## Abstract

The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg–de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross–Pitaevskii equation in Bose–Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non--symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.

## 1. Introduction

In the theory of quantum mechanics [1], the Hamiltonian is usually required to be Hermitian, i.e. , which describes the dynamics of the quantum system.^{1} It has been shown in pioneering works [2,3] that there exist some non-Hermitian Hamiltonians with the space–time reflection () symmetry (i.e. , which is said to be -symmetric) exhibiting entirely real spectra. Here, the parity operator (or space-reflection operator) is linear and has the effect
1.1and the time-reversal operator is antilinear and has the effect
1.2where *p* and *x* stand for the momentum and position operators, respectively. Moreover, there exist these properties:
1.3For more detail about symmetry, see [4–7].

The complex -symmetric Hamiltonians are regarded as a complex extension of the Hermitian Hamiltonians and have extended theoretically to some physical systems (see earlier studies [4–7] and references therein). More recently, -symmetric optical systems with complex potentials have been experimentally observed [8,9], which further excite the study of the -symmetric systems [10–14].

The rest of this review is arranged as follows. In §2, we simply introduce a family of non-Hermitian -symmetric Hamiltonians due to Bender & Boettcher [2]. In §3, we consider the one- and two-dimensional nonlinear Schrödinger (NLS) equations with different complex -symmetric potentials and give their exact solutions. In §4, we introduce some complex -symmetric extension principles and their applications in some nonlinear wave equations such as the -symmetric Korteweg–de Vries (KdV) equation and non--symmetric Burgers equation. In particular, we consider the solutions of the complex -symmetric Burgers equations in detail. Finally, we give some conclusions in §5.

## 2. Non-Hermitian -symmetric Hamiltonians

In 1998, Bender & Boettcher [2] first presented the general parametric family of non-Hermitian -symmetric Hamiltonians in the form
2.1In particular, for the case *m*=0, the spectrum of the non-Hermitian -symmetric Hamiltonians was discussed. For the real parameter *ϵ*, the spectrum of non-Hermitian -symmetric Hamiltonians given by equation (2.1) exhibits three distinct behaviours [2]: (i) when *ϵ*≥0, the spectrum is real and positive, and the energy levels rise with increasing *ϵ*. The lower bound of this region, *ϵ*=0, corresponds to the harmonic oscillator, whose energy levels are *E*_{n}=2*n*+1; (ii) when −1<*ϵ*<0, there are a finite number of real positive eigenvalues and an infinite number of complex-conjugate pairs of eigenvalues. When *ϵ* decreases from 0 to −1, the number of real eigenvalues decreases; when *ϵ*≤−0.57793, the only real eigenvalue is the ground-state energy. As *ϵ* approaches −1^{+}, the ground-state energy diverges; (iii) when *ϵ*≤−1, there are no real eigenvalues. When *ϵ*≥0, the symmetry is unbroken, but when *ϵ*<0, the symmetry is broken (figure 1).

Moreover, some extensions of non-Hermitian -symmetric Hamiltonians given by equation (2.1) have also been considered [15–20]. The notion of pseudo-Hermiticity has also been introduced [21–24] and it has been shown that every Hamiltonian with a real spectrum is pseudo-Hermitian [25].^{2} More types of pseudo-Hermitian Hamiltonians have also been studied (see [27–37] and references therein).

## 3. Complex -symmetric potentials

Consider the linear Schrödinger eigenvalue equation
3.1where *m* denotes the atomic mass, in which it has been shown that is -symmetric provided that *V* (*x*)=*V**(−*x*) [38], which implies that the real part *V* _{R}(*x*) of the complex -symmetric potential *V* (*x*)≡*V* _{R}(*x*)+i*V* _{I}(*x*) should be an even function of position *x*, whereas the imaginary part *V* _{I}(*x*) is odd, namely
3.2

In the following, we pay much attention to one- and two-dimensional NLS equations (or Gross–Pitaevskii (GP) equation in Bose–Einstein condensates (BECs)) with different complex -symmetric potentials such that their exact solutions have been obtained [39–42].

