AA tale of two Nekrasov’s integral equations
Nikolay KuznetsovOn the occasion of centenary of Nekrasov’s equation for deep water
Laboratory for Mathematical Modelling of Wave Phenomena,Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,V.O., Bol’shoy pr. 61, St. Petersburg 199178, Russian FederationE-mail: [email protected]
Abstract
Just 100 years ago, Nekrasov published the widely cited paper [42], in which hederived the first of his two integral equations describing steady periodic waves on thefree surface of water. We examine how Nekrasov arrived at these equations and hisapproach to investigating their solutions. In this connection, Nekrasov’s life after 1917is briefly outlined, in particular, how he became a victim of Stalin’s terror. Furtherresults concerning Nekrasov’s equations and related topicz are surveyed.
Theory of nonlinear water waves has its origin in the work of George Gabriel Stokes (1819–1903) dating back to the 1847 paper [59] (see also [60], pp. 197–219). His research in thisfield is well documented; see, for example, the detailed survey [14], where further referencesare given. The next major achievement in developing this theory was due to AleksandrIvanovich Nekrasov (1883–1957), who derived two integral equations now named after him;one describes Stokes waves on deep water, whereas the other one deals with waves on waterof finite depth. He investigated his equations by virtue of mathematical techniques availablein the 1920s, but a comprehensive theory of these and other integral equations arising in thewater-wave theory was developed much later after the invention of abstract global bifurcationtheory in the 1970s.The pioneering paper [42], published by Nekrasov in the Bulletin of the now non-existentIvanovo-Voznesensk Polytechnic Institute, is widely cited, but without any detail about itscontent. The reason for this was explained by John V. Wehausen (1913–2005) in his re-view of the book [46]; see MathSciNet, MR0060363. (This memoir summarising Nekrasov’sresults about both integral equations was published by the Soviet Academy of Sciences in1951.) Wehausen writes: “The author’s work [. . . ] appeared in publications with very smalldistribution outside the USSR [. . . ] and consequently has not been well known.” Thesepublications had very small distribution inside the USSR as well because the times whenNekrasov carried out this research and sent it to print were extremely hectic like thosedescribed by Charles Dickens in the opening sentence of his
A Tale of Two Cities .1 a r X i v : . [ m a t h - ph ] S e p t was the best of times, it was the worst of times, it was the age of wisdom,it was the age of foolishness, it was the epoch of belief, it was the epoch ofincredulity, it was the season of Light, it was the season of Darkness, it was thespring of hope, it was the winter of despair, we had everything before us, we hadnothing before us, we were all going direct to Heaven, we were all going directthe other way—in short, the period was so far like the present period, that someof its noisiest authorities insisted on its being received, for good or for evil, inthe superlative degree of comparison only.The start of 1917 was “the season of Light” for the privat-dozent Nekrasov who hadrecently turned 33. Indeed, on 18 February 1917, just a few days before the beginning ofthe February Revolution in Russia, he presented his talk “On waves of permanent type onthe surface of a heavy fluid” to the Moscow Physical Society. The corresponding paper [43]was published five (!) years later; however, its value should not be underestimated becausethe first step to Nekrasov’s integral equation for waves on deep water was made in it. Theequation itself had appeared in [42] in 1921, but its first widely available presentation waspublished by Wehausen only in 1960 (see the extensive survey [67], pp. 728–730).Before that, J. J. Stoker (1905–1992) included references to [42] and [46] into his classicaltreatise [58] on water waves, just to mention that the problem “of two-dimensional periodicprogressing waves of finite amplitude in water of infinite depth [. . . ] was first solved byNekrassov”; see p. 522 of his book. Shortly after that, the 4th edition of widely cited textbook[39] by L. M. Milne-Thompson (1891–1974) was published; a detailed account of Nekrasov’sequation for waves on deep water is given in it (see sect. 14 · The Great Terror: Stalin’sPurge of the Thirties .Finally, some results from the global theory of periodic water waves are surveyed as well asthe proof of Stokes’ conjectures about the wave of extreme form. (The latter has a stagnationpoint at each crest, where smooth parts of the wave profile form the angle π/ A biography of Nekrasov by J. J. O’Connor and E. F. Robertson is available online athttp://mathshistory.st-andrews.ac.uk/Biographies/Nekrasov.htmlHowever, it provides rather scanty information about him and even omits the fact of hisimprisonment in 1938–1943. This may be explained by the fact that the article is based onthe note [55] published in 1960; at that time the Soviet censorship was active, regardless ofthe fact that this was the period of the so-called Khrushchev Thaw. Furthermore, Nekrasovdid not teach and undertake his “research in Moscow for the rest of his life” after 1913 as iswritten in O’Connor and Robertson’s article. To fill in these omissions, the most essentialpoints of his activity after 1917 are outlined below.
In the book [66] published in 2001 (it covers all aspects of Nekrasov’s biography), the authorsdevoted 25 pages to sketching out major points of his life; the most interesting of themconcern the Soviet period and are outlined below.The tsarist regime was overturned in March 1917, and self-rule was granted to univer-sities. After that reelection of staff began because many progressive professors had beenfired by the old government. Nekrasov was reelected as a docent by the Council of MoscowUniversity and promoted to professorship in 1918. However, in the fall of that year he movedto Ivanovo-Voznesensk joining the newly organized Polytechnic Institute, and so he was onleave from Moscow University for four years. The reason to leave was scarce food rationsin overcrowded Moscow during the Civil War, whereas the situation was much better inIvanovo-Voznesensk due to food supply from regions down the Volga river. The staff of thefirst Soviet polytechnic (Lenin signed a decree establishing it on 6 August 1918) includedmany professors evacuated from Riga in 1915 (a consequence of World War I) and severalprofessors from Moscow University; Luzin, Nekrasov and Khinchin were the most notable.3igure 1: Professors and graduate students of the Faculty of Civil Engineering, Ivanovo-Voznesensk Polytechnic Institute. A. I. Nekrasov in the middle of front row.Along with professorship in Theoretical Mechanics, Nekrasov was Dean of the Faculty ofCivil Engineering for four years and headed the whole institute of six faculties for 13 months.He returned to Moscow University in 1922, holding simultaneously a professorship atthe Moscow Technical High School; until 1929 he combined teeching and administrativeduties at Narkompros (the Soviet ministry of education), but since that year, the CentralAero-hydrodynamic Institute (TsAGI founded in 1918) became his second affiliation. ThereNekrasov was drawn into mathematical problems related to aircraft design; moreover, he wasa deputy of Academician S. A. Chaplygin (1869–1942), who headed research at the institutein the 1930s. Several times Nekrasov travelled abroad, in particular, he was the head of theSoviet delegation at the 14th Air Show in Paris in 1934. Next year he spent six monthsin the USA together with A. N. Tupolev (1888–1972)—a prominent aircraft designer—andother researches from TsAGI; their aim was to overview aircraft production and commercialoperation of airlines in the USA. This was a period of intensive collaboration between theSoviets and Americans; on the one hand, the Soviet industry was interested in Americantechnologies, whereas selling production to the Soviet Union helped American corporationsto recover after the Great Depression.Tupolev’s design office separated from TsAGI in 1936, but ceased its existence next yearwhen Tupolev and many of his colleagues were arrested during the Great Terror organisedby the People’s Commissariat for Internal Affairs (the interior ministry of the Soviet Union)known as NKVD. Nekrasov’s turn came in 1938, when he was charged with “participationin anti-soviet, spy organisation in TsAGI”. He spent next five years behind the bars inTupolev’s “sharashka” (this is the argot term for “Special Design Bureau at the NKVD”).This “bureau” was one of hundreds existed within the special department created in 1938by Beria (he headed NKVD at that time) and disbanded in 1953, shortly after Stalin’sdeath. (For general information about sharashkas see, for example, the Wikipedia article4igure 2: A. I. Nekrasov in the 1950s.available at https://en.wikipedia.org/wiki/Sharashka. Tupolev’s sharashka is described indetail by L. L. Kerber [26], who was an aviation specialist and had the long professional andpersonal relationship with Tupolev; see also [23] for a review of [26], where some omissionsare mentioned.)Nekrasov was released in 1943 (exonerated only in 1955, two years before his death), andagain he returned to Moscow University, holding simultaneously a position at the Institutefor Mechanics of the Soviet Academy of Sciences (he was its Corresponding Member since1932 and promoted to a Full Academician in 1946). However, he continued to head thetheoretical department at Tupolev’s design office until 1949. (A characteristic feature of thetime: it was located in the same sharashka’s building, but now without armed escorts.) Afterthat, Nekrasov returned to his studies of water waves and compiled the monograph [47]. Hislast research note was published in 1953 and he stopped teaching the same year; the reasonwas asthma contracted during imprisonment.5 few lines about Nekrasov’s distinctions. In 1922 he was the first recipient of Joukowskiprize for the paper [42], and received prize from the Narkompros for the booklet
Diffusion ofa vortex ten year later. For the monograph [46] he was awarded Stalin prize in 1952. Nextyear, Nekrasov was decorated with the Order of Lenin on the occasion of his 70th birthday.
