## Abstract

This paper investigates the dispersive long wave equations that can be used to describe wide channels or open oceans with finite depth. By utilizing the Binary Darboux transform and the generalized Schur polynomial, the 2*N*-lump solutions are constructed. Subsequently, different dynamic behaviors are analyzed, such as (1) The 2*N* lumps can be divided into *N* groups, and each group is composed of a principal peak and an adjoint peak; (2) The deflection of peak trajectories in the *rs*-plane by \(90^\circ \) after collision, which can be explained by correlating the peak positions with the roots of the heat polynomials; (3) The peak locations are separated in the scales as when \(\left| t\right| \rightarrow \infty \); (4) For the peak heights of the principal peaks, we can find that their heights are time-dependent and will tend to the same constant value.

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## 1 Introduction

Nonlinear evolution equations, such as the nonlinear Schrödinger equation [1], Korteweg-de Vries equation [2], sine-Gordon equation [3], serve as mathematical models for a variety of complex phenomena. These equations typically entail nonlinear effects in time and space, and can exhibit a wealth of phenomena such as soliton solutions [4] and shock waves [5], providing important mathematical tools for understanding complex behavior in nature. In this article, we will focus on the \(\left( 2+1\right) \)-dimensional dispersive long wave equations(DLWEs)

where *u* and *v* are wave amplitude functions in relation to spatial variables (*x* and *y*) and time variable (*t*). Equation (1.1) is the mathematical model that describes the scenario of wide channels or open oceans with finite depth [6,7,8], which has substantial research significance in physics, engineering and earth science [9]. In the field of optics, the DLWEs are used to describe the dispersion effect of light waves propagating in the medium. It can give us a thorough grasp of the propagation characteristics of light waves, as well as useful references for optical communication, laser technology and optical device design [10, 11]. In marine science, Eq. (1.1) is widely used to study the propagation and interaction of ocean waves. Waves will be affected by the sea floor and coastline during their propagation, so Eq. (1.1) can describe this complex wave behavior and help to predict the changes of sea waves and coastal erosion [12,13,14]. In addition, the equation has been widely applied in seismology, plasma physics and fluid mechanics [11, 15, 16]. The study of DLWEs not only helps to deepen the understanding of wave phenomenon, but also provides valuable mathematical tools and theoretical support for scientific research and engineering applications in various fields.

In the last several years, a variety of fascinating properties of the \(\left( 2 + 1\right) \)-dimensional DLWEs have been investigated. The DLWEs was initially introduced by Boiti in 1987 as a compatibility condition for a weak Lax pair [9]. By employing the modified F-expansion method, new exact solutions expressed in terms of Jacobi elliptic functions were obtained [17]. In Ref. [18], the authors discovered a new method for separating variables in excitation using the extended mapping method. They also enhanced the diversity of non-propagating solitons by carefully selecting appropriate functions. In Ref. [19], the authors utilized Sato theory and the Hirota bilinear method to present solutions for both the Grammian and Wronskian types of determinants in the system. Additionally, in Ref. [20], by using truncated Painlevé expansion, the nonlocal symmetry and Bäcklund transformation are constructed, and the equation is further verified to be solvable by Riccati expansion. Furthermore, there have been numerous other studies conducted on DLWEs [21,22,23,24,25].

In addition, as a new research project, lump solutions have received extensive attention in recent years. A lump refers to a localized object or structure that can be described as a nonlinear wave, maintaining its shape and amplitude as it propagates through a medium [26, 27]. The study of lumps has been extensively conducted in the field of nonlinear dynamics and has shown potential applications in data transmission, energy storage, and quantum computing [28,29,30,31,32]. Therefore, further investigation and exploration of the potential of lump solutions will unveil more mysteries about nonlinear evolution equations and drive development and innovation in the scientific field. Take the Kadomtsev-Petviashvili(KP) equation for instance [33,34,35], Ref. [36] gave twelve classes of lump-kink solutions to the KP equation by using the Hirota bilinear methods. Ref. [37] used the simplified form of \(\tau \)-function to construct a class of generalized solutions of the KP equation, and described various distribution characteristics of lump chains. It is wonderful that Ref. [38, 39] have constructed multi-lump solutions of KPI equation through the Binary Darboux transform(BDT) and polynomial theory, and deeply analyzed the multi-lump solution by introducing the concept of partition, providing valuable insights into the phenomenon.

