On Solutions to the Nonlocal ∂ ¯ ¯ ¯ Problem and (2+1) Dimensional Completely Integrable Systems
aa r X i v : . [ n li n . S I] A ug ON SOLUTIONS TO THE NONLOCAL DBAR PROBLEM AND(2+1) DIMENSIONAL COMPLETELY INTEGRABLE SYSTEMS
PATRIK V. NABELEK
Abstract.
In this short note we discuss a new formula for solving the nonlocal dbarproblem, and discuss application to the Manakov–Zakharov dressing method. Wethen explicitly apply this formula to solution to the complex (2+1)D Kadomtsev–Petviashvili equations and complex (2+1)D completely integrable generalization ofthe (2+1)D Kaup–Broer (or Kaup–Boussinesque) system. We will also discuss howreal (1+1)D solutions are expressed using this formalism. It is simple to expressthe formalism for finite gap primitive solutions from [11], [9] using the formalismof this note. We also discuss how recent results on the infinite soliton limit for the(1+1)D Korteweg–de Vries equation and the (2+1)D Kaup–Broer system. In anappendix, the classical solutions to the 3D Laplace equation (2+1)D d’Alambertwave equation by Whittaker are described. This appendix is included to elucidatean analogy between the dressing method and the Whittaker solutions. Introduction
The KdV equation is(1) η t + cη x + αηη x + βη xxx = 0 . The KdV equation described weakly nonlinear long wave propagating in (1+1)D, andis the first known example of a completely integrable PDE. The KP equation(2) ( η t + cη x + αηη x + βη xxx ) x = γη yy is a two dimensional completely integrable generalization of the KdV equation, anddescribes weakly nonlinear long waves in (2+1)D that are primarily traveling in the x direction, with small deviations in the y dimension. Solutions to the KP equationthat do not satisfy this condition are not physically realized in weakly nonlinear longwave systems.The KP equation can be solved by the inverse scattering problem using a nonlocaldbar problem [1]. The structure of the nonlocal dbar problem was further used byZakharov and Manakov to discuss inverse scattering for general classes of problemsin dimension higher than (1+1)D [16, 17]. This formulation of the inverse scatteringproblem also leads to class of solutions to the KdV equation called primitive solutionthat have been studied in [5, 4, 6, 9, 10, 15, 18].The Kaup–Broer system is ϕ t + ε ( ϕ x ) + ε η = 0 , (3) η t + µ ϕ xx + µ ( ηϕ x ) x + µ ϕ xxxx = 0 . (4) Date : September 1, 2020.
The Kaup–Broer equation was introduced by Kaup and Broer as a completely non-linear system that described shallow water long waves. The Cauchy problem for theKaup–Broer system is ill-posed because of instabilities that lead to finite time blowup [1]. However, a class of solutions called primitive solutions introduced in [11] arestable and bounded, and can still be analyzed in practical models even if the fullspace of solutions is problematic.Its (2+1)D completely integrable generalization is ϕ t + α ϕ u − β ϕ v + γ Π = 0(5) η t + α ( ϕ u η ) u − β ( ϕ v η ) v + α ϕ uu − β ϕ vv + α ϕ uuuv − β ϕ uvvv = 0(6) α η uu − β η vv − γ Π uv = 0 . (7)A real version of the system was introduced in [12]. The nonlocal dbar dressingmethod for solving the fully complex version of this equation was discussed in [11].The real version introduced by [12] generalizes a single canonical scaling of the Kaup–Broer system, while the fully complex version generalizes all four scaling class of theKaup–Broer system [11].The KdV equation is an isospectral evolution of 1D Schr¨odinger operators. TheKP equation is an isospectral evolution of the (1+1)D Schr¨odinger equation witha fixed spectral curve. The (2+1)D generalization of the Kaup–Broer system is anisospectral evolution of 2D Schr¨odinger equations at fixed energy level in the sensethat it preserves the spectral curve [11].These equations can be solved using the dressing method. The dressing method isa name given to a whole family of methods for solving completely integrable partialdifferential equations. In this paper I will discuss the nonlocal dbar problem for theKP equation. The formulation of the dressing method addressed in this paper isbased on the nonlocal dbar problem method, and we summarize this as follows in ourmain theorem Theorem 1.