Prehistory.
In 1906, just before his graduation from Moscow University, Nekrasov hadcompleted a study of the motion of Jupiter’s satellites for which he was awarded a goldmedal, but then he chose the water-wave theory for his further studies. Presumably, thiswas due to the influence of N. E. Joukowski (1847–1921), whose own research was on thistheory at that time. In the paper [24] of 1907, he obtained an important result on the waveresistance of a ship; it occurred to be similar to that published by J. H. Michell (1863–1940)in 1898, but Joukowski found it independently using a different method.In the first paper dealing with water waves (see [40], also the first paper in the
CollectedPapers [47]), Nekrasov applied the method of power expansions, which he knew well fromhis work on Jupiter’s satellites and the knowledge of which he extended while translatinginto Russian the 2nd volume of the Goursat
Cours d’Analyse Math´ematique [18]. In fact,Nekrasov follows the Stokes paper [59], where expansions in powers of the wave amplitudewere used and the latter was implicitly supposed to be a small parameter. On the contrary,such a parameter is introduced by Nekrasov explicitly into the nonlinear Cauchy–Poissonproblem as it is referred to now; namely, the motion is generated by an impulsive pressureapplied to the surface of initially resting fluid. Assuming the pressure impulse given by aconverging series in powers of the parameter with the zero initial coefficient, Nekrasov derivesthe initial-boundary value problems for four initial harmonic functions in the expansion ofthe velocity potential and finds particular solutions for two of these problems.In his second paper on water waves (we recall that [43] was published with a long delay),Nekrasov radically changes the topic of research turning to the two-dimensional problem ofsteady gravity waves on deep water. This problem was 70 years old at that time, but hedeveloped a new approach to it. First, he formulated it in terms of the complex velocitypotential w ( z ) = ϕ + i ψ considered as a function of the complex variable z = x + i y in thedomain under the wave profile y = η ( x ). (It should be noticed that it was Michell, who firstused methods of complex analysis in the water-wave theory; see his pioneering paper [38]on the Stokes wave of the extreme form.) Second, Nekrasov proposed a new transformationof the problem which occurred one of the key points on the way to deriving the integralequation.In our presentation of Nekrasov’s results, his terminology and notation varying frompaper to paper is unified and updated; the velocity field is ∇ ϕ ( ∇ denotes the gradient) andso on (see, for example, [63] or [11], ch. 10, for the detailed statement of the problem).Nekrasov assumed that the fluid is infinitely deep andlim y →−∞ ∇ ϕ = ( c, , (1)whereas waves are periodic and symmetric about their troughs and crests and their wave-length is λ . If the origin in the z -plane is at a trough (this is convenient in what follows,but Nekrasov’s choice of the origin was at a crest; however, this is unimportant), then the6ppropriately chosen complex potential w ( z ) maps conformally the vertical semi-strip D z = {− λ/ < x < λ/ , −∞ < y < η ( x ) } with unknown upper boundary (it corresponds to the water domain under a single wave)onto the fixed semi-strip D w = {− cλ/ < ϕ < cλ/ , −∞ < ψ < } on the w -plane. The new transformation proposed in [43], namely, u ( w ) = exp { πw/ (i cλ ) } , (2)maps the latter semi-strip into an auxiliary u -plane and the image of D w is the unit disccut along the nonpositive real axis: D u = {| u | < r / ∈ ( − ,
0] when θ = ± π } ; here u = r e i θ . (3)Then Nekrasov noticed that the mapping D u → D z is defined by the relationd z d u = i λ π f ( u ) u , where f ( u ) = 1 + a u + a u + . . . with real coefficients a k ; (4)moreover, f is analytic in the unit disc. This allowed him to derive a formula for thepotential energy in terms of a k , and the following representation of the free surface profileparametrised by θ ∈ [ − π, π ]: η ( θ ) = λ π (cid:16) a cos θ + a θ + a θ + . . . (cid:17) , (5) x ( θ ) = − λ π (cid:16) θ + a sin θ + a θ + a θ + . . . (cid:17) . (6)Moreover, differentiating the Bernoulli equation with respect to θ , Nekrasov reduced it todd θ (cid:2) f (e i θ ) f (e − i θ ) (cid:3) − = gλπc d η d θ ( θ ) , (7)where g is the acceleration due to gravity. Finally, several examples were given, when theinitial coefficients a k can be found approximately, but they are of little interest.In his third paper [41] (the first one published in the Bulletin of Ivanovo-VoznesenskPolytechnic), Nekrasov extended his analysis to what he called “the limiting form” of peri-odic waves “the possibility of which was first predicted by Stokes” (no reference is given).Indeed, in the 1880 collection of his papers [60], Stokes published the three short appendicesand 12-page supplement to his 1847 paper [59]. In the second appendix, he coined the term“wave of greatest height” to characterize a certain kind of periodic wave. Since the trueheight η max − η min may not be maximised by it, this wave is referred to as the “wave ofextreme form” nowadays, because there is a 2 π/ sic !) was found by Michell”. Again, no reference is given, but it is clear that this concernsthe Michell’s paper [38], in which a procedure for calculating the coefficients in a series7escribing this wave was developed, but no convergence was proved. Then Nekrasov formu-lates the aim of his paper: “to show how the general mode of wave can be obtained”. Forthis purpose he applies a modification of the transformation used in the previous paper forsmooth waves. Instead of (4), the mapping D u → D z is described by the relationd z d u = − i λ π ˆ f ( u ) u √ − u , where ˆ f ( u ) = 1 + ˆ a u + ˆ a u + . . . with real coefficients ˆ a k , (8)in order to catch its singularity. The resulting representation of the free surface profile is asfollows, θ ∈ [ − π, π ]:dˆ η d θ = λ π (cid:112) θ/ (cid:32) sin π − θ ∞ (cid:88) k =1 a k sin π + (6 k − θ (cid:33) , (9)d x d θ = λ π (cid:112) θ/ (cid:32) cos π − θ ∞ (cid:88) k =1 a k cos π + (6 k − θ (cid:33) . (10)Here integration is not as simple as in the case leading to (5) and (6). Similar to (7), acomplicated nonlinear equation is obtained for determining the coefficients ˆ a k ; it equatesthe right-hand side term in (9) (with an extra constant factor) and the derivative of(sin θ/ / (cid:46)(cid:2) ˆ f (e i θ ) ˆ f (e − i θ ) (cid:3) , where ˆ f is given in (8). After tedious calculations Nekrasov obtained only two initial co-efficients, but this was insufficient to achieve Michell’s accuracy. Almost 40 years later,Nekrasov’s method was rediscovered by Yamada [68], whose calculations provided the sameaccuracy as Michell’s. The breakthrough.