This paper aims to study multi-lump solutions of DLWEs by using the first-step BDT, it is the 2*N*-lump solution in this case. The dynamics of these solutions will be analyzed in detail, including peak height, peak position, and long-term asymptotic evolution. The distribution of the lumps studied here shows a unique property where 2*N*-lump solutions can be divided into *N* groups, each containing a principal peak and an adjoint peak. In the co-moving frame, they appear as the distribution of two parallel lines approximately parallel to the *s*-axis and attract each other (the principal peaks and the adjoint peaks are approximately collinear respectively). When all peaks complete the interaction process at the origin, 2*N* lumps are collinear along the *r*-axis and each group is far away from each other. The co-moving frame moves at a constant speed in the *xy*-plane, while the internal dynamics of the whole system evolve at a slower time on the scale of . The evolution of the peak position as a dynamic system can be correlated with the zeros of the two-dimensional heat equation, which also reveals the reason for the \(90^\circ \) deflection motion of the 2*N*-lump solution near \(t=0\).

The paper is structured as follows: In sect. 2, we will introduce the BDT and the generalized Schur polynomials, which are used to construct the expression of the multi-lump solutions of DLWEs. Section 3 presents the representation of function *v* based on the first-step BDT, and discusses some examples of solutions for \(N = 1, 2, 3\). In the case of \(N=2\), we provide a detailed analysis of the motion law of the peak position and peak height of the 4-lump solution throughout the evolution process. Section 4 focuses on the asymptotic analysis of 2*N*-lump solution under large time by using two-dimensional heat polynomials. Finally, the conclusions will be summarized in Sect. 5.

## 2 Construction of multi-lump solutions

In this section, we will describe a method for constructing a class of multi-lump solutions by using the BDT and a special type of complex polynomial. The specific steps are outlined in the following.

### 2.1 Binary Darboux transform

The Darboux transform(DT) based on Lax pairs has become one of the effective algorithmic methods to get explicit solutions of nonlinear evolution equations [40,41,42]. The DT was first presented by Darboux in 1882 as a transformation that gave a relation between two solutions [43]. In 1990, Matveev and Salle first investigated the DT in integral form and presented BDT [40]. Subsequently, there have been many results related to this research method. Next, we will state the construction of the rational solution for Eq. (1.1) by using BDT [44,45,46].

As mentioned above and noting \(u_y\) = \(p_x\) for convenience, Estévez and Gordoa obtained the Lax pairs of DLWEs based on the singular manifold method as [47,48,49]

The common solution \(\phi (x, y, t)\) of Eq. (2.1) is usually referred to as the wave function, with the compatibility condition \(\phi _{xyt}=\phi _{txy}\) resulting in the potential *u*(*x*, *y*, *t*) and *v*(*x*, *y*, *t*) satisfying Eq. (1.1). In addition, there exists a set of adjoint Lax pairs associated with the same potential function *u*(*x*, *y*, *t*) and *v*(*x*, *y*, *t*) as well as the conjugate wave function \(\phi ^*(x, y, t)\)

where \(*\) represents the complex conjugation in this paper. The singular manifolds \(\psi \) and \(\delta \) are defined in an abbreviated form using the Lax pairs mentioned above [48]

where \(\varGamma \left( \phi ,\phi ^*\right) \) and \(\varOmega \left( \phi ,\phi ^*\right) \) have been defined as the solutions of the over-determined system

With these definitions, we have

and it is seen that \(\varGamma \left( \phi ,\phi ^*\right) \) and \(\varOmega \left( \phi ,\phi ^*\right) \) are related by

Further, arbitrary *N* sets of solutions \(\left( \phi _k,\phi ^*_k\right) \) for Eqs. (2.1) and (2.2) are used to obtain the expressions of *N*-order BDT as [48]

where \(k=1, 2, \cdots , N\),

and

In the following, we will mainly discuss the potential function *v*.

### 2.2 Generalized Schur polynomials

In this section, we will use the BDT to construct a class of rational solutions for DLWEs decaying as \(\sqrt{x^2+y^2}\rightarrow \infty \). We start with the simplest solution namely \(u=v=0\) and take \(p=2k^3\) by the notation \(u_y\) = \(p_x\), then choose the eigenfunctions \(\left( \phi ,\phi ^*\right) \) as well as a set of specific solutions \(\left\{ \left( \phi _j,\phi _j^*\right) \right\} _{j=1}^{n}\) which make Eqs. (2.1) and (2.2) stand under this condition.