### (a) Nonlinear Schrödinger equation with complex -symmetric potentials

We consider the NLS equation with complex -symmetric potentials in the form [39–41]
3.3where *ψ*≡*ψ*(*x*,*t*) is the envelope function field and the term *V* (*x*)+i*W*(*x*) denotes the complex -symmetric potential, which means that the real-valued functions *V* (*x*) and *W*(*x*) have symmetric properties: *V* (−*x*)=*V* (*x*), *W*(−*x*)=−*W*(*x*). In nonlinear optics, *V* (*x*) is associated with index guiding, whereas *W*(*x*) stands for the gain or loss distribution of the optical potential [43] and the real function *g*(*x*) denotes the varying nonlinear interaction as well as satisfying the symmetric property: *g*(−*x*)=*g*(*x*). It is easy to see that equation (3.3) differs from the usual NLS equation with varying coefficients because the gain or loss terms at most depend only on time in the usual NLS equation [44–47].

#### (i) Stationary solutions with complex -symmetric potentials

The stationary solutions of equation (3.3) can be obtained from *ψ*(*x*,*t*)=*ϕ*(*x*) e^{iεt}, where *ϕ*(*x*) is a complex function of space, and −*ε* denotes the chemical potential. As a consequence, we have the nonlinear ordinary differential equation for the complex function field *ϕ*(*x*)
3.4When the gain or loss term *W*(*x*) is zero, equation (3.4) becomes the usual stationary NLS equation, whose solutions have been analysed for some chosen potential *V* (*x*) and nonlinearity *g*(*x*) [48–50]. In the following, we focus on the study of equation (3.4) for the case with the external potential *V* (*x*), gain or loss distribution *W*(*x*) and the constant nonlinearity *g*(*x*)=*g*=const. [39–41].

The ansatz of the complex function in the form *ϕ*(*x*)=*Φ*(*x*) e^{iφ(x)} leads equation (3.4) to the coupled nonlinear system
3.5In the following, we consider the solutions of system (3.5) for some potentials *V* (*x*) and *W*(*x*).

#### (ii) Complex -symmetric potentials in the form of the hyperbolic or periodic functions

Here, we consider the Scarff II potential in the form [33]
3.6which are displayed in figure 2*a*, where *V* _{0} and *W*_{0} are the amplitudes of the external potential and gain or loss term. The linear case of equation (3.4) with *g*=0 and *V* (*x*),*W*(*x*) given by equation (3.6) has been verified to possess an entirely real spectrum for the condition [33]. Moreover, for the non-zero nonlinearity *g*=const≠0, the solution of equation (3.4) has been given in the form (also see case I in table 1) [41]
3.7where , and *g*=1, whose real and imaginary parts are depicted in figure 2*b*.

Similar to the earlier-mentioned Scarff II potential, with the aid of symbolic analysis, other types of potentials and corresponding solutions of equation (3.4) are listed in table 1 [40,41]. As another example, the complex potentials together with the solution given in case II of table 1 for the potential parameters *V* _{0}=0.5, *W*_{0}=0.6 and the different moduli *k*=0.8,0,1 are exhibited in figures 3–5, respectively.

#### (iii) The complex -symmetric potential in the form of harmonic and Hermite–Gaussian functions

In fact, equation (3.3) is also regarded as the dimensionless form of a quasi-one-dimensional GP equation with complex -symmetric potentials in one-dimensional BEC [42]. Here, *v*_{R}(*x*)=−*V* (*x*) denotes the external potential usually chosen as the harmonic trap [51,52] and *v*_{I}(*x*)=−*W*(*x*) stands for the gain or loss distribution (also called dissipative effect [53]), which is phenomenologically incorporated to account for the interaction of atomic or thermal clouds.

