Higher dimensional localized and periodic wave dynamics in a new integrable (2+1)-dimensional Kundu-Mukherjee-Naskar model

Dynamics of nonlinear systems reveal several interesting phenomena which leads to a better understanding and influential applications in different fields of science, engineering and technology. This is due to the existence of various nonlinear coherent structures such as elliptic waves, solitons/solitary waves, rogue waves, breathers, shock waves, compactons, etc. associated with those systems. For the past few decades, the dynamics of such nonlinear waves have shown promising results in theoretical research as well as their corresponding experimental realization through which several applications have been demonstrated in diverse areas of physics like nonlinear optics, fluid dynamics, Bose-Einstein condensates, plasma physics, biophysics, etc. In general, these nonlinear waves possessing systems are modelled by nonlinear evolution (ordinary/partial differential) equations with various control parameters corresponding to the physical system under consideration. This ranges from the simple Kortewe-de Vries, sine-Gordon, nonlinear Schrödinger, Hirota, Burgers, Benjamin-Ono, Boussinesq, Camassa-Holm, Degasperis-Procesi, nonlinear $\sigma$ model, Heisenberg ferromagnet equations in one-dimension to Davey-Stewartson, Kadomtsev-Petviashvili, Navier-Stokes, Landau-Lifshitz model, Zakharov-Schulman, Gross-Pitaevskii equations in higher dimensions to name a few in different contexts of physics which appear as either integrable or non-integrable[1]. The complications in the complete analysis of these and their related/extended version of nonlinear equations make them less well understood due to insufficient generalized mathematical tools. Hence, there is always a need for new integrable nonlinear systems due to their rich analytic beauty [1, 2].

Integrable systems show wider variety of nonlinear coherent structures including the much celebrated solitons which attract huge attention not only due to their remarkable stability but also due to their intriguing collision dynamics. This enhances the understanding and consequences of different interactions in physics, such as the interaction between large amplitude waves along with wave-wave and wave-particle interactions. It is well known that the occurrence/existence of solitons is bound only to integrable nonlinear systems, while solitary waves (not solitons) appear in certain non-integrable as well as nearly-integrable models. Also, one has to note that a higher dimensional extension of an integrable one-dimensional model can become non-integrable and its dynamics will be restricted with a lesser number of free parameters [1, 2, 3]. In view of the ever increasing interest on higher dimensional integrable nonlinear systems, the following new integrable (2+1) dimensional nonlinear evolution equation was proposed by Anjan Kundu, Abhik Mukherjee and Tapan Naskar to describe the dynamics two-dimensional oceanic rogue waves and two-dimensional ion-acoustic wave in a magnetized plasma [4, 5]:

where $q(x,y,t)$ is the envelope of nonlinear wave and asterisk corresponds to the complex conjugation. Here the first and second terms represent the temporal ($t$) evolution and the dispersion of the wave along two spatial ($x$ and $y$) dimensions. This equation was first derived in Ref. [4] to model two-dimensional oceanic rogue waves and later it was named as Kundu-Mukherjee-Naskar (KMN) equation [6, 7, 8, 9, 10, 11]. Importantly, it can be seen that the nonlinear terms are different from the conventional Kerr-type nonlinearity arising in the celebrated nonlinear Schrödinger (NLS) and other related/generalized models. Physically, it can be viewed as current-like nonlinearity arising from chirality as well as to study the phenomena of bending of light beams [12]. Recently, the above KMN equation (1) was shown to govern the dynamics of optical wave or soliton along an Erbium-doped coherently excited resonant waveguides [13]. Also, KMN equation (1) possesses a rich dynamical characteristic similar to the standard NLS equation, and hence it can also be treated as a (2+1)-dimensional integrable generalization of the NLS equation. Due to these reasons, it has attracted considerable interest among the researchers including us for the present investigation, and several reports are available in the literature.

It is important to highlight a few interesting observations made on KMN equation (1). Its integrability nature and rogue wave dynamics are reported in [4]. Also, Hirota method was applied to obtain soliton and lump solutions of KMN model including its Lax pair, modulation instability and rogue waves in [5, 6]. Higher order rational/rogue wave solutions are also obtained with N-fold Darboux transformation method [7, 8] and a power series solution along with bright/dark soliton solutions have been studied by ansatz method [9, 10, 11, 12, 13, 14]. Still we can look for more general analytical solutions with different features resulting from more number of free/arbitrary parameters. In this aspect, one can study the periodic nonlinear waves and hyperbolic functions (elliptic waves) which give tremendous applications in nonlinear optics, plasma physics, hydrodynamics, etc. [15, 16, 17, 18] as they can explain many physical situations. One of the advantages of elliptic/periodic solutions is that they can be employed both for integrable as well as non-integrable evolution equations with different types of nonlinearities (for example focusing, defocussing and mixed type nonlinearities) [19, 20, 21, 22, 23, 24].