We choose \(\phi \left( k\right) =e^{i\sigma }\) and \(\phi ^*\left( k^*\right) =e^{-i\sigma ^*}\) where \(\sigma \left( k\right) =kx+k^3y+k^2t+\gamma \left( k \right) \) and \(\gamma \left( k\right) \) is an arbitrarily differentiable function of *k*. Here \(k\in {\mathbb {C}}\) is an arbitrary complex parameter contained in each solution \(\phi \left( k\right) \) and \(\phi ^*\left( k^*\right) \). Since the Eqs. (2.1) and (2.2) are independent of *k* and \(k^*\), \(\frac{\partial ^n\phi }{k^n}\) and \(\frac{\partial ^n\phi ^*}{k^{*^n}}\) satisfy the same equation. Further, we can generate a series of linearly independent solutions \(\left\{ \left( \phi _j,\phi _j \right) \right\} _{j=1}^{n}\) by

where \(k_0=a+ib\in {\mathbb {C}}\) and \(\sigma \left( x,y,t,k\right) =kx+k^3y+k^2t+\gamma \left( k \right) .\) To ensure the above solutions are linearly independent, \(1\le m_1<\cdots <m_n\) are distinct positive integers. It is evident from Eq. (2.4) that \(\phi _j\) will have the form of \(p\left( x, y, t\right) e^{i\sigma }\). The polynomial \(p\left( x, y, t\right) \) is known as generalized Schur polynomial and will play a key role in future discussions.

### Definition 2.1

The generic expression for the generalised Schur polynomials is defined by the relation [50,51,52]

where \(\gamma \left( k\right) \) is an arbitrarily differentiable function of *k*. Now we define \(\sigma _j\left( k\right) =\frac{i\partial _k^j\sigma \left( k\right) }{j!}\), which gives us

where \(\gamma _j\left( k\right) =\frac{\partial _k^j\gamma \left( k\right) }{j!}\) are independent complex parameters related to *k* only. It is worth noting that \(\frac{\partial \sigma _j}{\partial k}=\left( j+1\right) \sigma _{j+1}\) and \(p_n\) is related to the derivative of \(\sigma \) to *k*, which means that \(p_n=p_n\left( \sigma _1,\sigma _2,\cdots ,\sigma _n\right) \). The expression of the generalised Schur polynomials \(p_n\) concerning \(\sigma _1, \cdots , \sigma _n\) can be obtained by comparing Eqs. (2.6) and (2.7), which are Taylor expansions of \(e^{i\sigma \left( k+h\right) }\) in different orders

By comparing the two equations above in which the powers of *h* are equal, we obtain

Based on the above equation, we get the first few terms of \(p_n\) as

From Eq. (2.8), we know that \(p_n\) is a weighted polynomial of order *n* in \(\sigma _j, j=1,\cdots ,n\), where weights \(wt(\sigma _j)=j\) and each \(p_n\) has degree *n* in *x*, *y*, *t* since \(\sigma _1^n\) appears in the expansion.

Here are some simple properties of the generalized Schur polynomials [53].

### Property 2.1

Generalized Schur polynomials satisfy the recurrence formula

### Property 2.2

The derivative of the generalized Schur polynomial with respect to \(\sigma _j\) satisfies the recurrence formula

### Property 2.3

The frequency shifts of the generalized Schur polynomials satisfy the relation

### 2.3 The rational solutions of the DLWEs

In this subsection, we will make further constructions for the rational solution *v*. From Eq. (2.4), we have \(v^{\left( n\right) } = -2\frac{\partial ^2}{\partial x\partial y}\ln \left[ \det \left( \varTheta \left[ N\right] \right) \det \left( \varLambda \left[ N\right] \right) \right] \), notated in the following form for convenient expression:

Since the path independence of the line integral, we choose our path from an arbitrary initial point \(\left( x, y, t\right) \in {\mathbb {R}}^3\) and terminating at some point \(\left( \infty , y_0, t_0\right) \) defined by \(\gamma =\gamma _1 + \gamma _2\)

Taking \(\varGamma \left( \phi _i,\phi _j^*\right) \) as an example, it reduces to

From Eqs. (2.4) and (2.5), we can rewrite \(\phi _j\) and \(\phi _j^*\) as

Then the product \(\phi _i\phi _j^*\) contains an exponential term \(e^{i\left( k_0-k_0^*\right) x}\) arising from the exponent \(i\left( \sigma -\sigma ^*\right) \), the first integral in Eq. (2.14) will converge and the second integral will vanish if we choose Im\(\left( k_0\right) = b > 0\). As a result, we simply obtain that \(\varGamma \big (\phi _i,\phi _j^*\big )=\int _{x}^{\infty }{\phi _{i,x}\phi _j^*dx}\). Using the integral by parts repeatedly, it can be reduced to