Suppose that f ∈ D and R ∈ D , where D and D are certain spacesof distribution such that the choice of D can effect D . If f solves the equation (8) f ( ζ ) − Z Z C K ( ζ , ξ ) f ( ξ ) d ξ = g ( ζ ) , (9) K ( ζ , ξ ) = 1 π Z Z C R ( ζ , η ) η − ξ d η, (10) g ( ζ ) = Z Z C R ( ζ , η ) d η, and (11) L f ( ζ ) = f ( ζ ) − Z Z C K ( ζ , ξ ) f ( ξ ) d ξ has trivial nullspace, then (12) χ ( λ ) = 1 + Z Z C f ( ζ ) λ − ζ d ζ ONLOCAL DBAR PROBLEM AND (2+1)D INTEGRABLE SYSTEMS 3 is the unique solution to the nonlocal dbar problem (13) ∂χ∂ ¯ λ ( λ ) = Z Z C R ( λ, η ) χ ( η ) d η. satisfying χ ( λ ) = 1 + O ( λ − ) as λ → ∞ . The first sentence of the theorem is simply saying that we are free to choose adressing function R in a precise manner, and therefore we only need to pick the spacesof a distributions on a case by case basis and verify the Fredholm alternative for theoperator L . Our perspective of the dressing method gives the following solutions tothe KP equation.Applying the main theorem to the nonlocal dbar problem dressing method for theKP equation gives the following form(14) u ( x, y, t ; R ) = A ∂∂x
Z Z C f ( ξ, x, y, t ; R ) dξ of a solution to the KP equation, where f ( λ ; R ) is the solution to(15) f ( ζ ) − π Z Z Z Z C R ( ζ , η ) η − ξ f ( ξ ) d ηd ξ = Z Z C R ( ζ , η ) d η, and(16) R ( λ, η ) = e φ ( η,x,y,t ) − φ ( λ,x,y,t ) R ( λ, η )where φ ( λ, x, y, t ) = λx + αλ y + λ t and R is independent of x , y , t . We willalso discuss the computations of isospectral manifolds containing all potentials withthe same spectrum. The solutions to completely integrable systems are dynamicalsystems on the isospectral manifolds.2. The Nonlocal Dbar Problem
A nonlocal dbar problem is an integro-differential equation of the form(17) ∂χ∂ ¯ λ ( λ ) = Z Z C R ( λ, η ) χ ( η ) d η where R ( λ, η ) : C → C is a collection of Dirac masses, single layer potentials and 4potentials with compact support, and d η is the Lebesgue measure on R ≡ C . Welook for a solution satisfying the boundary condition χ ( λ ) = 1 + O ( λ − ) as λ → ∞ .Suppose the support of R ( λ, η ) is some embedded compact complex sub-manifold of C that is also an algebraic variety.The solution χ ( λ ) to the nonlocal dbar problem solves the integral equation(18) χ ( λ ) = 1 + 1 π Z Z Z Z C R ( ζ , η ) χ ( η ) λ − ζ d ζ d η. We look for a solution χ ( λ ) to this integral equation of the form(19) χ ( λ ) = 1 + 1 π Z Z C f ( ξ ) λ − ξ d ξ. PATRIK V. NABELEK
Then the generalized function f ( λ, ζ ) solves the singular integral equation(20) Z Z C f ( ξ ) λ − ξ d ξ − π Z · · · Z C R ( ζ , η ) f ( ξ )( λ − ζ )( η − ξ ) d ηd ξd ζ = Z Z Z Z C R ( ζ , η ) λ − ζ d ηd ζ where the support of R and f are generalized functions defined on a compact subsetof C that has regularity no worse than delta functions. This is an integral equationsatisfied by a charge density that is supported on a set of points in the plane, a setof contours in the plane, and a set of 2D subsets of the plane.If the integral transform(21) H C f ( λ ) = Z Z C f ( ξ ) λ − ξ d ξ is invertible on an appropriate space of generalized function for a specific application,then this singular integral equation take the form(22) f ( λ ) − Z Z C K ( λ, ξ ) f ( ξ ) d ξ = g ( λ )where the generalized function K ( λ, ξ ) is the kernel(23) K ( λ, ξ ) = 1 π Z Z C R ( λ, η ) η − ξ d η and the generalized function g ( λ ) is(24) g ( λ ) = Z Z C R ( λ, η ) d η. Each choice of R leads to a choice of K and g , so we only need to consider a case