Based on the above, our objective of the present study is to obtain certain new type of periodic/elliptic/ hyperbolic solutions by applying the traveling wave ansatz and Jacobian elliptic solutions. Here we demonstrate the occurrence of bright and dark optical soliton solutions for specific choices of parameters in addition to the nonlinear periodic waves. Also, we have employed Ricatti equation method and ansatz function approach to construct complexiton and soliton solutions, where we have shown the importance of different wave parameters in their dynamics. Additionally, we attempt to find a doubly-localized dynamic rational solution of a forced KMN model. One of the interesting advantages of the present KMN model (1) is that its nonlinear terms denoted by the derivatives provide additional features resulting into both bright as well as dark soliton solutions irrespective of its positive spatial dispersive effect/term comparing to the standard NLS equation which admits only bright (dark) soliton for focusing (defocusing) nonlinearity or with anamolous (normal) dispersion.

This article is structured in the following manner: In Sec. 2 a generalized traveling wave approach is adopted to obtain soliton, elliptic, periodic and unbounded solutions. Section 3 deals with complexiton solutions derived with the help of Riccati equation. In Sec. 4, an ansatz function approach is utilized to construct bright and dark solitons. Section 5 deals with the dynamic rational solutions while the final section is alloted for conclusions.

In this section, we derive exact analytical solutions of KMN equation (1) supporting different coherent structures. For this purpose, we apply the following wave transformation to Eq. (1)

Here the function $u(\varepsilon)$ determines the shape of the nonlinear wave and the quantity $\Phi$ denotes the phase of the wave. Further in Eq. (2) $k$, $l$, $m$, $a$, $p$, $n$, $r$ and $b$ are real parameters. Upon substitution of Eq. (2) into Eq. (1) and equating real/imaginary parts separately, we obtain

The above system of equations can be written in a convenient form as

where $\alpha=kl$, $\beta=4p$, and $\gamma=r-np$. Let us consider that the solution $u(\varepsilon)$ of equation (4a) satisfies the Jacobian elliptic equation of the form

where $h_{0},h_{2}$ and $h_{4}$ are real constants to be determined. By differentiating the above equation (5) we obtain

By comparing Eq. (4a) with Eq. (6), and equating the coefficients of different powers of $u(\varepsilon)$ to zero, we obtain the following relations among the arbitrary real constants:

For various choices/combinations of the arbitrary constants appearing in the above equation, we can end up with different type of traveling wave solutions to the desired KMN equation (1) which we are exploring in the following subsections.

Complexitons are nothing but non-singular periodic solitons and they are new type of exact solutions admitted by certain nonlinear evolution equations. In this section, we obtain complexiton solutions by using the Riccati equation method [25], which results into the form of periodic/trigonometric and hyperbolic function solutions. In order to compute such complexiton solutions, we use the following assumption:

where $\xi=l_{1}x+l_{2}y-vt+\theta_{1}$ is a real-valued function in which $l_{1}$, $l_{2}$, $v$ and $\theta_{0}$ are real constants to be determined. Substituting Eq. (18) into the KMN equation (1), we get the following nonlinear ODE:

where $A_{1}=-l_{1}l_{2}$, $A_{2}=i(v-\alpha_{2}l_{1}-\alpha_{1}l_{2})$, $A_{3}=\beta+\alpha_{1}\alpha_{2}$, and $A_{4}=-4\alpha_{1}$. By balancing the linear term arising as the highest order derivative $f^{\prime\prime}$ and the nonlinear term $f^{3}$ from the above ODE, we can obtain the solution $f$ as

where $\psi$ satisfies the Riccati equation $\psi^{\prime}=R+\psi^{2}$, which has the following four set of solutions for two choices of $R$ [25]:

Hence from the above, the resultant hyperbolic and trigonometric traveling wave solutions of KMN equation (1) can be obtained through Eq. (18) as below.

For different choices of arbitrary parameters in Eq. (23), we can evidence bounded regular as well as unbounded singular solutions. Particularly, we obtain dark solitons resulting for ‘tanh’ solution, while the other solutions lead to localized/unlocalized singular structures. We have illustrated such complexiton solutions exhibiting dark soliton and singular structures in Figs. 8 and 9 for appropriate parameters. To be precise, the bounded regular dark soliton (23a) possesses an amplitude defined as $\sqrt{\dfrac{-l_{1}l_{2}}{2\alpha_{1}}}$ while the velocity takes the form $v/l_{1}$ and $v/l_{2}$ during its propagation along $x-t$ and $y-t$ spatiotemporal dimensions, respectively.