In his celebrated paper [42], Nekrasov combined the approach devel-oped in [43] (see (2), (3) and (7) above) and modifications of two earlier results; one obtainedby M. P. Rudzki (1862–1916) (the main field of his research was geophysics; see [22]) andthe other one from the dissertation of N. N. Luzin (1883–1950) (a Soviet Academician notedfor his numerous contributions to mathematics and his famous doctoral students at MoscowUniversity, known as the “Luzitania” group). This allowed Nekrasov to derive his integralequation for waves on deep water described by the unknown slope of the free surface profile.Rudzki proposed a transformation of the Bernoulli equation in 1898 (see his paper [54]and also [67], pp. 727–728, where the essence is explained). In the transformed equation,the sine of the angle between the velocity vector in a two-dimensional flow and the positive x -axis (the function describing this angle is harmonic) is expressed in terms of the conjugateharmonic function. This allowed Rudzki to apply the so-called inverse procedure (see [48]for its description) for obtaining an exact solution for waves over a corrugated bottom; see[67], pp. 737–739, in particular, figure 52 b.Nekrasov mentioned this solution in his paper [42] with a remark that Rudzki failed toconsider the case of a horizontal bottom, whereas Nekrasov’s aim was to investigate waveson deep water. Therefore, he modified Rudzki’s transformation as follows. He introduced R ( θ ) and Φ( θ ) so that − πλ d x d θ ( θ ) = R ( θ ) cos Φ( θ ) , − πλ d η d θ ( θ ) = R ( θ ) sin Φ( θ ) , (11)8here η and x are defined by (5) and (6), respectively. Thus Φ( θ ) is the angle between thetangent to the wave profile η and the positive x -axis, and it is parametrised by θ ∈ [ − π, π ]over a single period between troughs. In new variables, the differentiated Bernoulli equation(7) takes the form d R − d θ = 3 gλ πc sin Φ = d exp {− R } d θ , (12)which, as Nekrasov emphasised in his paper, was a crucial point of his considerations. It isworth mentioning that formula (32.88) in Wehausen’s description of Rudzki’s transformation(see [67], p. 728) is similar to the second equality (12).The last but not least point in deriving an equation for Φ was to deduce another relationinvolving Φ and R simultaneously. It was Luzin—a Nekrasov’s colleague at Ivanovo-Vozne-sensk Polytechnic—who pointed the way to obtaining such a relation (this is acknowledgedin a footnote in [42]). In his outstanding dissertation [35] published in 1915, Luzin proved thefollowing theorem, which involves the singular integral operator C defined almost everywhereon [ − π, π ] by the formula C U ( θ ) = 12 π P V (cid:90) π − π U ( τ ) cot θ − τ τ , where P V stands for the Cauchy principal value . Let U be a function harmonic in the open unit disc D centred at the origin. If U ( θ ) = lim r → U ( r e i θ ) belongs to L ( − π, π ) , then V ( θ ) = C U ( θ ) is also in L ( − π, π ) and provides boundary values of V —the harmonicconjugate to U in D . This theorem provoked numerous generalisations (see, for example, [69] for a description),the first of which was due to Privalov (a member of “Luzitania”), who demonstrated [52]that for all α ∈ (0 ,
1) the operator C : C α [ − π, π ] → C α [ − π, π ] is bounded. Another resultfrom the same area of harmonic analysis as Luzin’s theorem is the so-called Dini’s formula(see, for example, [20], pp. 266–267). Presumably, Luzin realised that a consequence of thisformula would provide a required relation connecting Φ and R and recommended Nekrasovto consider this option. Indeed, the following corollary of Dini’s formula was obtained in [42]. Let U + i V be holomorphic in D , and let V ( θ ) = lim r → V ( r e i θ ) be absolutely continuouson [ − π, π ] and such that V (2 π − θ ) = V ( θ ) for all θ ∈ [ − π, π ] . Then U ( θ ) = 12 π (cid:90) π − π V (cid:48) ( τ ) log (cid:12)(cid:12)(cid:12)(cid:12) sin( θ + τ ) / θ − τ ) / (cid:12)(cid:12)(cid:12)(cid:12) d τ + const . (13)Two formulae similar to (13), but expressing V ( θ ) in terms of U (cid:48) ( θ ) were also given in [42]for symmetric and antisymmetric U ( θ ).Then Nekrasov applied his proposition to the function i log f ( u ); it is holomorphic in D ,has U ( θ ) = − Φ( θ ) and V ( θ ) = log R ( θ ), and the last function satisfies the symmetry condi-tion. It is clear that (13) turns intoΦ( θ ) = 12 π (cid:90) π − π d log R d τ log (cid:12)(cid:12)(cid:12)(cid:12) sin( θ − τ ) / θ + τ ) / (cid:12)(cid:12)(cid:12)(cid:12) d τ (14)9n this case; here the constant vanishes because Φ(0) = 0 for a symmetric wave. Excluding R from this relation by virtue of (12), Nekrasov arrived at his integral equation for waveson deep water which are symmetric about the vertical through a crest (and trough):Φ( θ ) = µ π (cid:90) π − π sin Φ( τ )1 + µ (cid:82) τ sin Φ( ζ ) d ζ log (cid:12)(cid:12)(cid:12)(cid:12) sin( θ + τ ) / θ − τ ) / (cid:12)(cid:12)(cid:12)(cid:12) d τ , θ ∈ [ − π, π ] . (15)Here µ is a non-dimensional parameter which, first, arose as the constant of integration whiledetermining R from (12). Subsequently, Nekrasov found an expression for this parameter interms of characteristics of the wave train; namely: µ = 32 π gcλq . (16)Here g , c and λ were defined above, whereas q is the velocity at a crest assumed to benon-zero.What else can be found in the paper [42]? First, a restriction on µ is obtained. As-suming that | Φ( θ ) | of a solution to (15) is bounded by M , Nekrasov demonstrated that thecorresponding µ must satisfy the inequality µ < (cid:20) πM + sin M M (cid:21) − . (17)Second, it is demonstrated that the equality Φ( θ ) = Φ( π − θ ) cannot hold for all θ ∈ (0 , π/ ,if Φ is a nontrivial solution of (15). This proves rigorously the observation made by Stokesthat his approximate wave profile has sharpened crests and flattened troughs; see the cor-responding figure in [60], p. 212, reproduced in [14], p. 31.Finally, several pages near the end of [42] are concerned with various modifications andsimplifications of equation (15) obtained under different assumptions about the wave type(gentle slope etc.). In the last paragraph, Nekrasov introduced the linearisation of (15):Φ( θ ) = µ π (cid:90) π − π Φ( τ ) log (cid:12)(cid:12)(cid:12)(cid:12) sin( θ + τ ) / θ − τ ) / (cid:12)(cid:12)(cid:12)(cid:12) d τ . θ ∈ [ − π, π ] , (18)Moreover, he claimed that the set of its characteristic values is { , , , . . . } , whereas thecorresponding eigenfunctions are { sin θ, sin 2 θ, sin 3 θ, . . . } . At the end of his paper [42], Nekrasov announced that a method of solution of the integralequation (15) would appear in a separate paper; its manuscript was prepared under thetitle “On steady waves, part 3”, but never published as such. According to the concludingremark in [42], the method is based on the theory developed for a certain class of nonlinearintegral equations; the latter was presented in [44] (the last Nekrasov’s publication in the