Similarly, there is

Observe that \(A^\textbf{H}=B\), where \(\textbf{H}\) denotes the conjugate transpose, thus *AB* is a hermitian matrix and the determinant \(\tau =\det \left( AB\right) \frac{e^{2in\left( \sigma -\sigma ^*\right) }}{\left( 2b\right) ^{2n}}\) is real. Furthermore, since \(\frac{\partial ^2}{\partial x\partial y} \ln \left[ \frac{e^{2in\left( \sigma -\sigma ^*\right) }}{\left( 2b\right) ^{2n}}\right] =0\), the solution in Eq. (2.13) reduces to \(v\left( x,y,t\right) = -2\frac{\partial ^2}{\partial x\partial y}\ln \big (\det \left( AB\right) \big )\), where \(\det \left( AB\right) \) is a polynominal in *x*, *y*, *t*.

## 3 Classification of lump solutions of the DLWEs under 1-Step BDT

In the previous section, we have constructed an expression for the multi-lump solution of the DLWEs, and then, we will focus on the case of \(n = 1\), namely, the first-step Binary Darboux transform is applied.

### 3.1 Notation and its properties

To begin, we will introduce the notations used in this paper and discuss their properties. Let

where the first term of the subscript of \(Q_{N,N}\) represents the maximum index of \(p_n\) inside the parentheses, and the second term of the subscript represents the index of \(p_n^*\) outside the parentheses. As previously mentioned, \(p_n\) is a generalized Schur polynomial with respect to \(\sigma _1,\cdots ,\sigma _n\), and \(p_n^*\) is a polynomial with respect to \(\sigma _1^*,\cdots ,\sigma _n^*\), then we can get the derivative of \(Q_{N,N}\) with respect to \(\sigma _j, \sigma _j^*, \left( j=1, \cdots , N\right) \) and *x*, *y* combined with the \(C-R\) condition, respectively.

### Property 3.1

Properties of the polynomial \(Q_{N,N}\).

**(a)** The derivative of \(Q_{N,N}\) with respect to \(\sigma _j, \sigma _j^*,\) where \(j=1, \cdots , N\), has the following representation:

**(b)** The derivative of \(Q_{N,N}\) with respect to *x* and *y* is expressed as follows:

where the third term of the subscript of \(Q_{N,N,l}\) denotes the degree to which the polynomial \(Q_{N,N}\) takes the derivative with respect to *x*.

**(c)** The polynomial \(Q_{N,N}\) with shifted indices can be expressed as

### 3.2 Expression of lump solution under 1-step BDT

Based on the previous conversation, we now investigate the lump solutions of DLWEs after the first-step BDT. In this case \(n=1\) and let \(m_1=N\). From Eq. (2.15), we can write \(\phi _1\) and \(\phi _1^*\) as follows

so that *A* and *B* are both \(1 \times 1\) matrices whose corresponding elements can be obtained from Eqs. (2.16) and (2.17). Then the function *v* has the following form

The expression for \(F_N\) in Eq. (3.5) can be cast as a sum of squares of polynomials, that means

The above expression is a Hermitian form, thus

where *C* is a real, symmetric \(\left( 2N+1\right) \times \left( 2N+1\right) \) matrix that has a unique decomposition

Here *U* is a \(\left( 2N+1\right) \times \left( 2N+1\right) \) upper-triangular matrix with \(1^{\prime }\)s along its main diagonal and the corresponding elements are represented by \(U_{rs}\), and *D* is a diagonal matrix. Then \(F_N\) can be expressed as a sum of squares \(F_N = q^\textbf{H}Dq, q = Up\), which can be explicitly expressed as:

where we used the Property 3.1. Observe that \(F_N\) is a positive definite polynomial of order 4*N* in *x*, *y* and *t*, whose highest order term is \(\left| Q_{N,N}\right| ^2\). In addition, from Eq. (3.6), we see that \(F_N\) can be expressed as a formula related only to *b* and \(Q_{N,N}\), and further it can be viewed as \(F_N(\sigma _1,\cdots , \sigma _N,\sigma _1^*,\cdots ,\sigma _N^*, b)\) with \(wt\left( \sigma _j\right) =\) \(wt\left( \sigma _j^*\right) =j\), \(j=1, 2, \cdots , N\) and \(wt\left( b\right) =-1\). Finally, by substituting Eqs. (3.7) into (3.5), we get the lump solution of DLWEs

### 3.3 Examples of multi-lump solutions

In this subsection, we will provide several examples of multi-lump solutions of the DLWEs after 1-step BDT. However, to make the study easier, we will introduce the following coordinate system

Recall that \(a=Re\left( k_0\right) \) and \(b=Im\left( k_0\right) \), let \(a=\omega b\) and parameter *b* be a constant, so that \(k_0=b\left( \omega +i\right) \) can be obtained. By changing the value of \(\omega \), we can know the effects of \(k_0\) on DLWEs’ lump solutions. Thus, the variable \(\sigma _j\) in Eq. (2.6) becomes the following

where we can set \(\gamma _j=0\) for simplicity.