Let us now consider the physically relevant complex -symmetric potential with the external potential *v*_{R}(*x*) and the gain or loss distribution *v*_{I}(*x*) to be of the harmonic trap and the Hermite–Gaussian distributions, respectively [42],
3.8where *ω*>0 and *w*_{0} are real-valued constants, the even integer index *n* (i.e. *n*=0,2,4,…) is required to make sure *v*_{I,n}(*x*) be the odd function for the complex -symmetric potentials, and *H*_{n}(*x*)=(−1)^{n}e^{x2}(d^{n}e^{−x2}/d*x*^{n}) stands for the Hermite polynomial. In particular, the first two gain or loss terms *v*_{I,0}(*x*) and *v*_{I,2}(*x*) can be written as
3.9

As a consequence, we find that for the earlier-mentioned -symmetric potentials, equation (3.4) possesses a family of exact Hermite–Gaussian solutions [42]
3.10where *n*=0,2,4,…, *g*>0, and the chemical potential satisfies the condition . It is easy to show that these solutions are localized because .

The external potential *v*_{R}(*x*) and the gain or loss distribution *v*_{I}(*x*) given by equation (3.8), as well as the intensity of the corresponding solution given by equation (3.10) are illustrated in figure 6*a*,*b* for *n*=0, 2, respectively.

### (b) Two-dimensional nonlinear Schrödinger (Gross–Pitaevskii) equations with complex -symmetric potentials

Similar to the earlier-mentioned one-dimensional case, we can also consider the two-dimensional NLS (GP) equation with complex -symmetric potentials [41]
3.11where stands for the two-dimensional Laplace operator and *ψ*≡*ψ*(*x*,*y*,*t*) denotes the two-dimensional envelope field. Note that for the two-dimensional -symmetric potentials, one might have different versions of symmetry (e.g. reflecting in one space or two spaces [54,55]). Here, we consider the complex -symmetric potential *V* (*x*,*y*)+i*W*(*x*,*y*) and nonlinearity *g*(*x*,*y*) in the form
3.12

#### (i) Stationary solutions

The stationary solution of equation (3.11) can be obtained from *ψ*(*x*,*y*,*t*)=*ϕ*(*x*,*y*) e^{iεt+iφ(x,y)}. Consequently, we have the coupled system of nonlinear partial differential equations
3.13by separating the real and imaginary parts of equation (3.11), where *ε* is the chemical potential.

#### (ii) Complex -symmetric potentials in the form of the hyperbolic or periodic functions

*Case 1.* If we consider the complex -symmetric potential with *V* (*x*,*y*) and *W*(*x*,*y*) being of the form [41]
3.14which are displayed in figure 7*a*,*b*, then it follows from equation (3.13) with the constant nonlinearity *g*(*x*,*y*)=1 that we have exact solutions of equation (3.11) in the form *ψ*(*x*,*y*,*t*)=*ϕ*(*x*,*y*) e^{iεt+iφ(x,y)} with the amplitude *ϕ*(*x*,*y*) and phase *φ*(*x*,*y*) given by
3.15whose profiles are depicted in figure 7*c*,*d*, where *V* _{0} and *W*_{0} are real-valued constants with , and *ε*=2.

*Case 2.* If we consider another complex -symmetric potential in the form (figure 8*a*,*b*)
3.16and the non-trivial nonlinearity (figure 8*c*)
3.17where *W*_{0} and *g*_{0} are real-valued constants, then it follows from equations (3.13), (3.16) and (3.17) that we have the solution of equation (3.11) in the form *ψ*(*x*,*y*,*t*)=*ϕ*(*x*,*y*) e^{iεt+iφ(x,y)} with the amplitude *ϕ*(*x*,*y*) and phase *φ*(*x*,*y*) given by
3.18where *ε*=2, whose profiles are illustrated in figure 8*d*,*e*.

In fact, the earlier-mentioned idea may also be extended to study the *d*-dimensional (*d*>2) NLS (GP) equations with the complex -symmetric potential
3.19and the *d*-dimensional two-component NLS (or GP) equations with complex -symmetric potentials
3.20where *ψ*≡*ψ*(**r**,*t*),*ψ*_{j}≡*ψ*_{j}(**r**,*t*)(*j*=1,2) denote the *d*-dimensional envelope field, *q*>*p*≥3 are real numbers, and denotes the *d*-dimensional Laplace operator. Notice that for the higher dimensional -symmetric potentials, one might have different versions of symmetry (e.g. reflecting in one space, two spaces or more spaces [54,55]), for example, the complex -symmetric potentials *V* (**r**)+i*W*(**r**), *V* _{j}(**r**)+i*W*_{j}(**r**),*u*_{j}(**r**)+i*w*_{j}(**r**) and nonlinearities *g*(**r**),*g*_{ji}(**r**) can be chosen the conditions *V* (**r**)=*V* (−**r**),*W*(**r**)=−*W*(−**r**),*V* _{j}(**r**)=*V* _{j}(−**r**),*W*_{j}(**r**)=−*W*_{j}(−**r**),*u*_{j}(**r**)=*u*_{j}(−**r**),*w*_{j}(**r**)=−*w*_{j}(−**r**) and *g*(**r**)=*g*(−**r**),*g*_{ji}(**r**)=*g*_{ji}(−**r**).