Bulletin of Ivanovo-Voznesensk Polytechnic, 1922). The unpublished manuscript was usedwhile preparing [46] for publication almost 30 years later, and in the course of that someimprovements were proposed by Y. I. Sekerzh-Zenkovich (1899–1985) (each acknowledged inthe text). Subsequently, he became the editor of Nekrasov’s
Collected Papers , I, [47], andtranslated [39] and [58] into Russian. 10igure 3: The title page of Nekrasov’s monograph [46].11et us outline key points of Nekrasov’s approach to the existence of nontrivial solutionsas they are presented in [46], mainly in Sections 5 and 6. First, he used a straightforwardcalculation based on Euler’s formula to transform the kernel in equation (15), namely:log (cid:12)(cid:12)(cid:12)(cid:12) sin( θ + τ ) / θ − τ ) / (cid:12)(cid:12)(cid:12)(cid:12) = 2 ∞ (cid:88) k =1 sin kθ sin kτk . This sum divided by 3 π was denoted by K ( θ, τ ) and equation (15) written in the form:Φ( θ ) = µ (cid:90) π − π sin Φ( τ )1 + µ (cid:82) τ sin Φ( ζ ) d ζ K ( θ, τ ) d τ , θ ∈ [ − π, π ] . (19)Moreover, Nekrasov noticed that the assertion about the set of solutions of the linearisedequation (18) immediately follows from the expression for K ( θ, τ ).Since the right-hand side in inequality (17) is equal to 3 when M = 0, he concludedthat there are no non-trivial solutions when µ ≤
3. Therefore, it is reasonable to introduce µ (cid:48) = 3 − µ > θ, µ (cid:48) ) = µ (cid:48) Φ ( θ ) + µ (cid:48) Φ ( θ ) + µ (cid:48) Φ ( θ ) + µ (cid:48) Φ ( θ ) + . . . , thus reducing (19) to an infinite system of equations for Φ k . Nekrasov also emphasised thatthe series would represent a solution only if its convergence is proved. In modern terms,his aim was to construct an initial part of the solution branch bifurcating from the firstcharacteristic value of the linearised operator.Another Nekrasov’s aim was to obtain Φ , Φ and Φ explicitly, and so he noticed thatsin Φ = µ (cid:48) Φ + µ (cid:48) Φ + µ (cid:48) (Φ − Φ /
6) + µ (cid:48) (Φ − Φ Φ /
2) + . . . and(3 + µ (cid:48) ) sin Φ( τ )1 + (3 + µ (cid:48) ) (cid:82) τ sin Φ( ζ ) d ζ = µ (cid:48) + µ (cid:48) (cid:104) + Φ − (cid:90) τ Φ( ζ ) d ζ (cid:105) + . . . . The last expansion was written down by Nekrasov with four terms as well, but this took threeextra lines plus three lines of additional notation. Substituting these expansions into (19)and equating the coefficients at every µ (cid:48) k ( k = 1 , , . . . ), he would obtain a recurrent sequenceof linear integral equations for Φ k with free terms depending on solutions of previous ones.The first two equations are as follows:Φ ( θ ) = 3 (cid:90) π − π Φ ( τ ) K ( θ, τ ) d τ , θ ∈ [ − π, π ] , (20)Φ ( θ ) = 3 (cid:90) π − π Φ ( τ ) K ( θ, τ ) d τ (21)+ (cid:90) π − π Φ ( τ ) (cid:104) − (cid:90) τ Φ ( ζ ) d ζ (cid:105) K ( θ, τ ) d τ . For obtaining Φ , Φ and Φ two more equations are needed and the corresponding formulaeare given in [46]; they look awful, but are still tractable.It is clear that a non-trivial solution of (20) is Φ ( θ ) = C sin θ with C (cid:54) = 0 to befound from the orthogonality of this function and the free term in (21)—the condition of12olubility of this equation. The corresponding value is C = 1 /
9, and so, after an obvioussimplification, (21) takes the form:Φ ( θ ) = 3 (cid:90) π − π Φ ( τ ) K ( θ, τ ) d τ + 108 − sin 2 θ , θ ∈ [ − π, π ] . (22)Hence Φ ( θ ) = C sin θ + 54 − sin 2 θ , and the same procedure as above yields C = − / ( θ ), Nekrasov concluded Section 5 with the formulaΦ( θ, µ (cid:48) ) = (cid:16) µ (cid:48) − µ (cid:48) + 11513122 µ (cid:48) + . . . (cid:17) sin θ + (cid:16) µ (cid:48) − µ (cid:48) + . . . (cid:17) sin 2 θ + (cid:16) µ (cid:48) + . . . (cid:17) sin 3 θ + . . . , (23)accompanied by the remark that it remains to snow that this series is convergent. On thebasis of this formula, Nekrasov found that the wave height is as follows: η max − η min = η (0) − η ( π ) = λπ (cid:20) µ (cid:48) − µ (cid:48) + 716561 µ (cid:48) + . . . (cid:21) . At the beginning of the next section dealing with the question of convergence, Nekrasovpointed out that his method of solution of nonlinear integral equations developed in [44] (itis reproduced in [46], Sections 13 and 14) is not applicable directly to (19). The reason is thepresence of (cid:82) τ sin Φ( ζ ) d ζ in the integrand and to overcome this difficulty “an artificial trickwas used in the original manuscript”. It is not clear how the trick worked, but Nekrasovpreferred to follow the method suggested by Sekerzh-Zenkovich “as more natural”.Its essence is to replace (19) by a system of two integral equations. PuttingΨ( θ ) = (cid:104) µ (cid:90) θ sin Φ( τ ) d τ (cid:105) − , one obtains Ψ (cid:48) ( θ ) = − Ψ ( θ ) sin Φ( θ ) , whereas (19) takes the form:Φ( θ ) = µ (cid:90) π − π Ψ( τ ) sin Φ( τ ) K ( θ, τ ) d τ , θ ∈ [ − π, π ] . (24)Since Ψ(0) = 1, the last differential equation yieldsΨ( θ ) = 1 − µ (cid:90) θ Ψ ( τ ) sin Φ( τ ) d τ , θ ∈ [ − π, π ] , (25)which together with (24) constitutes the required system for Φ and Ψ. Then, as in his paper[42], Nekrasov introduced the parameter µ (cid:48) , and a solution of this system was sought in theform of the following series:Φ( θ, µ (cid:48) ) = ∞ (cid:88) k =1 µ (cid:48) k Φ k ( θ ) , Ψ( θ, µ (cid:48) ) = 1 + ∞ (cid:88) k =1 µ (cid:48) k Ψ k ( θ ) . The subsequent Nekrasov’s considerations are rather vague and can hardly be considered asa rigorous proof. This concerns equations, which form the infinite system for { Φ k } and { Ψ k } ,13he question of solubility of this system and, especially, Nekrasov’s treatment of the methodof majorising functions that serves for proving the convergence of these series. Therefore,Sekerzh-Zenkovich, the editor of Nekrasov’s Collected Papers, I (see [47], where the article[46] occupies about one fifth of the content) added five pages of comments aimed at clarifyingeach of these points.Fortunately, a clear presentation of this approach is available in English; see [64], Sec-tion 37, where the following generalisation of equation (19)Φ( θ ) = µ (cid:90) π − π sin Φ( τ ) + P ( τ ) cos Φ( τ )1 + µ (cid:82) τ (cid:2) sin Φ( ζ ) + P ( ζ ) cos Φ( ζ ) (cid:3) d ζ K ( θ, τ ) d τ , θ ∈ [ − π, π ] , is investigated. This equation proposed by Sekerzh-Zenkovich [56] describes waves generatedby a small-amplitude periodic pressure applied to the free surface of an infinitely deep flow( P is related to pressure’s x -derivative). It is worth noticing that Φ ≡ (cid:104) µ (cid:90) θ sin Φ( τ ) d τ (cid:105) − = ∞ (cid:88) k =0 (cid:104) − µ (cid:90) θ sin Φ( τ ) d τ (cid:105) k . Further studies of equation (19) are outlined in Section 3, where methods of nonlinearfunctional analysis are intensely applied; these tools were actively developed since the 1950s.