#### 3.3.1 2-lump solution for N=1

For \(N=1\), Eq. (3.7) gives the form of \(F_1\):

Combining Eqs (2.9), (3.1), (3.8) and (3.10), we can get the 2-lump solution of the DLWEs,

Note that the above expression is independent of *t*, which means that the 2-lump solution is static in the *rs*-plane. In addition, it can be seen from Eq. (3.9) that in the *xy*-plane, the displacement will occur with the change of *t*, and this waveform moves in the *xy*-plane with a velocity \(\big (\frac{b\left( \omega ^2+1\right) }{\omega }, \frac{1}{3b\omega }\big )\) at an angle \(tan^{-1}\big (3b^2\left( \omega ^2+1\right) \big )^{-1}\) with the positive *x*-axis.

From Fig. 1d, we can see a smaller bulge at the right of the highest peak and they form a 2-lump solution together. In contrast to the situation in Fig. 1d, the highest peak in Fig. 1f points in the negative direction of the *v*-axis, with a smaller depression on the right, which is also a peak pointing in the negative direction. In Fig. 1e, the state is between the above two, and it is in the stage of the peak direction from upward to downward.

#### 3.3.2 4-lump solution for N=2

For \(N=2\), Eq. (3.7) gives the form of polynomial \(F_2\), which consists of five terms:

By substituting the above expression into Eq. (3.8), the rational solution *v* of Eq. (1.1) is obtained when \(N=2\). Note that, unlike \(F_1\), the polynomial \(F_2\) depends on *t* and the 4-lump solutions are not stationary in the *rs*-plane and will change their shapes and positions as *t* change.

As shown in Fig. 2a–c, when \(\left| t\right| \gg 1, t<0\), the 4 lumps are divided into two groups and distributed along the *r*-axis. When \(t\rightarrow 0\), they attract each other and appear deflection after fusion near the origin, then they will mutually exclusive and separate from each other along the *s*-axis when \(t\gg 1\). Meanwhile, the positions of the four peaks can be simplified into the structure shown in Fig. 3. When \(t<0\), the positions of the four peaks appear as two rows parallel to the *s*-axis, and when \(t>0\), all the peaks are located on the *r*-axis. From Fig. 2d–f, we can see that with the increase of time *t*, the four peaks of the 4-lump solution gradually divide into two groups and move away from each other, which can be regarded as two independent 2-lump solutions and are similar to the solution obtained in the case of \(N=1\). As shown in Fig. 4, when \(0<\omega <1\), the peaks of the 4 lumps point in the positive direction of the *v*-axis, and when \(\omega >1\), the peaks point in the opposite direction.

To analyze the properties of lumps, we can further explore their peak locations and heights. Taking the 4-lump solution as an example, combined with the previous content, it may be considered as two separate groups of the 2-lump solution in the process of \(t\rightarrow \infty \), where each group contains two peaks and we define the higher peak as the principal peak and the other peak as the adjoint peak. Firstly, the expression of the 4-lump solution is written as

When \(F_2\) takes the minimum value, \(F_{2x}=0\) and \(F_{2y}=0\), we can get \(\left| v_2\right| = \left| 2\frac{F_{2xy}}{F_2}\right| \) for the maximum value which is the peak height. Secondly, \(F_2\) contains 5 terms and the first term (the term of \(j=0\)) plays a major role when \(\left| t\right| \gg 1\), the term in its module is denoted as \(l_0\). And then, by using the Eqs. (3.3a) and (3.4), we get

Set \(l_0=0\) and find its zeros, which are the approximate peak positions of the 4-lump solution. Through the above method, we obtain the principal peak positions of the 4-lump solution for \(\left| t\right| \gg 1\) as follows

By substituting the above into Eq. (3.14), the peak heights of the principal peaks can be obtained as

Because the distance between the adjacent peaks and the principal peaks are close and the height differences are large, the peak positions and peak heights of the adjacent peaks are not easy to approximate and need to be further studied. Figure 5 represents the evolution of peak positions and peak heights of the 4-lump solution under a set of specific values. It can be seen from the figure that the change of the positions of the principal peak is consistent with the time evolution process described before. In addition, we can also find that the peak heights of the principal peaks are time-dependent and will tend to the same constant value when \(\left| t\right| \rightarrow \infty \).