bycase basis. The Neumann series solution is(25) f ( λ ) = ∞ X n =0 Z · · · Z C n K ( λ, η ) · · · K ( η n − , η n ) g ( η n ) d η · · · d η n which depends only on the choice of dressing function R ( λ, η ). (It is a nonlinearfunctional of the dressing function R ).The solutions to the nonlocal dbar problem(26) ∂χ∂ ¯ λ ( λ ) = Z Z C R ( λ, η ) χ ( η ) d η is then given by(27) χ ( λ ) = 1 + Z Z C f ( ξ ) λ − ξ d ξ. In summary, the nonlocal dbar problem is equivalent to the signular integral equation(28) f ( λ ) − Z Z C K ( λ, ξ ) f ( ξ ) d ξ = g ( λ )where the generalized function K ( λ, ξ ) is the kernel(29) K ( λ, ξ ) = 1 π Z Z C R ( λ, η ) η − ξ d η ONLOCAL DBAR PROBLEM AND (2+1)D INTEGRABLE SYSTEMS 5 and the generalized function g ( λ ) is(30) g ( λ ) = Z Z C R ( λ, η ) d η. This form of the integral equation produces complex and singular solutions to theKP equation. To produce a real solution, a reality condition needs to be introducedthat will imply that R ( λ, ζ ) has to be supported on a surface. Similarly, the di-mensional reduction of a (2+1)D integrable system to (1+1)D will involve a similarcondition that R ( λ, ζ ) has to be restricted to a surface. The intersection of thesesurfaces will be a real curve, and this is why the dressing function for solutions to real(1+1)D equations are supported on curves. This integral equation has the followinginteresting degenerations (in all cases, all the below functions used to define R and f are H¨older continuous with H¨older constant less than 1).In the following we make analogy to potential theory. This is because the connectionbetween the Cauchy–Riemann equations and the Laplace equation. The use of chargesis to get an intuitive picture of the sources for the Cauchy–Riemann equations.2.1. Rational Solutions to the Nonlocal ¯ ∂ Problem.
To produce rational solu-tions we make the assumption that(31) f ( ξ ) = N X n =1 f n δ ( ξ − ζ n ) , R ( ζ , η ) = N X n =1 r n δ ( ζ − ζ n )( η − η n ) . Then f n solves(32) f n − π N X m =1 r n η n − ζ m f m = r n . The values f n can then be determined by { r n , ζ n , η n } by solving this linear equation.2.2. Sectionally Holomorphic Solutions to the Nonlocal ¯ ∂ Problem.
In thiscase we will consider R a single layer potential. By this we mean(33) R ( λ, η ) = N X n =1 Z γ n ˜ R n ( s ) δ ( λ − ˜ ζ n ( s )) δ ( η − ˜ η n ( s )) ds and we consider f of the form(34) f ( ζ ) = N X n =1 Z γ n ˜ f n ( s ) δ ( ζ − ˜ ζ n ( s )) ds. The functions ˜ f n ( s ) solves the system of integral equations(35) ˜ f n ( s ) − π N X m =1 Z γ m ˜ R n ( s, s ′ ) ˜ f m ( s ′ )˜ η n ( s ′ ) − ˜ ζ m ( s ) ds ′ = ˜ R n ( s ) . These solutions can also be computed using a nonlocal scalars Riemann–Hilbertproblem, and an equivalent local vector Riemann–Hilbert problem.
PATRIK V. NABELEK
This case includes all primitive solutions to the KdV equation and the KB system[4, 5, 6, 9, 9, ? , 15, ? ], and thus all finite gap solutions of the KdV equation and likelyall physically relevant finite gap solutions to the KB system.2.3. Solutions to the Nonlocal ¯ ∂ Problem Supported on 2D Subsets of C . Let us consider(36) R ( ζ , η ) = N X n =1 Z Z Σ n ˘ R n ( p ) δ ( ζ − ˘ ζ n ( p )) δ ( η − ˘ η n ( p )) d p and(37) f ( ζ ) = N X n =1 Z Z Σ n ˘ f n ( p ) δ ( ζ − ˘ ζ n ( p )) d p. The functions ˘ f n ( p ) solve(38) ˘ f n ( p ) − π N X m =1 Z Z Σ n R n ( p, p ′ ) ˘ f m ( p ′ )˘ η n ( p ′ ) − ˘ ζ m ( p ) dp ′ = ˘ R n ( p ) . These include the case of a separable sum considered in [11].2.4.