There was a break in Nekrasov’s studies of water waves owing to teaching overload, preparingJoukowski’s papers for publication after his death in 1921 and administrative duties. Thislasted until 1927, when Nekrasov presented his second integral equation at the All-RussianCongress of Mathematicians; see the enlarged abstract [45] reproduced (with author’s per-mission) in the book [57] published in 1936.Presentation in [45] follows that in the paper [42], dealing with waves on deep water,but with appropriate amendments. Namely, the stream function ψ (the imaginary part ofthe complex velocity potential w ) is chosen so that it vanishes on the free surface y = η ( x )(however, the location of the origin is not specified), whereas the domain D z = {− λ/ < x < λ/ , − h < y < η ( x ) } , corresponding to a single wave, is mapped conformally on the annular one D u = { r < | u | < r / ∈ ( − r ,
0] when θ = ± π } , u = r e i θ , with log r = − πh. (26)Furthermore, on the basis of a formula analogous to (4) Nekrasov concluded thatd w d z = − c e Ω( u ) , where Ω( u ) = a + ∞ (cid:88) k =1 a k (cid:34)(cid:18) ur (cid:19) k + (cid:16) r u (cid:17) k (cid:35) (27)has real coefficients in view of the bottom boundary condition. (It should be mentioned thatthe meaning of c , referred to as the velocity of wave at the beginning of [45], was cleared up14nly in [46], where, at the end of Section 8, it was demonstrated that c is the mean velocityof flow at the bottom.) For Φ( θ ) and Ψ( θ )—imaginary and real part of Ω(e i θ )—Nekrasovobtained formulae similar to (11)d x d θ ( θ ) = − λ π e Ψ( θ ) cos Φ( θ ) , d η d θ ( θ ) = − λ π e Ψ( θ ) sin Φ( θ ) , which imply that Φ( θ ) is the angle between the tangent to the wave profile η and the positive x -axis. Moreover, since Ψ( θ ) = a + ∞ (cid:88) k =1 a k (cid:0) r − k + r k (cid:1) cos kθ ,a and a k (cid:0) r − k + r k (cid:1) are the cosine Fourier coefficients of Ψ, and soΩ( u ) = 1 π (cid:90) π − π Ψ( θ ) (cid:32)
12 + ∞ (cid:88) k =1 (cid:20) u k cos kθ r k + r k cos kθu k (1 + r k ) (cid:21)(cid:33) d θ . (28)Then Nekrasov noticed that ∞ (cid:88) k =1 r k cos kθu k (1 + r k ) = u u ∞ (cid:88) k =1 ( − k − log (cid:20) − r k u cos θ + r k u (cid:21) and a similar formula holds for the first sum in (28). The sum of logarithms is equal to thelogarithm of an infinite product which is naturally representable in terms of the Weierstrasssigma functions, whose periods ω and ω (cid:48) are such that ω (cid:48) / i ω = 4 h/λ . These considerationsreduced (28) to the following equalityΦ( θ ) = − π (cid:90) π − π Ψ (cid:48) ( τ ) log (cid:12)(cid:12)(cid:12)(cid:12) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) (cid:12)(cid:12)(cid:12)(cid:12) d τ , θ ∈ [ − π, π ] , (29)about which is said that it is a result of rather tedious calculations (the most part of themis omitted in [46] as well). It should be noted that (29) is analogous to (14) obtained in thecase of deep water.On the other hand, Bernoulli’s equation differentiated with respect to θ yieldsde − d θ = 3 gλ πc sin Φ , and so Ψ (cid:48) ( θ ) = − µ θ )1 + µ (cid:82) θ sin Φ( τ ) d τ , (30)where θ ∈ [ − π, π ] and µ is the same as for deep water; see (16).Combining (29) and (30), Nekrasov arrived at his integral equation for waves on waterof finite depthΦ( θ ) = µ π (cid:90) π − π sin Φ( τ )1 + µ (cid:82) τ sin Φ( ζ ) d ζ log (cid:12)(cid:12)(cid:12)(cid:12) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) (cid:12)(cid:12)(cid:12)(cid:12) d τ , (31)where θ ∈ [ − π, π ]. This equation differs from (15) obtained for deep water by the expressionunder the logarithm sign. 15t the end of the note [45], Nekrasov pointed out thatlog (cid:12)(cid:12)(cid:12)(cid:12) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) σ ( ω ( θ + τ ) /π ) σ ( ω ( θ − τ ) /π ) (cid:12)(cid:12)(cid:12)(cid:12) = 2 ∞ (cid:88) k =1 − r k r k sin kθ sin kτk , which was demonstrated in [46] with the help of some formulae from [61] (this book onelliptic functions was published in 1895, presenting the theory in an updated for that timeform). Moreover, the equality(1 − r k ) / (1 + r k ) = tanh(2 πkh/λ )expresses the sum in terms of geometric characteristics of the wave.Denoting by K ( θ, τ ) the last sum divided by 3 π , we see that equation (31) takes exactlythe same form as (19). This similarity was emphasised by Nekrasov and for this reason herestricted himself to calculating just two initial terms in the expansion analogous to (23),which expresses Φ—a solution of (31)—as a series in powers of a small parameter µ (cid:48) . For h < ∞ , he considered µ (cid:48) = µ − µ , where µ is the first characteristic value of K ( θ, τ ); itcoincides with the kernel of the integral operator obtained by linearisation of the nonlinearone in equation (31). The set of these characteristic values, namely { k coth(2 πkh/λ ) } ∞ k =1 ,was announced already in [45].In the monograph [46], Nekrasov concludes his considerations of equation (31) with thebrief Section 11, in which an important property of this equation is formulated. Namely, sincethe kernel is such that K ( θ, π − τ ) = − K ( θ, τ ), a solution satisfies the symmetry equalityΦ(2 π − θ ) = − Φ( θ ), and so it is sufficient to consider an equivalent equation on the interval[0 , π ]. At the beginning of Section 4, this property was mentioned for a solution of equation(19) as well. However, the advantage of reducing the equation to [0 , π ] demonstrated bysubsequent studies (see below) was not used in [46].Here ends the tale about Nekrasov and his work on water waves, but this is not the endof tale about Nekrasov’s integral equations. Along with Nekrasov’s equations, which are the oldest examples of nonlinear integral equa-tions describing water waves, there are many others and some of them will be mentionedbelow. Presumably, the first survey on this topic was published in 1964; see [21].
Existence theorems for equation (19).
It was Mark Alexandrovich Krasnosel’skii (1920–1997) (a pioneer of nonlinear functional analysis in the Soviet Union), to whom we owe thefirst consideration of equation (19) as an operator equation in a Banach space. His briefnote [29] on this topic was published in 1956, just five years after [46]. A general approachto bifurcation points of nonlinear operator equations can be found in his monograph [30];see, in particular, the following assertion applicable to (19) and (31).
Let A µ ( µ > be a family of operators defined in a neighbourhood of —the zeroelement of a Banach space X ; each operator is assumed to be completely continuous and uch that A µ = . Let also the Frechet derivative of A µ be µB , where B is linear (ofcourse, completely continuous) not depending on µ . If µ is an odd-multiple characteristicvalue of B , then the equation Φ = A µ Φ has a continuous branch of nontrivial solutions ( µ, Φ µ ) in a neighbourhood of ( µ , ) and (cid:107) Φ µ (cid:107) X → as µ → µ . In his note [29], Krasnosel’skii demonstrated the following about Nekrasov’s operator A µ defined by the right-hand side in either (19) or (31). In a neighbourhood of ∈ C [ − π, π ],it is completely continuous and its Frechet derivative is µB , where( Bφ )( θ ) = (cid:90) π − π φ ( τ ) K ( θ, τ ) d τ , θ ∈ [ − π, π ] . (32)Since all characteristic values of B , namely, { k } ∞ k =1 for infinite depth ( { k coth(2 πkh/λ ) } ∞ k =1 for finite depth) are odd-multiple, each of them is a bifurcation point of A µ ; that is, in aneighbourhood of this value equation (19) ((31), respectively) has a continuous branch ofsolutions Φ µ ∈ C [ − π, π ] satisfying the equation and such that (cid:107) Φ µ (cid:107) C → µ tends to thecorresponding characteristic value.Thus, the question of local solution branches was established, but the existence of globalones was proved for Nekrasov’s equation more than 20 years later. However, the first globalresult for periodic waves was proved by Krasovskii [31] in 1961. He reduced the Levi-Civitaformulation of the water-wave problem to a nonlinear operator equation and considered iton the cone of nonnegative functions in a Banach space. This allowed him to apply Kras-nosel’ski’s theorem about positive operators with monotonic minorants (see [30], Chapter 5,Section 2.6), thus demonstrating the existence of a wave branch such that each value in(0 , π/
6) serves as max Φ for some wave profile belonging to the branch.Further story is about the existence of a global branch of solutions to Nekrasov’s equation.The abstract bifurcation theory in the form developed by Rabinowitz [53] and Dancer [15]in the 1970s provided tools for proving this result. A self-contained analysis by Toland (see[63], in particular Sections 8 and 9) presents the reasoning of Keady and Norbury [25], whoobtained a continuum of solutions, that is, a maximal closed connected set of them.Their result is formulated in terms of a cone in a real Banach space. Since the originis chosen at a trough, it is reasonable to use the subspace of C [ − π, π ] that consists of oddfunctions vanishing at zero and π . The closed convex cone, say K , in this subspace consistsof functions satisfying: (i) f ( t ) ≥ t ∈ [0 , π ]; (ii) f ( t ) / sin( t/
2) is nonincreasing on [0 , π ];(iii) f ( t ) ≤ f ( s ) for all t ∈ [ π/ , π ] and s ∈ [ π − t, t ]. The crucial point is that the operator B maps the cone of functions nonnegative on [0 , π ] into itself as well as K to K . Now we arein a position to formulate the following theorem. An unbounded continuum C of solutions of equation (15) exists in [0 , ∞ ) × K and ( µ, ) belongs to C if and only if µ = 3 . Moreover, for ( µ, Φ) ∈ C the following properties hold: • if Φ does not vanish identically, then µ > • < Φ < π/ and Φ( θ ) /θ is nonincreasing on (0 , π ); • Φ (cid:48) ( θ ) ≤ for θ ∈ [ π/ , π ] . An interesting open question is whether there are solutions of (15) which do not belong to C , but lie in a wider cone, for example, defined by conditions (i) and (ii). From the resultsof Keady and Norbury it is not clear whether C is a curve as numerical computations of17raig and Nicholls demonstrate (see Figure 1 in their article [13]); an analytical proof is stillabsent.There are infinitely many continua of solutions of equation (15). Indeed, let n > { ( nµ, Φ( nθ )) : ( µ, Φ( θ )) ∈ C } is also a continuum of solutions; it bifurcates from (3 n,
0) and each element of this continuumyields a wave of minimal period λ/n .To the best author’s knowledge there are no analogous existence results for equation (31)describing waves on water of finite depth. However, another integral equation for this casewas investigated by Norbury [49]. Since it arises after the hodograph transform, its unknownis the angle of the flow velocity with the horizontal as a function of the potential along the freesurface profile. The corresponding nonlinear operator is positive and completely continuouson continuous functions, and so the same approach as in [25] yields an unbounded continuumof solutions; moreover, it includes those found in [31].
