ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear Schro¨dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.     

Whitham Equations, Bergman Kernel and Lax—Levermore Minimizer

We study multiphase solutions of the Whitham equations. The Whitham equations describe the zero dispersion limit of the Cauchy problem for the Korteweg—de Vries (KdV) equation. The zero dispersion solution of the KdV equation is determined by the Lax—Levermore minimization problem. The minimizer is a measurable function on the real line. When the support of the minimizer consists of a finite number of disjoint intervals to be determined, the minimization problem can be reduced to a scalar Riemann Hilbert (RH) problem. For each fixed x and t 0, the end-points of the contour are determined by the solution of the Whitham equations. The Lax—Levermore minimizer and the solution of the Whitham equations are described in terms of a kernel related to the Bergman kernel. At t = 0 the support of the minimizer consists of one interval for any value of x, while for t > 0, the number of intervals is larger than one in some regions of the (x,t) plane where the multiphase solutions of the Whitham equations develop. The increase of the number of intervals happens whenever the solution of the Whitham equations has a point of gradient catastrophe. For a class of smooth monotonically increasing initial data, we show that the support of the Lax—Levermore minimizer increases or decreases the number of its intervals by one near each point of gradient catastrophe. This result justifies the formation and extinction of the multiphase solutions of the Whitham equations. Furthermore we characterize a class of initial data for which all the points of gradient catastrophe occur only a finite number of times and therefore the support of the Lax—Levermore minimizer consists of a finite number of disjoint intervals for any x and t 0. This corresponds to give an upper bound to the genus of the solution of the Whitham equations. Similar results are obtained for the semi-classical limit of the defocusing nonlinear Schrödinger equation.     

Unified approach to KdV modulations

We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial‐value problem for the zero‐dispersion KdV as the steepest descent for the scalar Riemann‐Hilbert problem [6] and on the method of generating differentials for the KdV‐Whitham hierarchy [9]. By assuming the hyperbolicity of the zero‐dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)‐plane. The resulting system effectively solves the zero‐dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc.     

The generation,propagation, and extinction of multiphases in the KdV zero‐dispersion limit

We study the multiphases in the KdV zero‐dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi‐linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space‐time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler‐Poisson‐Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler‐Poisson‐Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler‐Poisson‐Darboux solution. © 2002 Wiley Periodicals, Inc.     

On the Semiclassical Limit of the General Modified NLS Equation

We study the semiclassical limit of the so-called general modified nonlinear Schrödinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity are proved for the dispersion limit of the cubic nonlinear case. The limiting transition from the MNLS equation to the NLS equation is also discussed.     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

Oscillations of the zero dispersion limit of the korteweg-de vries equation

We attack the multiphase averaged systems for the zero dispersion limit of the KdV equation. Attention is paid to the most important case—the single phase oscillations. A scheme is developed to solve the Whitham averaged system (single phase averaged system). This system, under our scheme, is transformed to a linear over-determined system of Euler-Poisson-Darboux type whose solution can be written down explicitly. We show that, for any smooth initial data which has only one hump or is a nontrivial monotone function, the weak limit has single-phase oscillations within a cusp in the x-t plane for a short time after the breaking time for the corresponding Burgers equation. Outside the cusp, the limit satisfies the Burgers equation. More surprisingly, we also show that the weak limit has global single-phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point. © 1993 John Wiley & Sons, Inc.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs

We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L² initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L² initial data. © 2001 John Wiley & Sons, Inc.     

Asymptotic Behavior of Solutions to Nonlinear Equations with Dissipation for Large x and t

In this paper we continue to study large time asymptotic behavior of solutions to the Cauchy problem for a class of nonlinear nonlocal equations with dissipation. When t → ∞ and x → ∞ simultaneously, the asymptotics of solutions for a generalization of the Kolmogorov-Petrovsky-Piscounov equation, a model equation studied by Whitham, and an equation introduced by Ott, Sudan, and Ostrovsky is found. The character of asymptotics obtained is quasilinear.     

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H² initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L^∞‐norm in terms of the L^q‐norms up to a logarithm of the L^p‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H² estimates for global regularity.© 2016 Wiley Periodicals, Inc.     

Existence of a global solution of the Whitham equations

The Cauchy problem for Whitham equations with monotonic analytic initial data is studied. If the initial data f(u) satisfies the condition f^(2N+1)(u)<0 for all u∈ℝ except a number of isolated points, then the genus of the solution of the Whitham equations is at most equal to N, where 1≤N∈ℕ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 58–71, January, 1999.     

Zero-dispersion limit to the Korteweg-de Vries equation: a dressing chain approach

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in x asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.      

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$) . We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$ . The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$ ; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$ , with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The L^r‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L¹ and L ^∞ decay rates of its first order derivatives with respect to space variables, are derived by using L^q ? L^r estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.     

Large time WKB approximation for multi-dimensional semiclassical Schrödinger-Poisson system

We consider the semiclassical Schrödinger-Poisson system with a special initial data of WKB type such that the solution of the limiting hydrodynamical equation becomes time-global in dimensions at least three. We give an example of such initial data in the focusing case via the analysis of the compressible Euler-Poisson equations. This example is a large data with radial symmetry, and is beyond the reach of the previous results because the phase part decays too slowly. Extending previous results in this direction, we justify the WKB approximation of the solution with this data for an arbitrarily large interval of R₊.     

Temporal asymptotics for soliton equations in problems with step initial conditions

In this survey, we present modern approaches to the construction and justification of large time asymptotics for solutions of main soliton equations with step-like initial condition whose boundary conditions as x ± are finite-gap, quasi-periodic solutions. The principal term of the asymptotic is also a finite-gap, quasi-periodic solution whose phase vectors are modulated with respect to the slow space-like variable. The Whitham equations describing this modulation are studied in detail. For the KdV equations, we construct and justify the principal term of the asymptotic for arbitrary finite-gap boundary conditions. By examining the sine-Gordon equation, we study the case of boundary conditions with complex-valued, self-conjugated quasi-periods. We prove the existence and uniqueness theorems in the case of the complex-valued group velocities that appear in the Whitham equations in the case considered. We present a complete picture (uniform in x) of the Whitham deformation for the case of 1-gap boundary conditions in the sine-Gordon equation.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.