The universal solution of the Korteweg-de Vries equation (KdV) introduced by Gurevich and Pitaevskii in order to describe the onset of dispersive shock waves is known to also obey the self-similar reduction of the next member in the KdV hierarchy. We show that, if this common solution obeys some lower order partial differential equation, its differential order must be one, and we provide its local representation as a converging Laurent series depending on both space and time.

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlevé VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, K_sech=1/(2 πcosh[(x-y)/2]). The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable sech kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlevé VI equation. The corresponding Painlevé VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlevé VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in ℝ^3. A folding transformation between such surfaces singles out the Painlevé VI equation with Manin coefficients [0,0,0,0], which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

The $N$-th order rational rogue wave of the nonlinear Schrödinger equation (NLS), which depends on $2 N-2$ real parameters, has been shown to be impossible to generate by the nonlinear superposition formula. We here generate this sequence by a three-term recurrence relation, each step only requiring the computation of three $N-1$-th order determinants of $2 N-2$ variables, while the previous method requires two determinants of order $2 N$ in $2 N$ variables. This allows us to obtain explicitly the seventh wave with its six arbitrary complex parameters. These very compact expressions open the possibility to investigate the possible existence of new patterns in addition to the already observed ones (concentric rings, polygonal configurations, \dots).

For each of the forty-eight exceptional algebraic solutions u(x) of the sixth Painlevé equation, we build the algebraic curve P(u,x)=0 of a degree conjectured to be minimal, and then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions.

The four-wave mixing (FWM) technique for optical phase conjugation (OPC) has been acknowledged for a real-time OPC mirroring. Its main advantage lies in the adjustment of wavefronts via self-diffraction of interacting waves on the phase dynamic grating occurring in reversible optical nonlinear media. The feature of wave-mixing is to create a periodic modulation of the refractive index, the so-called photo-induced dynamic grating, by acting with an interference light field. In this paper, we derive the dynamical system of equations for the FWM as an equivalent system of coupled wave equations that describes the nonlinear interaction between two lattices: the governing light lattice and the driven dynamic material grating. Further we use a new approach: nonlinear processes of self-organization in this system are considered from the point of view of dissipative soliton (DiS) formation. Thus, we study the properties of DiS of dynamic lattice interaction (DiSLI), which may be regarded as a new kind of spatial dissipative soliton. DiSLI exhibits a spatially inhomogeneous longitudinal envelope for the intensity in the maxima of the light interference field so as leads to inducing spatial localization of the dynamic grating amplitude along the thickness of the medium z. The physical mechanism for the emergence of a spatially localized envelope for the interference intensity is the energy transfer effect that occurs between interacting waves at wave-mixing in a bulk nonlinear medium. We analyze the conditions when a strongly localized DiSLI is formed exactly within the nonlinear medium. In this case the effect of considerable energy transfer from pump waves to the signal wave is realized, accompanied by a significant increase in the gain for the OPC-wave. Furthermore, we reduce the description of initial nonlinear system to a single complex Ginzburg-Landau equation (CGLE), by employing the multiscale expansion method.

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs to-gether with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.

We build several matrix Lax pairs of ${\rm q-P_{\rm VI}}$ valid even when the two eigenvalues of the residue of the monodromy matrix at infinity are equal. Their elements are rational functions of the dependent variables.

Bonnet has characterized his surfaces by a geometric condition. What is done here is a characterization of the same surfaces by two analytic conditions: (i) the mean curvature H of a surface in R3 should admit a reduction to an ordinary differential equation; (ii) this latter equation should possess the Painlev & eacute; property. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Nous établissons toutes les réductions du système de deux équations couplées de sine-Gordon introduit par Konopelchenko et Rogers à des équations différentielles ordinaires. Ces réductions sont toutes des dégénérescences d'une réduction maîtresse à une équation jugée par Chazy “curieuse en raison de [son] élégance”, transformée algébrique de la sixième équation de Painlevé la plus générale.

We consider the six-dimensional dynamical system in three components introduced by Ryan to describe the scenario of Belinskii, Khalatnikov and Lifshitz to the cosmological singularity when the spatial metric tensor is not diagonal. Despite its nonintegrability, recently proven by Goldstein and Piechocki, the three four-dimensional systems defined by canceling one of the three components happen to be integrable. We express their general solution as a rational function of, respectively, two exponential functions, a third Painlevé function, two exponential functions.