Properties of solutions belonging to C.
Let Φ µ ∈ C solve equation (15) for µ > for µ sufficiently large (cid:107) Φ µ (cid:107) > π/ (cid:107)·(cid:107) stands for the standard norm in C [0 , π ]. Subsequently, Amick [1] obtainedan upper bound of this norm; combining its slightly improved version (see [62], p. 36) andMcLeod’s result, one has π/ < sup {(cid:107) Φ (cid:107) : ( µ, Φ) ∈ C } < . ≈ (1 . π/ ≈ . ◦ ) . (33)It should be mentioned that this upper bound is very close to the value 0.530 ( ≈ . ◦ )calculated numerically in [34]. In the paper [1], Amick obtained also that (cid:107) Φ (cid:107) < . λ to infinity this bound is extended to solitary waves.It is clear that the following form of Nekrasov’s equation for deep waterΦ( θ ) = (cid:90) π − π sin Φ( τ ) µ − + (cid:82) τ sin Φ( ζ ) d ζ K ( θ, τ ) d τ , θ ∈ [ − π, π ] , (34)is equivalent to (15) and (19), but one can consider it with µ − = 0 (see below). Importantresults about Φ solving (34) with µ > Let Φ be a solution of (34) with µ > . If positive λ and c are such that (cid:20) gλ πc (cid:21) / = 12 π (cid:90) π − π cos Φ( τ ) d τ [ µ − + (cid:82) τ sin Φ( ζ ) d ζ ] / , (35) then there exist the water domain D z and a complex velocity potential defined on D z . Thedomain’s width is λ and it corresponds to a single wave, whose profile is characterised locallyby the angle Φ . The potential is periodic and satisfies the free-surface boundary conditionson the upper part of ∂D z , whereas c is involved in the limit condition (1) as y → −∞ .
18t is worth mentioning that a kind of explicit expression is obtained for η ( x ) describingthe upper boundary of D z . It is rather complicated and involves two integrals each havinga variable limit of integration; one of these integrals has the same integrand as that in (35).We conclude this section with the result of Kobayashi [27], which gives an answer, atleast partial, to the conjecture made by Toland at the end of his paper [63]; it says that C is a curve. Indeed, Kobayashi established the uniqueness theorem for solutions of equa-tion (34) provided µ ∈ (3 , µ ∈ (3 , . . , .
3] and [3 . , µ ∈ [3 . , . Let us consider equation (34) with µ − = 0 describing waves with stagnation points at wavecrests. Indeed, µ − = (2 π/ q / ( gcλ ) according to (16), where q is the velocity at a crest.The existence of an odd solution ofΦ ∗ ( θ ) = (cid:90) π − π sin Φ ∗ ( τ ) (cid:82) τ sin Φ ∗ ( ζ ) d ζ K ( θ, τ ) d τ , θ ∈ [ − π, π ] , (36)was proved by Toland [62], who based his approach on Keady and Norbury’s result concern-ing the existence of solutions of (34) with finite µ . Their method cannot be applied directlybecause the operator in (36) is not completely continuous unlike that in (34). Therefore,Toland considered a sequence { µ k } tending to infinity and the corresponding solutions { Φ k } of (34). Since the absolute values of Φ k are bounded, this sequence converges weakly in L to a nontrivial Φ ∗ . Then Toland demonstrated that Φ k converge to Φ ∗ strongly in L ,thus obtaining that Φ ∗ is a solution of (36); Φ ∗ was also shown to be continuous on [ − π, π ],except at zero which is a point of discontinuity, whose nature was not resolved in [62].This discontinuity was the topic of a conjecture made by Stokes in his “Considerationsrelative to the greatest height of oscillatory irrotational waves which can be propagatedwithout change of form” — an appendix added to [59] in [60], pp. 225–228. Since the originis chosen at a trough and the wavelength is 2 π , the conjecture takes the formlim θ ↑ π Φ ∗ ( θ ) = π/ . (37)In its support, a formal asymptotic argument near a stagnation point was provided and inconclusion was added:“[. . . ] whether in the limiting form the inclination of the wave to the horizoncontinually increases from the trough to the summit [. . . ] is a question which Icannot certainly decide, though I feel little doubt that .. [convexity].. representsthe truth.” 19herefore, the assertion that the free surface profile η is convex between its successive maxima is nowadays referred to as the second Stokes conjecture about waves of extreme form. So farit is known that there exists an extreme wave having this property; see [51] for the proof.Its idea is to study a parametrized family of integral equations each associated with a free-boundary problem. It occurs that for the parameter equal to 1 the equation has a uniquesolution with associated convex free boundary; moreover, a continuation argument showsthat the existence and convexity of solutions holds when the parameter goes from 1 to 1/3.Since the equation corresponding to the latter value is (36), its solution describes a Stokesextreme wave for which the property takes place. However, this approach leaves unansweredthe question whether every extreme wave is convex between its successive maxima.Let us turn to conjecture (37) which was proved in 1982, independently in [3] and [50].The authors of [3] discussed the difficulties of the proof (one finds its slightly amendedversion in [63]) in the paper’s Appendix; in particular, it is said that the operator in (36)lacks some properties needed for the application of fixed-point theorems used in the case of µ − (cid:54) = 0, that is, the singularity at τ = 0 is absent.To overcome these difficulties the following approximate integral equation φ ( θ ) = 13 π (cid:90) ∞ sin φ ( τ ) (cid:82) τ sin φ ( ζ ) d ζ log θ + τ | θ − τ | d τ , θ ∈ (0 , ∞ ) , (38)was used in [3]. It is straightforward to check that φ ≡ π/ this constant solution is the only pointwise bounded solution of (38) such that inf θ ∈ (0 , ∞ ) φ ( θ ) > and sup θ ∈ (0 , ∞ ) φ ( θ ) ≤ π/ . Finally, it was shown that this property of φ implies (37) for any odd solution Φ ∗ of (36)satisfying inequalities0 ≤ Φ ∗ ( θ ) < π/ θ ∈ ( − π, π ) and lim inf θ ↑ π Φ ∗ ( θ ) > . Since these properties of Φ ∗ were established in [5], (37) is true.The following asymptotic formulaΦ ∗ ( θ ) = π C ( π − θ ) β + C ( π − θ ) β + O (cid:0) [ π − θ ] β (cid:1) as θ ↑ π , refines the Stokes conjecture (37); here β ≈ . √ β ) = tan( πβ/ C < C > π -periodic Hilbert trans-form and this allowed him to simplify the proof of the first Stokes conjecture and to do thisfor its generalised form.We conclude this section with the result of Kobayashi [28], who presented a computer-assisted proof that equation (36) has a unique odd solution. Combining this fact and thatof Plotnikov and Toland [51], he concluded that every extreme wave is convex between itssuccessive maxima which settles the second Stokes conjecture about these waves.Kobayashi obtained his result in the same way as in [27]. He first proved that the uni-queness of a solution to (36) takes place if the supremum of a certain rather complicatedfunction is strictly less than one. Then this supremum was computed numerically withthe rounding mode controlled. The approximate result 0.99290443370699 guaranteed to besmaller than one, thus implying the uniqueness. The tale of Nekrasov’s integral equations would not be complete without mentioning furtherdevelopment during the past forty years. Amick and Toland initiated this development intheir seminal papers [4] and [5] published in 1981.Using a procedure resembling that applied by Nekrasov, an integral equation for solitarywaves was derived in [4]; it is analogous to (19), namely: (cid:98) Φ( θ ) = µ (cid:90) π/ − π/ sec τ sin (cid:98) Φ( τ )1 + µ (cid:82) τ sec ζ sin (cid:98) Φ( ζ ) d ζ (cid:98) K ( θ, τ ) d τ , θ ∈ ( − π/ , π/ . (39)Here, (cid:98) Φ denotes the angle between the real axis in the z -plane and the negative velocityvector at the free surface parametrised in a special way because it is infinite, whereas thekernel (cid:98) K ( θ, τ ) = 13 π ∞ (cid:88) k =1 sin 2 kθ sin 2 kτk resembles that in (19); finally, the equation’s parameter expressed in terms of characteristicsof the flow is as follows: µ = 6 ghcπq . This formula is analogous to (16), but involves h —the depth of flow at infinity.Unfortunately, sec τ is singular at τ = ± π/