Though this KMN model is integrable and more delicate methods such as standard Hirota bilinearization, Darboux transformation, etc. can be applied to find its soliton solutions, here we utilize a simplified approach to compare/analyze its limitations/advantages in exploring the integrable nonlinear evolution equations. In view of this, we apply an ansatz approach to find the soliton solutions of KMN equation (1) and compare them with already obtained solutions in the previous sections. For this purpose, we consider an ansatz

where $u(x,y,t)$ is a real function while $\theta$, $\psi$, $\phi$ are real parameters. By adopting the above form for $q$ (24), the complex KMN equation (1) can be transformed into the following two real equations after separating the real and imaginary parts:

Solving the above equations (25) can end up with various solutions for $u(x,y,t)$. Here we concentrate on the hyperbolic solutions to describe bright/dark solitons and briefly analyze their evolution.

In addition to the above discussed soliton and periodic solutions, in this section we examine the rational solution of KMN equation. Rational solutions are a branch of doubly-localized nonlinear wave structures with certain free/arbitrary parameters which define the nature of their evolution and dynamics. In one hand, they are closely related/resemble to rogue waves as well as peregrine solitons. On the other hand, in higher dimensional systems these rational solutions have strong conection with lump solutions [7]. Here one can also investigate the occurrence and dynamics of rational solution to the given system by considering the copropagation/coexistence of solitons/solitary waves which have attracted much attention in recent years. However, here we restrict our study to explore only the rational solution of KMN equation with an external time-dependent forcing. In fact, without the forcing term we shall have the presence of only static rational solution. In order to understand the features of a dynamical rational solution, we consider the following form of KMN model by including a forcing factor, which can also be called as modified KMN equation:

In the above equation $f(t)$ represents the external forcing term which can provide time dependent (dynamical) features of rational solution instead of a limited steady state (static) behaviour. Our aim in this section is to provide an infinite number of dynamic solution which can be obtained from a static solution through the following rotational transformation:

The above equation reduces (30) into a standard KMN model (1). From the static rational solution corresponding to Eq. (1) given in [4, 5] and with the help of rotational transformation (31) we can construct a dynamical rational solution for arbitrary/different choices of $f(t)$ in the form given below.

Here $a$ and $c$ are two arbitrary real constants which defines the nature of static lump/rational solution, while $f(t)\in C^{0}$ is a class of arbitrary continuous real function of time from which one can extract an infinitely many solutions of the given forced KMN model.

Thorough study on the above dynamical rational solutions does not express any real observable effect as the time-dependency lies as a phase factor which plays very less or no role in real dynamics. This rational solution in $x-t$ and $y-t$ planes takes structure similar to that of peregrine solitons [26]. However, this shows significant observable effects if one investigate the real and imaginary components separately that depicts the occurrence of rational wave structure on a periodically modulated/oscillating background due to the time-varying external forcing factor. By suitably tuning the arbitrary parameters $a$ and $c$ one can control the steepness, inclination and amplitude of rational/rogue wave in addition to an appropriate forcing function. To elucidate the understanding, we have graphically demonstrated such rational wave structure in Figs. 11 and 12. Importantly, the rational solution exhibit oscillations in its amplitude and background for a given choice of external forcing $f(t)$.

To conclude, we have constructed bright/dark optical soliton, complexiton and periodic solutions of the integrable (2+1)-dimensional Kundu-Mukherjee-Naskar equation by adopting Jacobian-elliptic function, Riccati equation and ansatz methods and analyzed their evolution for obtained arbitrary parameters. Interestingly, the present KMN model provides a nonlinear effect which can results from the mixture of both focusing and defocusing nonlinearities with spatial derivatives and this has proved the admittance of both bright and dark soliton solutions irrespective of the same type of dispersion. Additionally, a dynamical rational solution of a modified KMN model with an external forcing has been studied which attracts considerable interest in recent years. Consequently, we have shown that the propagation dynamics of these solitons having more number of arbitrary parameters can find added advantage over the solutions reported earlier in oceanic waves, nonlinear optical waves in coherently excited resonant waveguides, etc. As a further study, the present investigation can be extended to any other integrable systems. The present results will be an important contribution on higher dimensional nonlinear waves associated arising under different physical contexts. Also, an interesting future outlook is to explore the collision dynamics of multiple solitons which can be studied by constructing exact multi-bright and -dark soliton solutions of the KMN model and can look for possible non-trivial interaction scenario. Works are in progress along these directions and the results will be reported separately.

One of the authors SS would like to thank MHRD and National Institute of Technology, Tiruchirappalli, India for financial support through institute fellowship. KS was supported by DST-SERB National Post-Doctoral Fellowship (File No. PDF/2016/000547).