Nonintegrable Equations Research Articles

Weak Multi-Phase Asymptotics for Nonintegrable Equations

We describe an approach that allows us to construct multi-soliton asymptotic solutions for essentially nonintegrable equations and analyze the type of wave collision. The general idea is demonstrated by the example of a generalization of Korteweg-de Vries equation with small dispersion in the cases of two and three waves. We also discuss the phenomenon of nonuniqueness that occurs in the framework of the weak description of a multi-soliton interaction.

Number of Solitons Generated from an Intense Initial Pulse at Asymptotically Large Time

A method to derive formulas for the number of solitons into which a sufficiently intense initial pulse in the form of a localized simple wave evolves at asymptotically large time has been proposed on the basis of the Gurevich–Pitaevskii theorem on the number of oscillations entering the dispersive shock wave region. In application to integrable equations, this method reproduces Karpman-type formulas following from the semiclassical approximation for a linear spectral problem associated with the considered equation. It has been shown on particular examples that expressions previously obtained by means of the formal continuation of the solution of modulation Whitham equations to the dispersionless region of the wave can be derived in the case of nonintegrable equations.

The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

Rotary dynamics of the rigid body electric dipole under the radiation reaction

Rotation of a permanently polarized rigid body under the radiation reaction torque is considered. Dynamics of the spinning top is derived from a balance condition of the angular momentum. It leads to the non-integrable nonlinear 2nd-order equations for angular velocities, and then to the reduced 1st-order Euler equations. The example of an axially symmetric top with the longitudinal dipole is solved exactly, with the transverse dipole is analyzed qualitatively and numerically. Physical solutions describe the asymptotic power-law slowdown to stop or the exponential drift to a residual rotation; this depends on initial conditions and a shape of the top.

Resurgent trans-series for generalized Hastings–McLeod solutions

We show that the physical Hastings–McLeod solution of the integrable Painlevé II equation generalizes in a natural way to a class of non-integrable equations, in a way that preserves many of the significant qualitative properties. The Hastings–McLeod solution of Painlevé II is an important and universal example of resurgent relations between perturbative and non-perturbative physics. We derive the trans-series structure of the generalized Hastings–McLeod solutions, demonstrating that integrability is not essential for the resurgent asymptotic properties of the solutions.

Numerical solution of a highly nonlinear and non-integrable equation using integrated radial basis function network method.

In this paper, we investigate a wide range of dynamical regimes produced by the nonlinearly excited phase (NEP) equation (a single sixth-order nonlinear partial differential equation) using a more advanced numerical method, namely, the integrated radial basis function network method. Previously, we obtained single-step spinning solutions of the equation using the Galerkin method. First, we verify the numerical solver through an exact solution of a forced version of the equation. Doing so, we compare the numerical results obtained for different space and time steps with the exact solution. Then, we apply the method to solve the NEP equation and reproduce the previously obtained spinning regimes. In the new series of numerical experiments, we find regimes in the form of spinning trains of steps/kinks comprising one, two, or three kinks. The evolution of the distance between the kinks is analyzed. Two different kinds of boundary conditions are considered: homogeneous and periodic. The dependence of the dynamics on the size of the domain is explored showing how larger domains accommodate multiple spinning fronts. We determine the critical domain size (bifurcation size) above which non-trivial settled regimes become possible. The initial condition determines the direction of motion of the kinks but not their sizes and velocities.

The nonlinear integro-differential Ito dynamical equation via three modified mathematical methods and its analytical solutions

AbstractIn this work, we construct traveling wave solutions of (1+1) - dimensional Ito integro-differential equation via three analytical modified mathematical methods. We have also compared our achieved results with other different articles. Portrayed of some 2D and 3D figures via Mathematica software demonstrates to understand the physical phenomena of the nonlinear wave model. These methods are powerful mathematical tools for obtaining exact solutions of nonlinear evolution equations and can be also applied to non-integrable equations as well as integrable ones. Hence worked-out results ascertained suggested that employed techniques best to deal NLEEs.

Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrodinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrodinger equation, formulated on upper right-quarter plane with initial data $$u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})$$ and Neumann boundary data $$u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)$$ given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.

Traveling wave solutions for the (3+1)-dimensional Davey-Stewartson equations

In this work, the extended tanh method is used to construct wave solutions for the Davey–Stewartson equations. The extended tanh method is a powerful solution method for obtaining different kind solutions of nonlinear evolution equations. This method can be applied to nonintegrable equations as well as to integrable ones.

Multidimensional inhomogeneous mixed initial-boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous mixed initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model we propose general method based on Laplace approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Schrödinger evolution group. The advantage of our approach is that it can also be applied to non-integrable equations and arbitrary boundary conditions. This approach is new and it is not standard. We believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis.

Variational Approximations for Intersite Soliton in a Ablowitz-Ladik-Cubic Discrete Nonlinear Schrödinger Equation

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The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

The approach by Painlevé to construct the general solutions of some nonlinear second-order ordinary differential equations is generalized for finding exact solutions of nonintegrable differential equations. We demonstrate that this approach generalizes some other well-known algorithms of finding exact solutions: the tanh-function method, the G′/G-expansion method, the Exp-function method and some variants of the simplest equation method. The method allows us to search for both solution in the form of solitary waves and periodic solution expressed through elliptic functions. We demonstrate possibilities of the method for finding exact solutions of nonintegrable ordinary differential equations which are obtained as reductions of the modified Korteweg–de Vries equation with source, the generalized Burgers equation, the generalized Fisher equation, the generalized Gardner equation and the generalized Kuramoto–Sivashinsky equation.

On the Whitham system for the radial nonlinear Schrödinger equation

AbstractDispersive 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.

Algebraic entropy of (integrable) lattice equations and their reductions

We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler–Bobenko–Suris equations and Viallet’s and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.

Dispersive shock wave theory for nonintegrable equations.

We suggest a method for calculation of parameters of dispersive shock waves in the framework of Whitham modulation theory applied to nonintegrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham's "number of waves conservation law" as well as on the conjecture of applicability of its soliton counterpart to the above mentioned class of initial conditions which is substantiated by comparison with similar situations in the case of completely integrable wave equations. This allows one to calculate the limiting characteristic velocities of the Whitham modulation equations at the boundary with the smooth part of the pulse whose evolution obeys the dispersionless approximation equations. We show that explicit analytic expressions can be obtained for laws of motion of the edges. The validity of the method is confirmed by its application to similar situations described by the integrable Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations and by comparison with the results of numerical simulations for the generalized KdV and NLS equations.

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

AbstractThe nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge.

Rogue waves, homoclinic breather waves and soliton waves for a (3 + 1)-dimensional non-integrable KdV-type equation

PurposeThe purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.FindingsThe results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.Originality/valueThese results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.

Helical solitons in vector modified Korteweg–de Vries equations

We study existence of helical solitons in the vector modified Korteweg–de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.

Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays**Project supported by the National Natural Science Foundation of China (Grant Nos. 11671255 and 11701510), the Ministry of Economy and Competitiveness of Spain (Grant No. MTM2016-80276-P (AEI/FEDER, EU)), and the China Postdoctoral Science Foundation (Grant No. 2017M621964).

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term.