Mixed Type Solution.
If we take dressing functions that are linear combi-nations of the dressing functions produced in the above case, then we can computenonlinear superpositions. From the above discussions it is easy to write out the appro-priate integral equations for the mixed solutions, however they become notationallymessy. We therefore simply mention this possibility, and leave it to the reader tocomplete the rest.2.5.
Existence and Uniqueness for Some Cases.
Let π ( λ, η ) = λ and π ( λ, η ) = η . The nonlocal overlap condition is the condition that(39) π (supp( R ( λ, η ))) ∩ π (supp( R ( λ, η ))) = ∅ . In the real case, if there is no nonlocal overlap then the kernel is a regular Hilbertkernel and the integral equation therefore has a unique solution. In singular cases,such as the finite gap solutions to the KdV equation, then this argument doesn’twork. However, it is likely possible to use a limiting argument to produce a solutionin the singular case. Then the only issue would be verifying the Fredholm alternativecondition.When there is a nonlocal overlap and the solution is supported and loops and arcs,then a nonlocal overlap can sometimes be solved using a Riemann–Hilbert problem.For example, this is the case when one produces the finite Gap Solutions to the KdVequation.
ONLOCAL DBAR PROBLEM AND (2+1)D INTEGRABLE SYSTEMS 7 Solutions to the KP and (2+1)D KB Equations by the DressingMethod
In the case that(40) R ( ζ , η, x, y, t ) = e φ ( η,x,y,t ) − φ ( ζ,x,y,t ) R ( ζ , η )with φ ( ζ , x, y, t ) = ζ x + ζ y + ζ t then(41) η ( x, y, t ) = A ∂∂x
Z Z C f ( ξ, x, y, t ) d ξ is a solution to the KP equation(42) ( η t + cη x + αηη x + βη xxx ) x = γη yy . We can also use the canonical scaling(43) ( u t − uu x + u xxx ) x = γu yy . These reduce to the primitive potentials and solutions to the KdV equation.The function ψ ( λ, x, y, t ) = e φ ( λ,x,y,t ) χ ( λ, x, y, t ) is known as the wave function forthe KP equation. If we instead consider(44) φ ( λ, t , t , . . . ) = ∞ X n =1 t n λ n then(45) u ( t , t , . . . ) = A ∂∂t Z Z C f ( ξ, t , t , . . . ) d ξ is a solution to the KP hierarchy, and ψ ( λ, t , t , . . . ) = e φ ( λ,t ,t ,... ) χ ( λ, t , t , . . . ) is awave function for the KP higherarchy. The KP hierarchy can be used to traverse theisospectral manifold of all finite gap or Bargmann potentials with the same spectrum.These isospectral manifolds are finite dimensional. The isospectral manifolds of pe-riodic and primitive potentials are infinite dimensional. The KP1 and KP2 are thetwo scalings of the real KP equation, and are the two real reductions of the complexKP equation. The KdV is a dimensional reduction of the KP equation.In the case that φ ( λ, u, v, t ) = λu + aλ − v + ( αλ + βλ − ) t then(46) ϕ = A + B log (cid:18)Z Z C f ( ξ ) ξ dξ (cid:19) , η = C ∂∂x
Z Z C f ( ξ ) dξ + Dϕ xx solves the (2+1) dimensional completely integrable generalization of the Kaup–Broersystem ϕ t + αϕ u − βϕ v + Π = 0(47) η t + 2 α ( ϕ u η ) u − β ( ϕ v η ) v − aαϕ uu + 2 aβϕ vv + α ϕ uuuv − β ϕ uvvv = 0(48) Π uv = 2 αη uu − βη vv . (49) PATRIK V. NABELEK
This one complex equation has dimensional reductions to all four scaling classes ofthe real Kaup–Broer system in (1+1)D(50) η t + µ ϕ xx + µ ( ηϕ x ) x + µ ϕ xxxx = 0 , (51) ϕ t + ε
12 ( ϕ x ) + ε η = 0 . A solution to the full complex (2+1)D Kaup–Broer hierarchy(52) ϕ = A + B log (cid:18)Z Z C f ( ξ ) ξ dξ (cid:19) , η = C ∂∂t Z Z C f ( ξ ) dξ + Dϕ t t can be computed if we consider(53) φ ( λ, t , s , t , s , . . . ) = ∞ X n =1 t n λ n + s n λ − n . The Kaup–Broer hierarchy allows the manifolds of certain 2D Schr¨odinger equationswith the same rational spectral curves to be computed. In this case the isospectralmanifolds are manifolds of 2D Schr¨odinger operators with electromagnetic fields atfixed level. 4.