2, and so global bifurcation theory is not ap-plicable directly to equation (39). To overcome this difficulty Amick and Toland consideredthe following sequence of Nekrasov type equations: (cid:98) Φ n ( θ/
2) = 2 µ n (cid:90) π f n ( τ ) sin( J (cid:98) Φ n ( τ / µ n (cid:82) τ f n ( ζ ) sin( J (cid:98) Φ n ( ζ/ ζ K ( θ, τ ) d τ , θ ∈ (0 , π ) . (40)Here K is the same kernel as in (19), J : R → R is the following continuous function Ja = a if | a | ≤ π,π if a ≥ π, − π if a ≤ − π, { f n ( τ ) } is defined as follows: f n ( τ ) = (cid:26) − sec( τ /
2) if | τ | ≤ π − n − , − sec([ π − n − ] /
2) if π − n − ≤ | τ | ≤ π. It occurred that equation (40) is tractable in the same way as (19) (see Section 3.1); thatis, for every integer n ≥ C n , whoseelements ( µ n , (cid:98) Φ n ) are solutions of (40). Furthermore, Amick and Toland demonstrated thatthe sets C n converge in a certain sense to an unbounded, closed and connected set, whoseelements ( µ, (cid:98) Φ) are solutions of (39); moreover, to any µ ∈ [6 /π, ∞ ) there correspondsa solution. Properties of these solutions are investigated in detail and the existence of asolitary wave of extreme form is also proved in [4].Amick and Toland [5] developed further the technique of Nekrasov type integral equationsproposed in [5]; their new results were summarized as follows:A detailed discussion of Nekrasov’s approach to the steady water-wave problemsleads to a new integral equation formulation of the periodic problem. This deve-lopment allows the adaptation of the methods of Amick and Toland (1981) toshow the convergence of periodic waves to solitary waves in the long-wave limit.In addition, it is how the classical integral equation formulation due to Nekrasovleads, via the Maximum Principle, to new results about qualitative features ofperiodic waves for which there has long been a global existence theory (Krasovskii1961, Keady & Norbury 1978).The references mentioned here are [4], [31] and [25], respectively.It was Konstantin Ivanovich Babenko (1919–1987), who invented an equation alternativeto Nekrasov’s fpr describing steady periodic waves on deep water and proved a local existencetheorem for his equation; see the brief notes [6] and [7] published in 1987. A detailedanalysis of this quasi-linear equation given in [9] and [10] is based on properties of the 2 π -periodic Hilbert transform involved in the equation and its variational structure; indeed, itis the Euler–Lagrange equation of a simple functional introduced in [9]. A relation betweensolutions of Nekrasov’s and Babenko’s equations established in [11], Section 10.4, allowedthe authors to obtain some properties of the latter equation.Recently, it was demonstrated that the approach used in [6] yields an equation of the sameform as Babenko’s in the case when water has a finite depth; see [32]. The only difference isthat the 2 π -periodic Hilbert transform is perturbed by a compact operator in the latter case,and so the existence of local solution branches follows analogously to the case of deep water.Moreover, this equation and its modification proposed in [16] are convenient for numericalcomputation of global bifurcation diagrams, in particular, of secondary bifurcations whichmay be multiple as is shown in [16], Figure 4. References [1] Amick, C. J., Bounds for water waves.
Arch. Ration. Mech. Anal. (1987), 91–114.[2] Amick, C. J., Fraenkel, L. E., On the behavior near the crest of waves of extreme form. Trans.Amer. Math. Soc. (1987), 273–298.
3] Amick, C. J., Fraenkel, L. E., Toland, J. F., On the Stokes conjecture for the wave of extremeform.
Acta Math. (1982), 193–214.[4] Amick, C. J., Toland, J. F., On solitary water-waves of finite amplitude.
Arch. Ration. Mech.Anal. (1981), 9–95.[5] Amick, C. J., Toland, J. F., On periodic water-waves and their convergence to solitary wavesin the long-wave limit. Phil. Trans. Roy. Soc. London A (1981), 633–669.[6] Babenko, K. I., Some remarks on the theory of surface waves of finite amplitude.
Soviet Math.Doklady (1987), 599–603.[7] Babenko, K. I., A local existence theorem in the theory of surface waves of finite amplitude. Soviet Math. Doklady (1987), 647–650.[8] Benjamin, T. B., The alliance of practical and analytical insights into the nonlinear problemsof fluid mechanics. Appl. Methods Funct. Anal. Problems Mech., Proc. Joint Symp. Marseille .Berlin, Springer-Verlag, 1975, pp. 8–29.[9] Buffoni, B., Dancer, E. N., Toland, J. F., The regularity and local bifurcation of steady periodicwaves.
Arch. Ration. Mech. Anal. (2000), 207–240.[10] Buffoni, B., Dancer, E. N., Toland, J. F., The sub-harmonic bifurcation of Stokes waves.
Arch.Ration. Mech. Anal. (2000), 241–271.[11] Buffoni, B., Toland, J. F.,
Analytic Theory of Global Bifurcation: An Introduction . Princeton,Princeton University Press, 2003.[12] Conquest, R.,
The Great Terror: A Reassessment . Oxford, Oxford University Press, 2008.[13] Craig, W., Nicholls, D. P., Traveling gravity water waves in two and three dimensions.
EuropeanJ. Mech. B/Fluids (2002), 615–641.[14] Craik, A. D. D., George Gabriel Stokes on water wave theory. Ann. Rev Fluid Mech. (2005),23–42.[15] Dancer, E. N., Global solution branches for positive mappings. Arch. Ration. Mech. Anal. (1973), 181–192.[16] Dinvay, E., Kuznetsov, N., Modified Babenko’s equation for periodic gravity waves on waterof finite depth. Quart. J. Mech. Appl. Math. (2019), 415–428.[17] Fraenkel, L. E., A constructive existence proof for the extreme Stokes wave. Arch. Ration.Mech. Anal. (2007), 187–214.[18] Goursat., ´E.,
Cours d’Analyse Math´ematique, I, II . Paris, Gauthier–Villars, 1905.[19] Guyou, E., Sur le clapotis.
C.R. Acad. Sci. Paris . (1893), 722–724.[20] Henrici, P., Applied and Computational Complex Analysis . Vol. . New York, Wiley-Inter-science, 1986.[21] Hyers, D. H., Some nonlinear integral equations from hydrodynamics. Nonlinear Integral Equa-tions . (P. M. Anselone, ed.), Madison, University of Wisconsin Press, 1964, pp. 319–344.[22] Jackowski, A., Krzemie´n, K., Maurycy Pius Rudzki and the birth of geophysics.
