Analytic iteration procedure for solitons and traveling wavefronts with sources
AAnalytic iteration procedure for solitons and traveling wavefrontswith sources
Jonas Berx and Joseph O. Indekeu Institute for Theoretical Physics, KU Leuven, B-3001 Leuven, Belgium (Dated: July 26, 2019)
Abstract
A method is presented for calculating solutions to differential equations analytically for a varietyof problems in physics. An iteration procedure based on the recently proposed BLUES (BeyondLinear Use of Equation Superposition) function method is shown to converge for nonlinear ordinarydifferential equations. Case studies are presented for solitary wave solutions of the Camassa-Holmequation and for traveling wavefront solutions of the Burgers equation, with source terms. Theconvergence of the analytical approximations towards the numerically exact solution is exponen-tially rapid. In practice, the zeroth-order approximation (a simple convolution) is already usefuland the first-order approximation is already accurate while still easy to calculate. The type ofnonlinearity can be chosen rather freely, which makes the method generally applicable. a r X i v : . [ n li n . PS ] O c t olving differential equations (DEs) is of general interest in physics, since they are thelanguage in which many physical laws are formulated. Standard summation or integrationmethods for linear problems, such as the Fourier transform and the Green function method[1], are not indicated when nonlinearity is present. For nonlinear problems [2] numericalcomputation is often the only way to go and although this is satisfactory from the point ofview of precision, the physical insight and control gained by having explicit analytical formsat one’s disposal, is missing. For example, the way parameters in a DE affect its solutionis conspicuous in an analytical expression. Analytic methods for solving nonlinear DEs aresparse. The method we propose is complementary to the Simplest Equation method [3].It is also complementary to the soliton perturbation approach [4, 5] and to the HomotopyAnalysis Method [6]. In contrast with these methods, our approach appears easier to applywithout giving in on accuracy.In this Letter we pose and answer, from a physicist’s point of view, the following funda-mental questions. i) Can methods for solving a linear ordinary differential equation (ODE)serve as a basis for a useful iteration procedure for solving a nonlinear ODE? ii) Can oneobtain explicit analytical solutions that are simple and accurate in a nonlinear physicalproblem? iii) For which types of nonlinearity can one hope that the strategy might work?We provide positive answers to the first two questions by developing the recently proposedBeyond-Linear-Use-of-Equation-Superposition (BLUES) function method [7] into a usefuliteration procedure with exponentially rapid convergence of the successive analytical ap-proximations. To answer the third question we start from a linear ODE for which the Greenfunction is known, and show that a convergent iteration can be obtained when a nonlinearterm is added. The type of nonlinearity can be chosen rather freely. This demonstratesthe general applicability of the method, beyond the range of applications envisaged in [7].Possible extensions of the method to nonlinear partial differential equations are beyond thescope of the present work.We briefly recall the method proposed in [7]. Let N z be a nonlinear differential oper-ator and suppose a piecewise analytic function B ( z ) is known that solves (under suitableboundary conditions) N z B ( z ) = δ ( z ) , (1)where the Dirac delta function has been added as a source term. This term cancels a possiblediscontinuity of the highest derivative of B in z = 0. Examples have been given in [7, 8]2n the form of traveling wave solutions of reaction-diffusion-convection DEs related to theFisher equation of population biology [9].Next, one looks for a related linear operator L z so that B ( z ) also solves L z B ( z ) = δ ( z ) , (2)which makes B the Green function for L z . The proposition in [7] is to derive the linearoperator from the nonlinear one using linearization about z = 0. While this can provide thedesired linear DE with the property (2) in some cases, in other cases the desired