1. The two-dimensional Gross-Pitaevskii equation. We consider the model studied in sec-tion 4.1 and apply the deflated continuation algorithm with the symmetry invariant deflation op-erator defined by (4.12). Figure 7 demonstrates the efficiency of the new deflation operator todeflate the whole orbits of solutions while Figure 8 shows the solutions obtained by the algorithmat µ = 1.03.

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Figure 6. Interpolation of the time cost of the algorithm to compute a matrix of transformation

(a) (b) (c)

Figure 7. Density profiles |φ(x, y)|2 of solutions to the Gross-Pitaevskii equationat µ = 0.6.

We use the diagnostic

N =

∫R2

|φ(x, y)|2dxdy,

to summarize the dependence of each state branch on µ. This integral represents the number ofatoms in the Bose-Einstein condensate, considered as a function of µ. The bifurcation diagramof the solutions of the two-dimensional Gross-Pitaevskii equation has been computed with thisdiagnostic and is represented in Figure 9.

5.2. The two-component nonlinear Schrodinger system. In this section, we study the caseof the two-component nonlinear Schrodinger system (NLS) in two dimensions. As described in [9],this system is invariant under SO2 rotations in the coordinate and function spaces, which allowsus to deflate the roots by the same technique of the previous section.

We consider the coupled defocusing GP/NLS system written in two dimensions [15] as

i∂tφ− = −D−2∇2φ− + γ(g11|φ−|2 + g12|φ+|2)φ− + V (r)φ−,(5.1)

i∂tφ+ = −D+

2∇2φ+ + γ(g21|φ−|2 + g22|φ+|2)φ+ + V (r)φ+,(5.2)


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

Figure 8. Density profiles |φ(x, y)|2 of the solutions of the Gross-Pitaevskii equa-tion at µ = 1.03.

where ∇2 = ∂2x + ∂2

y stands for the Laplace operator in two dimensions, D± are the dispersioncoefficients, γ is the overall nonlinearity strength and the interaction coefficients are gij > 0 (∀i, j =1, 2) with g21 = g12. The external potential V (r) assumes the form

V (r) =1

2Ω2|r|2,

with |r|2 = x2 + y2 and normalized trap strength Ω. We study the case of equal interactioncoefficients gij = 1, ∀i, j = 1, 2 and use rescaling to fix D− = γ = 1, while D+ = D > 0 is therelative dispersion coefficient in the second component. Stationary solutions to (5.1)-(5.2) withchemical potentials µ± are found by assuming

φ±(x, y, t) = φ±(x, y) exp(−iµ±t).

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Figure 9. Bifurcation diagram of the two-dimensional Gross-Pitaevskii equation

This reduces (5.1)-(5.2) to

−1

2∇2φ− + (|φ−|2 + |φ+|2)φ− + V (x, y)φ− − µ−φ− = 0,(5.3)

−D2∇2φ+ + (|φ−|2 + |φ+|2)φ+ + V (x, y)φ+ − µ+φ+ = 0.(5.4)

The operating principle to “trap” bound states in the second component φ+ (in the complex functionsense) deals with the fact that the first component φ+ creates a potential well for φ+. To do so,we further first fix µ− = 1 and D = 1, which sets the background for the first component. Then,there is a one-parameter family of solutions for µ+. Explicit bounds for the emergence of statescan be found in [7] for the one-dimensional case, although such bounds can be determined onlynumerically for the two-dimensional case [8].

We study this problem on D = (−20, 20)2 with homogeneous Dirichlet boundary conditions. Ifwe denote U = U− ×U+, where U− (resp. U+) is the function space of the solution φ− (resp. φ+),the symmetry groups of the GP/NLS system are given by

G1 ≡ SO2(R) acts on U− by g · ((x, y), (φ−, φ+)) = ((x, y), (eiθφ−, φ+)),

G2 ≡ SO2(R) acts on U+ by g · ((x, y), (φ−, φ+)) = ((x, y), (φ−, eiθφ+)),

G3 ≡ SO2(R) acts on X by g · ((x, y), (φ−, φ+)) =


)·(xy

), (φ−, φ+)

).