Hist. GeoSpace Sci. (2016), 23–25.[23] Johnson, M. D. R., Review of [26]. Aerospace Power J. no. 4 (2000), 100–102.[24] Joukowski, N. E., On the wake wave. Trudy Otdelenia Fiz. Nauk O-va Lyubit. Estestvoznan. , no. 1 (1907), 1–24 (in Russian).[25] Keady, G., Norbury, J., On the existence theory for irrotational water waves. Math. Proc.Cambridge Philos. Soc. (1978), 137–157. Proceedings of the Branch of Physical Sciences, Society of Amateurs of Natural Sciences
26] Kerber, L. L.,
Stalins Aviation Gulag: A Memoir of Andrei Tupolev and the Purge Era . Wash-ington, D.C., Smithsonian Institution Press, 1996.[27] Kobayashi, K., Numerical verification of the global uniqueness of a positive solution forNekrasov’s equation.
Japan J. Ind. Appl. Math. (2004), 181–218.[28] Kobayashi, K., On the global uniqueness of Stokes’ wave of extreme form. IMA J. Appl. Math. (2010), 647–675.[29] Krasnosel’skii, M. A., On Nekrasov’s equation from the theory of waves on the surface of aheavy fluid. Doklady Akad. Nauk SSSR (1956), 456–459 (in Russian).[30] Krasnosel’skii, M. A.,
Topological Methods in the Theory of Nonlinear Integral Equations . Ox-ford, Pergamon Press, 1964.[31] Krasovskii, Yu. P., On the theory of steady-state waves of finite amplitude.
USSR Comput.Math. Math. Phys. (1961), 996–1018.[32] Kuznetsov, N., Dinvay, E., Babenko’s equation for periodic gravity waves on water of finitedepth: derivation and numerical solution. Water Waves (2019), 41–70.[33] Lewy, H., A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math.Soc. (1952), 111–113.[34] Longuet-Higgins, M. S., Fox, M. J. H., Theory of the almost highest wave: the inner solution. J. Fluid Mech. (1977), 721–742.[35] Luzin (also Lusin), N. N., Integral and Trigonometric Series . Moscow, Lissner & Sobko, 1915;also in
Matem. Sbornik (1916), 1–242; also in Collected Papers , I. Moscow, Izdat. Akad.Nauk SSSR, 1953, pp. 48–212 (in Russian).[36] McLeod, J. B., The Stokes and Krasovskii conjectures for the wave of greatest height.
Stud.Appl. Math. (1997), 311–334; also University of Wisconsin M.R.C. Report no. 2041, 1979.[37] McLeod, J. B., The asymptotic behavior near the crest of waves of extreme form.
Trans. Amer.Math. Soc. (1987), 299–302.[38] Michell, J. H., The highest waves in water.
Philos. Mag. (5) (1893), 430–437.[39] Milne-Thompson, L. M., Theoretical Hydrodynamics . 4th ed. London, Macmillan, 1962.[40] Nekrasov, A. I., Theory of waves on the surface of a rather shallow heavy fluid.
Trudy OtdeleniaFiz. Nauk O-va Lyubit. Estestvoznan. , no. 2 (1913), 1–20; also in [47] pp. 7–23 (in Russian).[41] Nekrasov, A. I., On Stokes’ wave. Izvestia Ivanovo-Voznesensk. Politekhn. Inst. (1919), 81–89;also in [47] pp. 26–34 (in Russian).[42] Nekrasov, A. I., On steady waves. Izvestia Ivanovo-Voznesensk. Politekhn. Inst. (1921), 52–65; also in [47] pp. 35–51 (in Russian).[43] Nekrasov, A. I., On waves of permanent type on the surface of a heavy fluid. Nauch. IzvestiaAkad. Tsentra Narkomprosa (1922), 128–138; also in [47] pp. 69–78 (in Russian).[44] Nekrasov, A. I., On steady waves. Part 2, On nonlinear integral equations. Izvestia Ivanovo-Voznesensk. Politekhn. Inst. (1922), 155–171 (in Russian).[45] Nekrasov, A. I., On steady waves on the surface of a heavy fluid. Trudy Vseross. S”ezda Matem-atikov , Moscow-Leningrag, Gosizdat, 1928, pp. 258–262 (in Russian).[46] Nekrasov, A. I.,
The Exact Theory of Steady Waves on the Surface of a Heavy Fluid . Moscow,Izdat. Akad. Nauk SSSR, 1951; also in [47] pp. 358–439 (in Russian); translated as Universityof Wisconsin MRC Report no. 813 (1967). Scientific Bulletin of the Academic Centre of the Education Ministry Proceedings of the All-Russian Congress of Mathematicians
47] Nekrasov, A. I.,
Collected Papers , I. Moscow, Izdat. Akad. Nauk SSSR, 1961 (in Russian).[48] Nem´enyi, P. F., Recent developments in inverse and semi-inverse methods in the mechanics ofcontinua.
Adv. Appl. Mech. (1951), 123–148.[49] Norbury, J., The existence of periodic water waves. Nonlinear Partial Differential Equationsand Applications. Lecture Notes in Math. , Berlin, Springer, 1978, pp. 117–128.[50] Plotnikov, P. I., A proof of the Stokes conjecture in the theory of surface waves.
DinamikaSplosh. Sredy (1982), 41–76 (in Russian); translated in Stud. Appl. Math. (2002),217–244.[51] Plotnikov, P. I., Toland, J. F., Convexity of Stokes Waves.
Arch. Ration. Mech. Anal. (2004), 349–416.[52] Privalov, I. I., Int´egrale de Cauchy.
Izvestia Saratov Univ. , no. 1 (1918), 1–94 (in Russian).[53] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems. J. Funct. Anal. (1971), 487–513.[54] Rudzki, M. P., Ueber eine Classe hydrodynamischer Probleme mit besonderen Grenzbedingun-gen. Math. Ann. (1898), 269–281.[55] Sekerzh-Zenkovich, Y. I., Aleksandr Ivanovich Nekrasov. Uspekhi matem. nauk , no. 1 (1960),153–162 (in Russian).[56] Sekerzh-Zenkovich, Y. I., Theory of steady waves of finite amplitude generated by a periodicpressure on the surface of an infinitely deep flow of heavy fluid. Doklady Akad. Nauk SSSR (1968), 304–307 (in Russian).[57] Sretenskii, L. N.,
The Theory of Wave Motions of a Fluid . Moscow–Leningrad, Ob (cid:48)(cid:48) edinennoeNauchno-Technicheskoe Izdat., 1936 (in Russian).[58] Stoker, J. J.,
Water Waves: The Mathematical Theory with Applications . New York, Inter-science, 1957.[59] Stokes, G. G., On the theory of oscillatory waves.
Trans. Camb. Phil. Soc. (1847), 441–455.[60] Stokes, G. G., Mathematical and Physical Papers . . Cambridge, Cambridge University Press,1880.[61] Tikhomansritzky, M., Th´eotie des Int´egrales et des Fonctions Elliptiques . Kharkov, Silberberg,1895 (in Russian).[62] Toland, J. F., On the existence of a wave of greatest height and Stokess conjecture,
Proc. Roy.Soc. London A (1978), 469–485.[63] Toland, J. F., Stokes waves.
Topol. Methods Nonlinear Anal. (1996), 1–48; Topol. MethodsNonlinear Anal. (1997), 412–413 (Errata).[64] Vainberg, M. M., Trenogin, V. A., Theory of Branching of Solutions of Nonlinear Equations .Leyden, Noordhoff International Publishing, 1974.[65] Varvaruca, E. Singular points on Bernoulli free boundaries.
Comm. Part. Diff. Equat. (2006), 1451–1477.[66] Volgina, V. N., Tyulina, I. A., Aleksandr Ivanovich Nekrasov 1883–1957 . Moscow, Nauka, 2001(in Russian).[67] Wehausen, J. V., Laitone, E. V., Surface waves.
Handbuch der Physik . . Berlin, Springer-Verlag, 1960, pp. 446–778.[68] Yamada, H., Highest waves of permanent type on the surface of deep water. Report Res. Inst.Appl. Mech. , Kyushu University no. 18 (1957), 37–52.[69] Zygmund, A., Trigonometric Series , I & II. Cambridge, Cambridge University Press, 1959., I & II. Cambridge, Cambridge University Press, 1959.