linear DEcan be found heuristically and it is not necessary to linearize the original DE, as we willdemonstrate in an example.Now we ask whether there is a systematic way to solve (1) for an arbitrary source ψ ( z ),knowing that the convolution product of B and ψ solves the linear DE with source ψ , L z ( B ∗ ψ ( z )) = ψ ( z ) (3)The proposal of [7] is to make use of the nonlinear differential operator R z ≡ L z − N z , calledresidual, and to calculate the solution to the nonlinear problem in the form B ∗ φ , so that N z ( B ∗ φ ( z )) = ψ ( z ) (4)where φ can be calculated making use of the identity φ ( z ) = ψ ( z ) + R z ( B ∗ φ ( z )) , (5)which can be iterated.To zeroth order, φ (0) ( z ) = ψ ( z ) (6)and the n th-order approximation ( n ≥
1) is found through [12] φ ( n ) ( z ) = ψ ( z ) + R z ( B ∗ φ ( n − ( z )) (7)Note that the zeroth-order solution to (4) is simply the convolution B ∗ ψ . This amounts tousing superposition beyond the domain of the linear DE, whence the name BLUES functionfor B ( z ). Testing the iteration procedure is the task to which we now turn.Our first example is in fluid mechanics. We study the propagation of waves of height u ( x, t ) on shallow water, as described by a variety of DEs among which the Korteweg-de3ries equation is the best known. In this context a source term represents an externalforce and DEs of this type are under active study [10]. Here we focus on a related DE, the(dimensionless) Camassa-Holm equation [11] with dispersion parameter κ ≥ u t + 2 κu x − u xxt + 3 uu x = 2 u x u xx + uu xxx , (8)where u t = ∂u/∂t etc. For κ > z = x − ct , with c the velocity. For vanishingdispersion ( κ →
0) the solitons become piecewise analytic with a jump in slope in their peak(“peakons”), which we place at z = 0. The dispersionless nonlinear ODE, in the co-movingframe, can be integrated once and one obtains − c ( U − U zz ) + U − U z − U U zz = 0, with U ( z ) ≡ u ( x, t ) and we have used the boundary conditions U ( | z | → ∞ ) = 0.In order for a peakon solution of arbitrary amplitude to be exact in every point including z = 0, we add a co-moving Dirac delta function source of strength s to cancel the jump in U z , which leads to the nonlinear ODE N z U ≡ s (cid:18) − c ( U − U zz ) + 32 U − U z − U U zz (cid:19) = δ ( z ) (9)This is solved exactly by the peakon solution U ( z ) = A e −| z | ≡ B ( z ) , (10)provided the strength s of the source, the wave speed c and the peakon amplitude A satisfy s = 2 A ( A − c ). Note that for s = 0 we retrieve the peakon with A = c of the dispersionlessCamassa-Holm equation without source [11]. We now calculate analytically a solution foran arbitrary co-moving source ψ ( z ). Following [7] we recall three lines of motivation for thissource: a) to smoothen the singularity of the peakon, b) to include the effect of a co-movingagent or substance, or c) to add a co-moving external field or probe.We now look heuristically for a related linear ODE for which the peakon is the Greenfunction. It can be found by neglecting the difference ( U − U z ) / z = 0 to zeroth order in z , setting U ( z ) = A , but keeping thederivatives. Indeed, the linear ODE obtained in this ad hoc way, L z U ≡ s ( A − c ) ( U − U zz ) = δ ( z ) , (11)with the same boundary conditions U ( | z | → ∞ ) = 0, is again solved by (10). Since thepeakon solves the nonlinear DE with a Dirac delta source and is also the Green function forthe related linear DE, it is a BLUES function according to the definition proposed in [7].4e now consider the DE with an arbitrary source ψ localised about z = 0, which forconvenience we normalize to unity, (cid:82) ψ ( z )d z = 1. We can take, e.g., the exponential cornersource (cf. [7]), ψ ( z ) = e −| z | /K K , (12)which is itself a peakon with tunable decay length, tending to a delta function in the limit K →
0. The zeroth-order solution is( B ∗ ψ )( z ) = AK − (cid:0) K e −| z | /K − e −| z | (cid:1) (13)and the first-order solution takes the form( B ∗ φ (1) )( z ) = ( B ∗ ψ )( z ) − A Ks ( K − (cid:34) (cid:0) K e − | z | /K − −| z | (cid:1) K −