We establish a one-to-one correspondence between,on one hand the four types of transcendental meromorphic solutions of the autonomous Schwarzian differential equations which are elliptic,on the other hand the four binomial equations of Briot and Bouquet possessing an elliptic solution.

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

The various regimes observed in the one-dimensional complex Ginzburg-Landau equation result from the interaction of a very small number of elementary patterns such as pulses, fronts, shocks, holes, and sinks. Here we provide three exact such patterns observed in numerical calculations but never found analytically. One is a quintic case localized homoclinic defect, observed by Popp et al. [S. Popp et al., Phys. Rev. Lett. 70, 3880 (1993)10.1103/PhysRevLett.70.3880], and the two others are bound states of two quintic dark solitons, observed by Afanasyev et al. [V. V. Afanasyev et al., Phys. Rev. E 57, 1088 (1998)10.1103/PhysRevE.57.1088].

We provide a Lax pair for the surfaces of Voss and Guichard, and we show that such particular surfaces considered by Gambier are characterized by a third Painlevé function.

В компактном замкнутом виде записано выражение для трехпараметрического бризера нелинейного уравнения Шредингера. Это позволяет аналитически доказать удвоение периода решения по времени, наблюдаемого в экспериментах. Как результат, значительно упрощается экспериментальная проверка того, что некоторые импульсы, генерируемые в оптических волокнах, действительно являются такими обобщенными бризерами.

We perform the analytic study of the buoyancy-drag equation with a time-dependent acceleration γ(t) by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of γ(t) define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of γ(t) including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exhibit two régimes for the asymptotic (large time t) solution of the buoyancy-drag equation. It is shown that they describe a mixing zone driven by the Rayleigh—Taylor instability and the Richtmyer—Meshkov instability, respectively.

The problem addressed here is, given an ODE which may admit a single-valued solution, to find it explicitly in closed form. Two kinds of ODEs are considered. The first kind is made of ODEs which pass the Painlevé test, in which case the goal is to find their general solution, and our three main examples will be: the four cases of the Lorenz model isolated in Sect. 2.1.1 , the traveling wave reduction of the Korteweg-de Vries equation and of the nonlinear Schrödinger equation. In these three examples, the general solution can indeed be found and is represented either by elliptic functions or by Painlevé functions. Elliptic function Painlevé function The second kind is made of ODEs which fail the Painlevé test, but which do not fail it too badly, leaving open the possibility for particular single-valued solutions. A powerful result of Nevanlinna theory leads us to split this second kind in two classes. The first class is made of ODEs which possess no arbitrary constant in their Laurent series and fulfill another easy to check condition (Eremenko's theorem); then all their solutions are either elliptic or rational in a single exponential or rational, and they can be and indeed are obtained in closed form. For the second class, made of all other ODEs, there only exist sufficient methods able to yield particular single-valued solutions, mainly those known as "truncation" methods. Several physical examples (Kuramoto-Sivashinsky, complex Ginzburg-Landau, Duffing-van der Pol, Bianchi IX, etc.) are presented in detail for both classes.

We present the Painlevé test on various examples of nonlinear ODEs, Painlevé test for ODEs 2.1 $$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle E(x,u^{(N)},\ldots,u^{\prime},u)=0,\, ^\prime = \frac{\mathrm{d}}{\mathrm{d} x}, \, \ u^{(N)}=\frac{\mathrm{d}^N u}{\mathrm{d} x^N}, {} {} \end{array} \end{aligned} $$ This is a local analysis which can be implemented as an algorithm to provide necessary conditions for the Painlevé property. Painlevé property for ODEs This method is historically due to Sophie (Sonya) Kowalevski and Bertrand Gambier, with many elements already in Hoyer [30]. In the motion of a rigid body around a fixed point, Kowalevski required the general solution to be a single-valued function of time not only on the real axis but also in the complex plane and, by applying only a subset of the method which we are going to describe, she isolated a fourth case (the "Kowalevski case") of possible single-valuedness and succeeded to prove its singlevaluedness [33]. Gambier developed a direction initiated by Appelrot and made the method algorithmic. This method was later rediscovered by Ablowitz, Ramani and Segur [2]. Hoyer Kowalevski Gambier As to Painlevé, he himself never used "le procédé connu de Madame Kowalevski …dont le caractère nécessaire n'était pas établi" [41, pp. 10, 83] [42, pp. 196, 269] but developed his own "α-method". Next, we introduce two methods to handle cases when the Kowalevski and Gambier method is indecisive. Finally, we recall the classical methods to process first order equations.