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Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev-Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the twodimensional Benjamin-Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich-Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.Recently, however, several studies have been devoted to Whitham theory for (2+1)-dimensional systems. In particular, [3] demonstrated the derivation and physical relevance of the Whitham systems for certain reductions of the KP and two-dimensional Benjamin-Ono (2DBO) equations. These reductions lead to the cylindrical KdV and cylindrical BO equations, hence they are quite different from the standard (1+1)dimensional KdV and BO reductions of the KP and 2DBO equations, respectively. From the point of view of Whitham theory, the reductions of these (2+1)-dimensional PDEs can be considered on the same footing as the (1+1)-dimensional ones. Subsequently Whitham theory has been considered for the (2+1) dimensional KP [1] and 2DBO [2] without using any one-dimensional reductions. We also mention that the underlying structure of solutions that will develop dispersive shocks in the small dispersion limit of the generalized KP equations has been studied in [11]. We refer the reader to these papers for additional background and references.In this work we present an approach that applies equally well to integrable and non-integrable PDEs, since at the heart of it lies Whitham's WKB type expansion and separation of fast and slow scales. We consider (2+1)-dimensional PDEs of the form

In [36], we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of β. Using this general result, the case β = 6 is further considered here. This is the smallest even β, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases β = 2 and 4. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ordinary differential equation (ODE) for the logarithmic derivative of Tracy-Widom distribution for β = 6 involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local singularity analysis yields series solutions with exponents in the set 4/3, 1/3 and −2/3.

Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g. there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system -a closed system of two nonlinear PDEs of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.

Starting from the diffusion equation at beta random matrix hard edge obtained by Ramirez and Rider (2008), we study the question of its relation with Lax pairs for Painleve III. The results are in many respects similar to the ones found for soft edge by Bloemendal and Virag (2010). In particular, the values beta = 2 and 4 (but not beta = 1) allow for a simple connection with Painlevé III solutions and Lax pairs. However, there is an additional surprise for a special relation of parameters where a simple solution of the diffusion equation can be obtained, which is a one-parameter generalization of Gumbel distribution. Our considerations can be extended to the other Painleve equations since the corresponding diffusions are in fact known as nonstationary (imaginary time) Schrödinger equations for quantum Painlevé Hamiltonians. We also track the hard-to-soft edge limit transition in terms of our Lax pairs. *

Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences. K E Y W O R D S nonlinear waves, nonlinear optics, water waves and fluid dynamics The applicability of the 2d NLS equation for Bose-Einstein condensate (BEC) wave functions in the appropriate geometries with the third coordinate axis being the axis of symmetry is well known and Stud Appl Math. 2019;142:269-313. wileyonlinelibrary.com/journal/sapm Company 269 270 ABLOWITZ ET AL. can be derived from many-body quantum dynamics, see, eg, Ref. 1. This equation can also describe deep water waves with sufficiently high surface tension. 2 A motivation to study the rNLS equation comes from the BEC experiments and the analysis in Ref. 3 where it was found numerically that the rNLS equation provides a good approximation to the dynamics.In the experiments of Ref. 3, an initial hump of the BEC density in the form of a ring was created by the laser beam effectively pushing the particles away from the center. Then, the BEC expanded radially. An additional radial parabolic potential that is absent in Equation 1 caused the BEC in experiments of Ref. 3 to move away from the center. Relative to this motion, however, the BEC expanded radially toward the inside that is what we consider analytically and numerically here.The dimensionless parameter characterizing dispersion (see Ref.3) was very small in these experiments, ≈ 0.012. With such a small value of Whitham theory, which is a nonlinear WKB-expansion in , is expected to be relevant. As pictures of the experiments show, in the expansion, large oscillations of the BEC density were created, which implies a dispersive shock wave (DSW) started from the initial density jump. The oscillations formed a number of concentric rings, ie, the DSW propagation was indeed radial with high degree of accuracy. Similar concentric rings forming a radial DSW were observed in nonlinear optics experiments. 4 However, in Ref.3, only the one-dimensional NLS equation was treated analytically. Whitham modulation theory for the (1 + 1)-dimensional NLS (1d NLS) equation (see Refs. 5 and 6) was applied to this problem. While qualitative agreement with experiments and direct simulations was found, and the solution/experiments exhibit character of a DSW, the analytical results did not always correspond closely (see, eg, figure 24 of Ref. 3). N...

Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and τ -functions of ASvM are found, for the example of finite size Gaussian matrices. Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship. * e-mail: igorrumanov@math.ucdavis.edu 1 I. INTRODUCTIONRandom matrices were introduced in the middle of last century by Wigner and Dyson as a description of complex systems that were difficult to study since they did not admit any exact or even approximate solutions. So the fact, thoroughly realized in the early nineties, is still amazing: many interesting random matrix (RM) models are closely related with and conveniently described by integrable systems -integrable hierarchies of partial differential equations (PDE). The first such result by Jimbo, Miwa, Môri and Sato 13 appeared earlier but remained unrecognized as an instance of a general rule for a decade. Matrix integrals, giving the probability distributions for eigenvalues of random matrices, proved to be τ -functions of integrable hierarchies. The ground-breaking works of Tracy and Widom 16,17 followed, which generalized and extended the results of Ref. 13 for the sine kernel to a number of other cases. The authors derived integrable PDE or Painlevé ODE as equations satisfied by the probabilities for the spectrum of various random matrices as well as by their large size asymptotics.Another major approach was developed by Adler, Shiota and van Moerbeke 1−6 . It is based directly on the correspondence between matrix integrals over the spectral domains and τ -functions -the "generating functions" of integrable PDE. Using both methods, different integrable PDE have been obtained not only for single matrix ensembles but also for coupled Hermitian matrices 4,5,8,18 . In the latter case equations derived by different methods seem to be so dissimilar that the only indication of the existence of relations between them is that they are equations for the same probabilities.In this paper, we compare the "algebraic" approach of Adler, Shiota and van Moerbeke with the "functional-theoretic" approach of Tracy and Widom and show the common structure behind them. This structure is (not surprisingly) the Toda lattice 19,1,3 . The relation between these two approaches has not been heretofore completely clarified. The reason lies in the fact that the authors of Refs. 2, 3 used the KP hierarchy rather than one-dimensional Toda lattice hierarchy (1-Toda) to derive their PDE for single Hermitian matrices. This led to equations very different in form from those derived in Refs. 16, 17. J.Harnad 11 demonstrated for the special cases of Airy and Bessel kernels, how the "KP"-equation of ASvM can be obtained by taking the appropriate combinations of TW equations. We show that there is a more direct and complete correspondence of structure...

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.