4) + e −| z | (1+1 /K ) K + 1 − e −| z | /K (cid:21) (14)Higher-order approximations can be calculated using (7).We now compare the approximate analytical solutions with the numerically exact so-lution of the nonlinear DE (9) with source (12). The zeroth-order approximation (simpleconvolution) follows the numerical solution well and higher-order approximations systemati-cally improve upon this. In Fig.1a the BLUES function, and the zeroth-order and first-orderapproximations are shown, together with the numerical solution, for K = 1 /
2. In Fig.1b azoom is presented near the peak at z = 0 and solutions are shown up till 3rd order. Evenfor this relatively broad source the first-order approximation (14) is already accurate.The fast convergence to the correct answer is conspicuous by inspecting the peak values(at z = 0) in Fig.1b, as Fig. 2 shows. The increments | ∆ U ( n,n − ψ (0) | ≡ | U ( n ) ψ (0) − U ( n − ψ (0) | decay to zero exponentially rapidly, as is seen in the semi-log plot in the inset of Fig. 2.In each iteration almost an order of magnitude is gained in precision. We verified that thisholds uniformly for all z .We have verified that the iteration procedure converges exponentially rapidly to thecorrect answer for values of K in the range 10 − ≤ K < . So we find that the methodworks well far beyond the scope envisaged in [7] and this suggests that the approach doesnot require perturbation theory.Our second example pertains to the propagation and diffusion of disturbances in liquids,gases, ..., traffic, etc. We start from one of the most widely used linear DEs in physics, the5 ( z ) U num U ψ ( ) U ψ ( ) - - (a) U num U ψ ( ) U ψ ( ) U ψ ( ) U ψ ( ) - - (b) Figure 1: (a)
Soliton solution to the nonlinear Camassa-Holm DE (9) with the exponentialcorner source (12). The numerical solution U num (red, full line), the zeroth-orderapproximation U (0) ψ (black, dashed line) and the first-order one U (1) ψ (black, wider spaceddashed line) are compared. The BLUES function (gray) is also shown. On this scale U (1) ψ ison top of the numerical solution. Parameter values are c = − / A = 1 /
2, and K = 1 / (b) A zoomed-in view around the maximum. The approximations are shown up to andincluding 3rd order. On this scale U (3) ψ is on top of the numerical solution.equation describing the diffusion of a (generalized) density u ( x, t ), being u t − ν u xx = 0, with ν the diffusion coefficient. We are interested in traveling wave solutions for general source6 U ψ ( n ) ( ) - - - Log | Δ U ψ ( n , n - ) ( ) | Figure 2: Peak value of the approximation versus order n for the nonlinear Camassa-HolmDE. The numerically exact peak value (red, dashed line) is also shown. Inset:
A log semi-log plot of the increments | ∆ U ( n,n − ψ (0) | of the approximations, and a linear fit.Parameter values are c = − / A = 1 /
2, and K = 1 / z = x − ct . After scaling the variables (setting ν = 1) we solve the ODE L z U ≡ − U z − c U zz = δ ( z ) , (15)with wavefront boundary conditions U z ( z → −∞ ) = 0 (and U ( z → −∞ ) >
0) and U ( z →∞ ) = 0. The exact solution (in every point including z = 0) is the piecewise analyticexponential tail, B ( z ) = , z < − z/k , z ≥ c ( k ) = 1 /k .Now we propose to include nonlinearity of any kind , as long as the solution satisfies theboundary conditions. For example, including the convective nonlinearity of the equations offluid motion, we arrive at Burgers’ equation, u t + u u x − ν u xx = 0 (17)which has proven useful in a variety of problems in physics and continues to be the subjectof intensive research [13]. Its extensions in the form of Euler and Navier-Stokes equationswith source terms that represent mass, momentum and energy sources, are relevant, e.g.,7n nonlinear acoustics [14]. For traveling wavefronts in the co-moving frame, and with anarbitrary (normalized) source ψ , Burgers’ equation can be rewritten as the nonlinear ODE N z U ≡ − U z + kU U z − kU zz = ψ ( z ) , (18)which is compatible with our boundary conditions provided 0 < k < / B ( z ) now does not satisfy (1) since the nonlinear term is notcompensated. Notwithstanding this fact, we can still apply the iteration procedure becausecondition (1) turns out not to be necessary. Indeed, (4) and (5) only require (3). Therefore,we can still use B ( z ) as BLUES function (although it does not satisfy the strict definitiongiven in [7]). We investigate the convergence of the analytic approximations that result fromiterating the residual operator, which now acts as R z U = − kU U z . Obviously, R z B (cid:54) = 0.Consequently, in the limit that ψ ( z ) approaches a Dirac delta source, the solution B ∗ φ converges to a function different from, but isomorphic to B ( z ), B ∗ φ ∼ B + B ∗ R z ( B + B ∗ R z ( B + B ∗ R z ( B + ... ))) , (19)which is straightforward to calculate.For an arbitary source ψ , the zeroth-order approximation is B ∗ ψ . Choosing the expo-nential corner source (12), we obtain( B ∗ ψ )( z ) = 12 − KK + k e z/K , z < KK − k e − z/K − k K − k e − z/k , z ≥ z = 0 andsolutions are shown up till 3rd order. The parameters are k = K = 1 /
