Elliptic General Analytic Solutions

To find analytically the traveling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method,""tanh-method,""exponential method,""Jacobi expansion method,""new … ," etc. The common default to all these "truncation methods" is that they provide only some solutions, not all of them. By implementing three classical results of Briot, Bouquet, and Poincaré, we present an algorithm able to provide in closed form  all  those traveling waves that are elliptic or degenerate elliptic, i.e., rational in one exponential or rational. Our examples here include the Kuramoto–Sivashinsky equation and the cubic and quintic complex Ginzburg–Landau equations.     

Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach

In this paper, we propose a modified generalized transformation for constructing analytic solutions to nonlinear differential equations. This improved unified ansätze is utilized to acquire exact solutions that are general solutions of simpler equations that are either integrable or possess special solutions. The ansätze is constructed via the choice of an integrable differential operator or a basis set of functions. The technique is implemented to obtain several families of exact solutions for a class of nonlinear evolution equations with nonlinear term of any order. In particular, the Klein–Gordon, the Sine–Gordon and Landau–Ginburg–Higgs equations are chosen as examples to illustrate the method.     

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schrödinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schrödinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1+2)-dimensional cubic Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.     

Integrability and Linear Stability of Nonlinear Waves

It is well known that the linear stability of solutions of \(1+1\) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general \(N\times N\) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for \(N=3\) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.     

New Miura type transformations between integrable dispersive wave equations

In this paper, an algebraic method is devised to construct new Miura type transformations between integrable dispersive wave equations. The characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be determined by the solutions of a simpler equation directly. Our work is an attempt in searching for travelling solutions of complicate nonlinear equations.     

Global existence of null-form wave equations in exterior domains

We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel–Smith–Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish the scaling vector field from the other admissible vector fields.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

Characteristics of rogue waves on a soliton background in the general three-component nonlinear Schrödinger equation

Under investigation in this work is the general three-component nonlinear Schrödinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.     

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.     

Nonlinear Volterra integral equations and the Schröder functional equation

We show an interesting connection between a special class of Volterra integral equations and the famous Schröder equation. The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented.     

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.     

To the problem of the recovery of nonlinearities in equations of mathematical physics

The general construction for finding all “essentially different” nonlinearities in equations of mathematical physics is exemplified by the inverse problem of the Grad–Shafranov equation. We present an algorithm that allows one to recover relatively fast all essentially different sought-for nonlinear right-hand sides of the Grad–Shafranov equation. We present the first example of a domain with smooth boundary for which the inverse problem has at most one solution in the class of affine functions, and also in the class of exponential functions. We select some subset of simply connected domains that model, in some sense, the so-called doublet configurations, for which the inverse problem has at least two essentially different solutions in the class of analytic functions. In the concluding subsection of the paper, we indicate a method of recovery, from boundary data, of all essentially different nonlinearities in equations of mathematical physics of considerably general kind, which includes a system of equations that describes combustion and detonation processes.     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.     

On the existence of integrable solutions for a nonlinear quadratic integral equation

In this paper, we investigate the existence of integrable solution to a nonlinear integral equation, which includes many important integral and functional equations that arise in nonlinear analysis and its applications. Our results are obtained under rather general assumptions. In particular, the solvability of the well known Chandrasekhar equation is discussed under appropriate assumptions. Our analysis uses the technique of measures of weak noncompactness and rely on a variant of Schauder’s fixed point theorem.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Lax pair, cyclic basis, and a new integrable system

A direct approach to zero-curvature representation, as introduced by Marvan is applied to generate a new class of integrable equations by starting with the generic Lax operator (x-part) for a coupled set of nonlinear equations. The class of equation so obtained is more general than those usually obtained with the help of standard recursion operator. Finally some particular type of equations are identified by the special choice of the arbitrary functions occurring in the final solution of the time part of the Lax equation. The methodology is specially useful when the x-part of the Lax operator does not contain any spectral parameter.     

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.     

Induced Surfaces and Their Integrable Dynamics

A method is considered to induce surfaces in three-dimensional (pseudo) Euclidean space via the solutions to two-dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + 1-dimensional nonlinear integrable equations associated with these 2D LPs. Coordinates X_i of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions ψ of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three-dimensional Riemann spaces associated with 2 + 1-dimensional nonlinear integrable equations are considered also.