## 4. Complex -symmetric extensions of nonlinear wave equations

### (a) Complex -symmetric extensions of the -symmetric wave equations

In the field of nonlinear science, there exist types of nonlinear wave equations, which are -symmetric (i.e. ), for example:

— generalized KdV equation:

*u*_{t}+*u*^{n}*u*_{x}+*u*_{xxx}=0 (),— Benjamin–Bona–Mahony equation:

*u*_{t}+*uu*_{x}+*u*_{xxt}=0,—

*K*(*m*,*n*) equation:*u*_{t}+(*u*^{m})_{x}+(*u*^{n})_{xxx}=0 (),— Boussinesq equation:

*u*_{tt}+(*u*^{2})_{xx}+*u*_{xxxx}=0,—

*B*(*m*,*n*) equation:*u*_{tt}+(*u*^{m})_{xx}+(*u*^{n})_{xxxx}=0 (),— Kadomtsev–Petviashvili equation: (

*u*_{t}+*uu*_{x}+*u*_{xxx})_{x}+*u*_{yy}=0,— nonlinear cubic-quintic Schrödinger equation: i

*u*_{t}+*u*_{xx}+*μ*|*u*|^{2}*u*+*ν*|*u*|^{4}*u*=0 (),— Klein–Gordon-type equation:

*u*_{tt}−*u*_{xx}+*F*(*u*)=0, where*F*(*u*) is a polynomial of*u*with real-valued coefficients,— Camassa–Holm equation:

*w*_{t}+2*ku*_{x}+*uw*_{x}+2*u*_{x}*w*=0,*w*=*u*−*u*_{xx}(),— Degasperis–Procesi equation:

*w*_{t}++*uw*_{x}+3*u*_{x}*w*=0,*w*=*u*−*u*_{xx}, etc.

Bender *et al.* [56] presented the family of the new complex -symmetric extensions of the KdV equation in the form
4.1by applying the complex -symmetric extension principle to the -symmetric KdV equation *u*_{t}+*uu*_{x}+*u*_{xxx}=0. Furthermore, some properties were also considered for the obtained complex -symmetric equation (4.1).

Subsequently, Fring [57] applied the earlier-mentioned -symmetric deformation principle [56] () to the third-order term of the KdV equation, i.e. 4.2As a consequence, another family of the complex -symmetric deformations of the KdV equation was found in the form 4.3In addition, the Painlevé analysis and solutions of the complex -symmetric equation (4.3) were also studied [58].

Several -symmetric deformations of super-derivatives have been presented and applied to consider the -symmetric extensions of the super-symmetric KdV equation [59]. Recently, the compactons were presented for the -symmetric extension for the even number *m* [60]
4.4by using the variational principle with the generalized non-Hermitian Hamiltonian density
4.5

More recently, symmetry breaking in the KdV equation with complex coefficients 4.6and Ito-type system with complex coefficients 4.7as well as their deformations have been investigated [61].

By using the earlier-mentioned -symmetric extension principle, one can obtain the corresponding complex -symmetric extensions of other complex -symmetric nonlinear wave equations such as the earlier-mentioned ones.