5. Note that in Fig.3athe simple convolution (zeroth-order) asymptotically approaches unity for z → −∞ , as doesthe BLUES function. This differs from the exact asymptotic value, found by integratingthe DE, U ψ ( −∞ ) = (1 − √ − k ) /k (which equals 1.1270... for k = 1 / U (1) ψ ( −∞ ) = 1 + k/
2, is already a significant improvement.8n order to assess the convergence we inspect the values near the shoulder of the wavefront(at z = 0) in Fig.3b. The results are shown in Fig. 4. The convergence to the numericallyexact value is fast and monotonic. The increments | ∆ U ( n,n − ψ (0) | ≡ | U ( n ) ψ (0) − U ( n − ψ (0) | decay to zero exponentially rapidly, as the semi-log plot in the inset of Fig. 4 shows. Ineach iteration almost an order of magnitude is gained in precision. Apparently, whether ornot condition (1) is satisfied does not affect the fast convergence of the method.We verified that the iteration procedure converges exponentially rapidly to the correctanswer for values of K in the range 10 − ≤ K < . Furthermore, for other types ofnonlinearity which we tried, and for different choices of sources, the method also works(under the same boundary conditions).In conclusion, an analytic iteration procedure based on the recently proposed BLUESfunction method holds promise to provide useful explicit solutions to a presumably largevariety of nonlinear DEs of general interest for physicists. The class of equations which canbe discussed with the method consists at this stage of ordinary DEs that are deterministicand integrable. Furthermore, the boundary conditions must require the solution, or itsderivative, to decay to zero asymptotically (for large | z | ). This conclusion has been reachedalong three lines of progress.Firstly, for a nonlinear ODE with a delta source that possesses an exact solution, whichis at the same time a Green function for a related linear ODE, an iteration has been set upwhich converges exponentially rapidly to the correct solution for an arbitrary source. Thishas been demonstrated for the Camassa-Holm equation for traveling wave pulses (solitons)on shallow water.Secondly, and of general interest, is our observation that a Green function of a linearproblem can be used as BLUES function in an analytic approach for solving a nonlinearODE. Nonlinear terms can be added rather freely to the ODE as long as the solutionrespects the boundary conditions. For the nonlinear ODE with an arbitrary source anexponentially rapidly convergent sequence of analytic solutions can be calculated. Thezeroth-order approximation is a simple convolution product and is already useful in practice.The first-order approximation is more interesting because it contains the effects induced bythe nonlinear terms in the ODE. The first-order approximation can be calculated withmoderate effort and provides already an accurate solution as compared with the numericallyexact one. This has been demonstrated starting from the linear diffusion equation applied9 ( z ) U num U ψ ( ) U ψ ( ) - - (a) U num U ψ ( ) U ψ ( ) U ψ ( ) U ψ ( ) - - (b) Figure 3: (a)
Travelling wavefront solution to the nonlinear Burgers DE (18) with anexponential corner source (12). The numerical solution U num (red, full line), thezeroth-order U (0) ψ (black, dashed line) and the first-order approximation U (1) ψ (black, widerspaced dashed line) are compared. The BLUES-function (gray) is also shown. (b) Azoomed-in view around the shoulder of the wavefront. The approximations are shown upto and including 3rd order. On this scale U (3) ψ is practically on top of the numericalsolution. Parameter values are k = K = 1 / U ψ ( n ) ( ) - - - Log | Δ U ψ ( n , n - ) ( ) | Figure 4: Wavefront values U ( n ) ψ (0) versus order n for the nonlinear Burgers DE. Thenumerically exact value (red, dashed line) is also shown. Inset:
A log semi-log plot ofthe increments | ∆ U ( n,n − ψ (0) | of the approximations, and a linear fit. Parameter values are k = 1 / K = 1 / [1] See for example, D.G. Duffy, “Green’s function with applications”, Chapman and Hall/CRC,New York (2015)[2] See, for example, A.R McGurn, “Nonlinear differential equations in physics and their soli-ton solutions”, in “Nonlinear Optics of Photonic Crystals and Meta-Materials”, Morgan andClaypool Publishers, California, USA (2015)[3] N.K. Vitanov, Z.I. Dimitrova and K.N. Vitanov, Modified method of simplest equation forobtaining exact analytical solutions of nonlinear partial differential equations: further devel-opment of the methodology with applications , Applied Mathematics and Computation ,363 (2015)[4] J.P. Keener and D.W. McLaughlin,
Solitons under perturbations , Physical Review A , 777(1977)[5] A.H. Bhrawy, A.A. Alshaery, E.M. Hilal, W.N. Manrakhan, M. Savescu and A. Biswas, Dis-persive optical solitons with Schr¨odinger-Hirota equation , Journal of Nonlinear Optical Physicsand Materials , 1450014 (2014)[6] S.J. Liao, “Homotopy Analysis Method in Nonlinear Differential Equations”, 1st ed. (Springer-Verlag Berlin Heidelberg, 2012)[7] J.O. Indekeu and K.K. M¨uller-Nedebock, BLUES function method in computational physics ,Journal of Physics A Mathematical and Theoretical , 165201 (2018)[8] J.O. Indekeu and R. Smets, Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations , Journal of Physics A Mathematical and Theoretical , 315601 (2017)
9] J.D. Murray, “Mathematical Biology: I. an Introduction”, 3rd edn (Berlin: Springer) (2002)[10] M. Chen,
Periodic and almost periodic solutions for the damped Korteweg-de Vries equation ,Mathematical Methods in the Applied Sciences , 7554 (2018)[11] R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons , Phys-ical Review Letters , 1661 (1993)[12] Our present formulation corrects an error in Eqs. 35 and 36 in the original formulation pre-sented in [7][13] M.P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukandan and V.S. Aswin, A systematic literaturereview of Burgers’ equation with recent advances , Pramana - Journal of Physics (2018)[14] K. Maeda and T. Colonius,
A source term approach for generation of one-way acoustic wavesin the Euler and Navier-Stokes equations , Wave Motion , 36 (2017)[15] T.J. Kippenberg, A.L. Gaeta, M. Lipson and M.L. Gorodetsky, Dissipative Kerr solitons inoptical microresonators , Science , 6402 (2018)[16] Y. Yang, Z. Yan and D. Mihalache,
Controlling temporal solitary waves in the generalizedinhomogeneous coupled nonlinear Schr¨odinger equations with varying source terms , Journal ofMathematical Physics , 053508 (2015)[17] M. Arshad, A. R. Seadawy and D. Lu, Modulation stability and dispersive optical solitonsolutions of higher-order nonlinear Schr¨odinger equation and its applications in mono-modeoptical fibers , Superlattices and Microstructures , 419 (2018).[18] X. Ma, O.A. Egorov and S. Schumacher,
Creation and manipulation of stable dark solitonsand vortices in microcavity polariton condensates , Physical Review Letters , 157401 (2017)[19] Z. Yan, X.-F. Zhang and W.M. Liu,
Nonautonomous matter waves in a waveguide , PhysicalReview A , 023627 (2011)[20] M.S. Jutley and V.S. Ajaev, Stability and nonlinear evolution of electrolyte films on substrateswith spatially periodic charge density , Physical Review E , 032803 (2018)[21] M.V. Berry, Minimal analytical model for undular tidal bore profile; quantum and Hawkingeffect analogies , New Journal of Physics , 053066 (2018)[22] J.A. Gonz´alez, A. Bellor´ın, M.A. Garc´ıa-Nustes, L.E. Guerrero, S. Jim´enez and L. V´azquez, Arbitrarily large numbers of kink internal modes in inhomogeneous sine-Gordon equations ,Physics Letters A , 1995 (2017)[23] Z. Gul, A. Ali and A. Ullah,
Localized modes in parametrically driven long Josephson junctions ith a double-well potential , Journal of Physics A: Mathematical and Theoretical , 015203(2019)[24] K.A. Takeuchi, An appetizer to modern developments on the Kardar-Parisi-Zhang universalityclass , Physica A , 77 (2018), 77 (2018)