As a solution to (5.3)-(5.4) has two components (φ−, φ+), we change the deflation operator definedby (4.12).

Definition 5.1 (GP/NLS deflation operator). Let G1, G2 and G3 be the three symmetry groupsof the GP/NLS system. For each r = (r−, r+) ∈ U , for each u = (u−, u+) ∈ U\r, define the


following deflation operator invariant under the action of G1, G2 and G3,

M(u; r) ≡

1∥∥∥(|u−|2 − |r−|2)(|u+|2 − |r+|2)∥∥∥p

U

+ σ

IW .Figures 10, 11 and 12 show the solutions found for different values of µ with the deflated contin-

uation algorithm and the GP/NLS deflation operator, while Figure 13 represents the superpositionof the two components φ−, φ+ in the domain. Contrary to the two-dimensional Gross-Pitaevskiiequation (Figure 8 (a)), ground states are not computed. In fact, the GP/NLS deflation operatorremoves the solutions φ = (φ−, φ+) if φ− = 0 or φ+ = 0.

(a) (b) (c) (d)

(a) (b) (c) (d)

Figure 10. Density profiles |φ−(x, y)|2 (in blue) and |φ+(x, y)|2 (in red) of thesolutions (a), (b) and (c) to the GP/NLS system at µ = 0.6. Solution (d) isrepresented at µ = 0.7.

We use the diagnostics

N− =

∫R2

|φ−(x, y)|2dxdy, N+ =

∫R2

|φ+(x, y)|2dxdy,

to summarize the dependance of each state branch on µ− and µ+ (here µ− is fixed) and plot thebifurcation diagram of φ− and φ+ in Figure 14. Both integrals represent the number of atoms inthe Bose-Einstein condensate, considered as a function of (µ−, µ+).

6. Conclusion and future challenges

In this work, we studied Lie groups theory in order to understand their relations with partialdifferential equations. According to Olver, symmetry groups of a given partial differential equationcan be identified by solving a system of equations and their invariants can be computed as well.This led us to adapt deflation operators, taking into account the symmetries of the equation.

27 n. boulle

(e) (f) (g) (h)

(e) (f) (g) (h)

(i) (j) (k) (l)

(i) (j) (k) (l)

Figure 11. Density profiles |φ−(x, y)|2 (in blue) and |φ+(x, y)|2 (in red) of thesolutions to the GP/NLS system at µ = 1.345.

The idea of deflated continuation algorithm has been introduced in [13] by Farrell, Birkisson andFunke and is based on the work of Wilkinson [21] and the generalization of Brown and Gearhart [4].Deflated continuation has now shown its efficiency in a wide range of problems [1, 5, 6, 11, 12, 13].

However, deflated continuation did not take into account the symmetries of the partial differentialequation and calculated every solutions, even those in the same orbit. Thus, we adapted thedeflation operators and found new operators which are invariant under the action of given symmetrygroups. This technique has been applied to the two-dimensional Gross-Pitaevskii equation and thetwo-component nonlinear Schrodinger system. As shown in section 5, the new deflation operatorscompute only solutions which lie in the same orbit and removed every rotations of solutions. This


(m) (n) (o) (p)

(m) (n) (o) (p)

(q) (r)

(q) (r)

Figure 12. Other solutions to the GP/NLS system at µ = 1.345.

improved the efficiency of the deflated continuation algorithm by decreasing the number of steps.In fact, we now use only one solution per orbit as an initial guess in order to seek new branches.

Nevertheless, we only studied problems with SO2 or SO3 symmetries and implemented theoperators with these symmetry groups. The G-invariant transformation can be used to defineother deflation operators to find solutions of partial differential equations with different symmetrygroups. In this work, we found solutions of the two-components nonlinear Schrodinger systemin two dimensions but it is also possible to study this system in three dimensions or with morecomponents and find solutions as well. Finally, another challenge would be to adapt the deflatedcontinuation algorithm in order to solve numerically time-dependent partial differential equations.Thus, it would be very interesting to compute multiple solution paths, which converge to differentstationary states.