#### 3.3.3 6-lump solution for N=3

For \(N=3\), Eq. (3.7) gives the form of polynomial \(F_3\), which consists of seven modulo-square terms,

Substituting the above into Eq. (3.8), we can obtain the solution for \(N=3\), which also depends on *t*. As shown in Fig. 6a–c, with the change of *t* from \(-\infty \) to \(\infty \), the six peaks of the 6-lump solution were distributed parallel to the *s*-axis, attracted to each other and deflated near the origin of the coordinates, and were collinear and far away from each other on the *r*-axis, which was similar to the situation of 4-lump solution. However, the difference is that two of the peaks are always near the origin of the *r*-axis during the above process, and their positions do not change greatly. Figure 6d and e show the simple schematic diagram of 6-lump solution peaks in the *rs*-plane. In addition, when we take different values for \(\omega \), the 6-lump solution also has properties as shown in Fig. 4, which will not be shown here.

In the same way, we can get the peak positions of the principal peaks of 6-lump solution for \(\left| t\right| \gg 1\) as follows

By substituting the above coordinates into Eq. (3.14), we can get the evolution of the principal peaks.

## 4 2N-lump asymptotics of the DLWEs under 1-Step BDT

From the example of the 2*N*-lump solution given in Sect. 3 where \(N= 2, 3\), we can see that 2*N*-lump will disperse into *N* sets of independent 2-lump solutions as \(\left| t\right| \rightarrow \infty \), and each group can be regarded as an independent 2-lump solution when *t* is infinite. In addition, the evolution of peak positions evolve on the scale of \(\left| t\right| ^\frac{1}{2}\) and admit an asymptotic expansion as \(\left| t\right| \rightarrow \infty \) in the form of

where \(z_j\) is the \(j^{th}\) principal peak, and \(z_{j+N}\) is the \(j^{th}\) adjoint peak, \(j=1, 2, \cdots , N\).

### 4.1 Asymptotic peak locations

Using the correlation property of the polynomial \(Q_{N, N}\), namely Property 3.1, the \(l_0\) in Eq. (3.7) can be simplified as

where \({\tilde{\sigma }}_j=\sigma _j+h_j\) and the coefficient \(h_j\) can be calculated by Eqs. (3.3a) and (3.4). Taking the leading term \(l_0=0\) in \(F_N\), we can get \({\tilde{p}}_N=0\) and \({\tilde{p}}_{N-1}+k {\tilde{p}}_N=0\), the peak positions of principal peaks and adjacent peaks can be approximated by the zeros of these two equations, respectively.

First of all, recall from Eq. (3.10), the *t*-dependence in \({\tilde{p}}_N\) occurs via \(\sigma _2\) and \(\sigma _3\) which are linear in *t*. Then from Eq. (2.8),

since \(\sigma _j+h_j\) for \(j>3\) are constants. In order to get the highest order in the polynomial concerning *t*, we need to maximize \(m_2+m_3\) in the case of \(m_1+2m_2+3m_3=N\). Next, we will consider the scenarios where *N* is odd or even, respectively. For the generalized Schur polynomial \({\tilde{p}}_N\), when \(N=2m\), the maximization is \(\left( m_1, m_2, m_3\right) = \left( 0, m, 0\right) \), where the highest order of \(\left| t\right| \) is \(\left| t\right| ^m\) generated by \(\frac{1}{m!}\sigma _2^m\). When \(N=2m+1\), there are two groups of maximization, which are \(\left( m_1, m_2, m_3\right) = \left( 1, m, 0\right) \) and \(\left( m_1, m_2, m_3\right) = \left( 0, m-1, 1\right) \), respectively. The highest order of \(\left| t\right| \) corresponding to them is also \(\left| t\right| ^m\), which originates from the terms \(\frac{1}{m!}\sigma _1\sigma _2^m\) and \(\frac{1}{\left( m-1\right) !}\sigma _2^{m-1}\sigma _3\), respectively. Then the dominant balance required to solve for \(\sigma _1\) from the equation \({\tilde{p}}_N=0\) asymptotically for \(\left| t\right| \gg 1\) arises from the following two cases:

Case \(\left( a\right) \) gives the scaling , whereas case \(\left( b\right) \) implies \(\sigma _1\sim O\left( 1\right) \). Collecting the dominant terms of case \(\left( a\right) \) from \({\tilde{p}}_N\) which are , thus

with \(m_1+2m_2=N\). The second expression above is obtained by retaining the dominant terms in \(\sigma _1, \sigma _2\) from Eq. (3.10) and defining \(z\left( t\right) = r\left( t\right) + i s\left( t\right) \) which is . Similarly, we have the following expression for case \(\left( b\right) \)

where \(2m+1=N\), \({\tilde{\sigma }}_j=\sigma _j+h_j\), \(j=1, 2, 3\) and \(z\left( t\right) = r\left( t\right) -r_0+\frac{1}{2b}+ i\big (s\left( t\right) -s_0\big )\). Note that since \(\sigma _1\sim O\left( 1\right) \) in this case, all constant terms in the expression \(z\left( t\right) \) are preserved. For the case \(\left( b\right) \) above, where *N* is odd and \(z\left( t\right) \sim O\left( 1\right) \), we have \(z\left( t\right) \sim \frac{m}{3b}+O\big (\frac{1}{\left| t\right| }\big )\) from \({\tilde{p}}_N=0\) combined with Eq. (4.2b). The constants \(r_0\), \(s_0\) are determined by the parameter \(\gamma _1\). Therefore, when \(N=2m+1\), one of the peak positions for the 2*N*-lump can be obtained as

for \(\left| t\right| \gg 1\), which is the principal peak near the origin in the *rs*-plane.

Secondly, similar to the first step, we now analyze the formula \({\tilde{p}}_{N-1}+k {\tilde{p}}_N=0\). When \(N=2m\),

in the case of \(\left| t\right| ^{m-\frac{1}{2}}\), then we have

where scale \(\sigma _1\sim O\big (\left| t\right| ^{\frac{1}{2}}\big )\). For \(N=2m+1\), both \({\tilde{p}}_{N}\) and \({\tilde{p}}_{N-1}\) are \(O\big (\left| t\right| ^{m}\big )\) and the scaling is \(\sigma _1\sim O\left( 1\right) \), thus obtained that

Combining Eq. (4.3c), we can obtain \(z=\frac{m}{3b}+ik_0\) and thus

which is the peak position of the adjacent peak located at the center of the *rs*-plane when *N* is odd. The remaining positions of principal peaks and adjacent peaks which scale as can be obtained by Eqs. (4.2a), (4.3a) and (4.3b). In order to obtain the leading order contribution, we put \(z = \left| t\right| ^{\frac{1}{2}}\rho \) in Eq. (4.5a) to obtain

where \(H_N\left( \rho ,\nu \right) \) is the two-dimensional heat polynomial in \(\rho \) and \(\nu \) although here \(\nu \) plays the role as a parameter. The heat polynomial can be obtained via the generating function [33, 34, 54,55,56]

and can be rewritten as

### Property 4.1

Heat polynomials have the following properties for \(n\ge 1\) [53]

If we set \({\tilde{p}}_N=0\) in Eq. (4.2a), then the principal peak locations are given by

where \(\rho _j, j=1, 2, \cdots , N\), are the roots of \(H_N\) which depend only on \(\rho \).

### Property 4.2

When *N* is odd, \(\rho =0\) is a root of the heat polynomial \(H_{n}\left( \rho ,\nu \right) \). In addition, \(H_{n}\left( \rho ,\nu \right) \) has *N* distinct real roots for \(\nu <0\) and *N* distinct pure imaginary roots for \(\nu >0\) [55].

### Proposition 4.1

For \(\left| t\right| \gg 1\) and \(\omega >0\), the 2*N*-lump solutions of DLWEs have *N* distinct principal peaks along the *r*-axis located at

and along the *s*-axis at

for each \(j=1,2,\cdots ,N\).

In the above study for the principal peak, we analyze the terms of \(O\big (\left| t\right| ^{N}\big )\) in Eq. (4.2a), and our calculation for the peak positions can be more precise through the expansion \(\rho =\rho _0+\epsilon \rho _1+\cdots \), where \(\epsilon =\left| t\right| ^{-\frac{1}{2}}\). For solving the peak positions of adjacent peaks, we can use the similar method to calculate, but the calculation and analysis will be more complicated, and the relevant results will be presented in future work.