### (b) Complex -symmetric extensions of non--symmetric wave equations

In fact, there also exist another type of nonlinear wave equations, which are not -symmetric (i.e. ), in the field of nonlinear science, for example:

— generalized Burgers equation:

— (

*N*+1)-dimensional Burgers equation:— KdV–Burgers equation: with

*b*≠0),— Kolmogorov–Petrevsky–Piskunov-type equation:

*u*_{t}−*u*_{xx}+*f*(*u*)=0, where*f*(*u*) is a polynomial of*u*with real-valued coefficients,— (2+1)-dimensional Broer–Kaup–Kupershimidt equation:

*u*_{ty}−*u*_{xxy}+ 2(*uu*_{x})_{y}+ 2*v*_{xx}= 0,*v*_{t}+ 2(*uv*)_{x}+*v*_{xx}=0, etc.

#### (i) The first type of complex -symmetric extensions of the Burgers equation

Based on the complex -symmetric extension principle () and these nonlinear transformations applied to the Burgers equation *u*_{t}+*uu*_{x}+*u*_{xx}=0, which is not -symmetric,^{3} we have the complex -symmetric extension of the Burgers equation in the form [62]
4.8where .

In the following, we consider the complex -symmetric extension of the Burgers equation for the different real-valued parameters *δ*,*ϵ*,*m*,*n*.

*Case 1.* *δ*=*ϵ*=*n*=*m*=1. Equation (4.8) becomes the complex Burgers equation
4.9which can be further reduced to the linear Schrödinger equation, *iϕ*_{t}+*ϕ*_{xx}=0, via the complex Cole–Hopf transformation *u*(*x*,*t*)=−2i*ϕ*_{x}/*ϕ*. Therefore, we have many types of solutions of equation (4.9) by solving the earlier-mentioned complex linear Schrödinger equation [63] and the complex Cole–Hopf transformation. The multiple exact wave solutions of equation (4.9) can be written as
4.10where *M*≥1, . For the case *M*=1 and *a*_{1}>0, we have the complex shock-like wave solution of equation (4.9)
4.11which is illustrated in figure 9*a*.

*Case 2.* *δ*=*ϵ*=*n*=1, *m*≠0. Equation (4.8) reduces to the complex generalized -symmetric Burgers equation
4.12whose complex kink-shaped wave solution can be given by
4.13which is invariant under the reflection. When *m*=1, *k*=*k*_{1}/2, , the solution (4.13) reduces to the solution (4.11). The profile of the solution (4.13) is exhibited in figure 9*b*,*c* for the parameter *m*=0.9 or *m*=2.

*Case 3.* *ϵ*=0. Equation (4.8) becomes the complex nonlinear wave equation with three real-valued parameters
4.14

*Case 3a.* *δ*=*m*=*n*=1. Equation (4.14) reduces to the linear Schrödinger equation with energy offset
4.15which can be reduced to the complex linear heat equation i*w*_{t}+*w*_{xx}=0 via the transformation *u*(*x*,*t*)=*w*(*x*,*t*)e^{it}. Therefore, we can obtain many types of solutions of equation (4.15) using this transformation and solutions of the linear heat equation [63].

*Case 3b.* *δ*=1, *m*=*n*≠0,1. Equation (4.14) becomes the complex -symmetric nonlinear wave equation
4.16

The separation transformation *u*(*x*,*t*)=[(i/(1−*n*))(*t*−*t*_{0})]^{1/(1−n)}*w*(*x*) can reduce equation (4.16) to the nonlinear ordinary differential equation (*w*^{n}(*x*))′′−(*n*−1)^{−2}*w*(*x*)+*w*^{n}(*x*)=0, which possesses the compacton solution
4.17for *n*>1, whose profiles are displayed in figure 10*a* for the parameters *n*=1,2. Equation (4.16) also admits two families of local conservation laws in the form *T*_{j,t}+*X*_{j,x}=0, where *T*_{j} and *X*_{j} denote the conserved density and the corresponding flux, respectively, and given by

*Case 3c.* *δ*=2*N*+1, *m*=*n*≠0. Equation (4.14) reduces to the complex nonlinear wave equation
4.18

Based on the separation transformation , equation (4.18) can be reduced to the real nonlinear ordinary differential equation
4.19where the prime denotes the derivative with respect to *x*, and *λ*,*t*_{0} are constants.