29 n. boulle

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r)

Figure 13. Superposition of the components |φ−(x, y)|2 (negative and blue) and|φ+(x, y)|2 (positive and red) of the solutions shown in Figures 10, 11 and 12.


(a) (b)

Figure 14. Bifurcation diagram of the GP/NLS system. Left and right panelshow the bifurcations of the solutions φ− and φ+ respectively.

References

[1] J. H. Adler, D. B. Emerson, P. E. Farrell, and S. P. MacLachlan, A deflation technique for detecting

multiple liquid crystal equilibrium states, SIAM Journal on Scientific Computing, 39 (2017), pp. B29–B52.

[2] C. Beentjes, Computing bifurcation diagrams with deflation, Master’s thesis, University of Oxford, 2015.

[3] A. Birkisson, Numerical Solution of Nonlinear Boundary Value Problems for Ordinary Differential Equations

in the Continuous Framework, PhD thesis, University of Oxford, 2014.[4] K. M. Brown and W. B. Gearhart, Deflation techniques for the calculation of further solutions of a nonlinear

system., Numerische Mathematik, 16 (1971), pp. 334–342.

[5] S. J. Chapman and P. E. Farrell, Analysis of Carrier’s problem, SIAM Journal on Applied Mathematics, 77(2017), pp. 924–950.

[6] E. G. Charalampidis, P. G. Kevrekidis, and P. E. Farrell, Computing stationary solutions of the two-

dimensional Gross-Pitaevskii equation with deflated continuation, Communications in Nonlinear Science andNumerical Simulation, 54 (2018), pp. 482–499.

[7] E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed, Dark-bright solitonsin coupled nonlinear Schrodinger equations with unequal dispersion coefficients, Physical Review E, 91 (2015),

p. 012924.

[8] E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed, Vortex-soliton complexesin coupled nonlinear Schrodinger equations with unequal dispersion coefficients, Physical Review E, 94 (2016),p. 022207.

[9] E. G. Charalampidis, W. Wang, P. G. Kevrekidis, D. J. Frantzeskakis, and J. Cuevas-Maraver,SO(2)-induced breathing patterns in multicomponent Bose-Einstein condensates, Physical Review A, 93 (2016),

p. 063623.

[10] E. S. Cheb-Terrab and K. Von Bulow, A computational approach for the analytical solving of partial differ-ential equations, Computer Physics Communications, 90 (1995), pp. 102–116.

[11] D. B. Emerson, P. E. Farrell, J. H. Adler, S. P. MacLachlan, and T. J. Atherton, Computing equi-

librium states of cholesteric liquid crystals in elliptical channels with deflation algorithms, arXiv preprintarXiv:1706.04597, (2017).

[12] P. E. Farrell, C. H. L. Beentjes, and A. Birkisson, The computation of disconnected bifurcation diagrams,

arXiv preprint arXiv:1603.00809, (2016).

[13] P. E. Farrell, A. Birkisson, and S. W. Funke, Deflation techniques for finding distinct solutions of nonlinearpartial differential equations, SIAM Journal on Scientific Computing, 37 (2015), pp. A2026–A2045.

[14] A. Haar, Der massbegriff in der theorie der kontinuierlichen gruppen, Annals of Mathematics, 34 (1933),pp. 147–169.

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[15] P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez, The defocusing nonlinear Schrodingerequation: from dark solitons to vortices and vortex rings, SIAM, 2015.

[16] A. Logg, G. N. Wells, and J. Hake, Dolfin: A C++/Python finite element library, Automated Solution of

Differential Equations by the Finite Element Method, (2012), pp. 173–225.[17] A. Massing, M. G. Larson, and A. Logg, Efficient implementation of finite element methods on nonmatching

and overlapping meshes in three dimensions, SIAM Journal on Scientific Computing, 35 (2013), pp. C23–C47.

[18] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986.[19] L. B. Rall, A note on the convergence of Newton’s method, SIAM Journal on Numerical Analysis, 11 (1974),

pp. 34–36.[20] W. C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Banach Center

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summer internship 2017 bifurcation analysis with symmetry

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