Conclusions
In this note we introduced a family of simple singular linear integral equations thatcan be used to solve many cases of the nonlocal ¯ ∂ problem. These solutions allowthe calculations of large families of solutions to the KP equation and a (2+1)D gen-eralization of the KB system, which we describe explicitly. In principle, the methoddiscussed in this note will be applicable to any (2+1)D completely integrable equa-tions for which the dressing function is known. Appendix A. Classical Examples of Local Solutions to Linear PDEs
Classically, mathematicians studied ordinary and partial differential equations fromthe point of view of calculating local solutions. That is, they were looking for formulasfor analytic functions that solve the equations. Due to the advent of computers,it has become popular to think of ODEs and PDEs in therms of functions spacesof solutions. One of the most elementary classical solution is the local solution isd’Alambert’s solution to the (1+1)D wave equation.The (1+1)D wave equation u tt = cu xx has the local d’Alambert solution(54) u ( x, t ) = f ( x − ct ) + g ( x + ct ) . In imaginary velocity case c = i , this equation becomes the elliptic 2D Laplace equa-tion u ( x, y ) = f ( x + iy )+ g ( x − iy ) where f and z solve the Cauchy–Riemann equations(55) ∂f∂ ¯ z = 0 . ONLOCAL DBAR PROBLEM AND (2+1)D INTEGRABLE SYSTEMS 9
This approach generalizes to Whittaker’s solutions to the (2+1)D wave equation u tt = c ∆ u and the 3D Laplace equation ∆ u = 0 is the case of imaginary time [13, 14].Whittaker’s solution to the (2+1)D wave equation is(56) u ( x, y, t ) = Z π f (cos( θ ) x + sin( θ ) y + ct, θ ) dθ and Whittaker’s solution to the 3D Laplace equation is(57) u ( x, y, z ) = Z π f (cos( θ ) x + sin( θ ) y + iz, θ ) dθ where f ( z, θ ) solves the Cauchy–Riemann equations(58) ∂f∂ ¯ z ( z, θ ) = 0 . The function f ( z, θ ) also has the Fourier series(59) ∞ X n = −∞ f n ( z ) e inθ . This means that the series can be written as(60) f ( z, θ ) = ∞ X n,m = −∞ a nm z m e inθ and therefore the general solution to the Laplace equation is the Whittaker series(61) u ( x, y, z ) = ∞ X n,m = −∞ a nm Z π (cos( θ ) x + sin( θ ) y + iz ) m e inθ dθ. The Whittaker series solution to the (2+1)D wave equation is(62) u ( x, y, t ) = ∞ X n,m = −∞ a nm Z π (cos( θ ) x + sin( θ ) y + ct ) m e inθ dθ. This series solution is a linear function of the coefficients a nm . The Whittaker seriesallows the computation of solutions without boundary conditions.The Whittaker series also generalize to the wave equation and Laplace equationhigher dimensions. The Whittaker series has the down side that the coefficients a nm do not uniquely determine the solution, however it was used by Whittaker to justifythe Bessel series solutions to the Laplace equation before the advent of spectral theory.The usual spectral theory does not apply to nonlinear equations, so a local solutionto the KP equation could help unify the inverse scattering transform for variousboundary conditions. References [1] M. J. Ablowitz, D. Bar-Yaacov, A. S. Fokas,
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