(i) When

*n*>2*N*+1, we find that equation (4.20) admits the following compacton solution: 4.20whose profile is displayed in figure 10*b*for the parameters*N*=1,*n*=4 (or*N*=2,*n*=7), where*x*_{0}is an arbitrary constant.(ii) When

*n*=1, equation (4.20) reduces to the nonlinear ordinary differential equation −*λv*^{2N+1}(*x*)+*v*(*x*)+*v*′′(*x*)=0, which possesses the periodic wave solution 4.21where*x*_{0}is an arbitrary constant.

*Case 3d.* *δ*=2. Equation (4.14) reduces to the nonlinear wave equation
4.22

(i) When

*n*=2,*m*=4, we find that equation (4.22) admits the solitary wave solution 4.23where*x*_{0}is an arbitrary constant, whose profile is displayed in figure 11*a*.(ii) When

*n*=2,*m*=3, we know that equation (4.22) has the solitary wave solution 4.24where*x*_{0}is an arbitrary constant, whose profile is displayed in figure 11*b*.

#### (ii) The second type of complex -symmetric extensions of the Burgers equation

Here, we use the earlier-mentioned -symmetric extensive principle () to generate the second-order term of the Burgers equation such that we have another family of complex -symmetric extensions of the non--symmetric Burgers equation [62] 4.25

Equation (4.25) with *ϵ*=1 reduces to the complex -symmetric Burgers equation (4.9) again. For the case *ϵ*=2(*N*+1), equation (4.25) becomes the real nonlinear wave equation with the symmetry
4.26Here, we consider the steady progressing wave *u*(*x*,*t*)=*ϕ*(*ξ*) with *ξ*=*k*(*x*−*ct*), where *c* and *k* are the wave speed and numbers, respectively. Therefore, integrating its travelling wave reduction equation with respect to *ξ* once leads to
4.27where *c*_{0} denotes an integration constant.

If we choose the special integration constant as , then it follows from equation (4.27) that we have the solution of equation (4.26).

*Case 1.* *N*≠0. In this case, we have the solution of equation (4.26) in the form

*Case 2.* *N*=0. In this case, equation (4.26) becomes *u*_{t}+*uu*_{x}−2*u*_{x}*u*_{xx}=0, which has the exponential function solution in the form with *a*_{0} being a constant.

Similarly, we also extend these obtained complex -symmetric extensions of the Burgers equation and other nonlinear wave equations to higher dimensional spaces by using the earlier-mentioned -symmetric extensive principles (see Yan [62,64] for the details).

## 5. Conclusions

To conclude, we have reviewed the recent development of the complex -symmetric nonlinear waves in differential fields of nonlinear science. In particular, we focus on the exact solutions of the one-dimensional and two-dimensional NLS (or GP) equation with the different types of complex -symmetric potentials. In addition, we also review the complex -symmetric extensions arising from both -symmetric nonlinear wave equations and non--symmetric nonlinear wave equations. In particular, we consider the solutions of some representative cases of the complex -symmetric Burgers equation in detail. The methods used can also be extended to generate other complex -symmetric nonlinear wave systems such that new physical phenomena and applications may be found.

## Acknowledgements

I thank both referees for their valuable suggestions. This work was partially supported by NSFC grants (nos. 11071242 and 61178091).

## Footnotes

One contribution of 17 to a Theme Issue ‘𝒫𝒯 quantum mechanics’.

↵1 Here † denotes the usual Dirac Hermitian conjugation, i.e. the combined operations of matrix transposition and complex conjugation.

↵2 A Hamiltonian is said to be pseudo-Hermitian if its adjoint satisfies , where

*η*_{+}is a positive-definite (metric) operator. In other words, the reality of the energy spectrum of a Hamiltonian is the quasi-Hermiticity, i.e. there exists an invertible operator such that is Hermitian with respect to the inner product 〈⋅|⋅〉 [25,26].↵3 Because the Burgers equation is not -symmetric, i.e. it is not invariant under , which differs from the KdV equation with the -symmetric property, we cannot directly use the -symmetric extension principle (i.e. ) in §4

*a*. Here, for the non--symmetric Burgers equation, we use the -symmetric extension principle for both*u*_{x}and*u*_{t}, i.e. and .

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.