### 4.2 Asymptotic peak heights

We will now calculate the approximate peak heights of the *N* principal peaks for the 2*N*-lump solutions. The peak heights can be obtained by calculating \(u_j=u\big (z_j\left( t\right) \big )\), where \(z_j\left( t\right) \) is given by \({\tilde{p}}_N=0\) as the principal peak position already obtained in the previous subsection. We decompose \(F_N\) in Eq. (3.7) as \(F_N=l_0+l\) where *l* represents all square terms starting from \(j=1\). Then the potential function *v* at the principal peak position \(z_j(t)\) can be written as

Since \(l_0=0\) is the minimum value at \(z_j(t)\), there is \(l_{0x}\big (z_j\left( t\right) \big ) =l_{0y}\big (z_j\left( t\right) \big )=0\), \(j=1,\cdots ,N\), and the above equation is simplified as

Next, we will further expand *l* in the above formula. Combined with the approximation of generalized Schur polynomial \(p_N\) at asymptotic peaks \(z_j\left( t\right) \)

for any positive *N*, even or odd, and

when *N* is odd. The squared terms of \(j = 1\) and \(j = 2\) play a significant role in *l*, denoted as \(l_1\) and \(l_2\), respectively. The formula for *v* is given as

Similar to Eq. (4.1), \(l_1\) and \(l_2\) can expressed as polynomials with shifted parameters as

where \({\hat{\sigma }}_j=\sigma _j+c_j\), \({\dot{\sigma }}_j=\sigma _j+d_j\) and the shifts \(c_j\) and \(d_j\), \(j=1,\cdots ,N\) are determined by Eq. (3.4).

From Property 4.2, we take \(\omega >0\) as an example, that means, \(t>0\). In the previous subsection we showed that for \(\left| t\right| \gg 1\), \({\tilde{p}}_N\sim \left| t\right| ^ {\frac{N}{2}}H_N\left( \rho ,\nu \right) \) for any positive integer *N*. Thus,

where the shift \(\iota _1\), \(\varpi _1\) are obtained as follows. From Eq. (4.11), one can see that \(\sigma _1\) is shifted by different amounts, namely, \(h_1 = \frac{i}{2b}\), \(c_1 =\frac{i}{b}\) and \(d_1 = \frac{3i}{2b}\) in \({\tilde{p}}_N\), \({\hat{p}}_N\) and \({\dot{p}}_N\), respectively. This indicates that there is an additional shift \(z\left( t\right) \rightarrow z\left( t\right) +\frac{1}{2b}\) in \({\hat{p}}_N\), and another one \(z\left( t\right) \rightarrow z\left( t\right) +\frac{1}{b}\) in \({\dot{p}}_N\). Therefore, after the scaling \(z\left( t\right) =\left| t\right| ^\frac{1}{2}\rho \) in \({\hat{p}}_N\), we can get \(\rho \rightarrow \rho +\iota _1\), where \(\iota _1=\frac{\epsilon }{2b}, \epsilon =\left| t\right| ^{-\frac{1}{2}}\). Similarly, we have \(\varpi _1=\frac{\epsilon }{b}\).

Using Property 4.1 and the fact \(H_N\left( \rho _j\right) =0\), we can expand \(H_k\left( \rho _j+\iota _1\right) , k=N-2, N-1, N\) to obtain that

and \(H_k\left( \rho _j+\varpi _1\right) \) can have similar expansions after replacing \(\iota _1\) with \(\varpi _1\).

By substituting the above expressions into Eq.(4.9), the approximate peak heights of the *N* principal peaks can be obtained as

where \(j=1,2,\cdots ,N\). From the above results, note that for the central peak, when \(\rho _j = 0\), and the case of \(\omega ^2=1\), it is necessary to calculate the contribution of \(O\big (\epsilon ^2\big )\) for \(v_N\big (z_j\left( t\right) \big )\).

## 5 Conclusions

In this paper, through the study of multi-lump solutions of DLWEs, we have gotten the following results. Firstly, we have introduced the simple properties of Binary Darboux transform and generalized Schur polynomials, and on this basis, we have constructed multi-lump solutions for DLWEs. Further, taking \(n=1\) and \(m_n=N\), we have derived the representation of mult-lump solutions (Here are 2*N*-lump solutions) in the case of the first-step Binary Darboux transform. Secondly, when *N* was set to 1, 2, 3 respectively in the above formula, 2-lump, 4-lump, and 6-lump solutions have been obtained. Furthermore, we also have deduced the evolution law of the position and height of the peaks in these situations. Finally, through the analysis of the peak positions and peak heights of the 2*N*-lump solution, we have obtained that 2*N* lumps can be divided into *N* groups in the process of time evolution, and the positions and heights of the peaks are related to *t*, in which the peak position of the non-central peak has the scale as . To sum up, this study extensively analyzes the multi-lump solutions of DLWEs, especially in terms of time evolution, which expands our understanding and application of these solutions. However, there are still some unexplained aspects in this paper, such as the detailed discussion of the adjoint peaks of 2*N*-lump solutions and the forms of multi-lump solutions under higher-order BDT. These will be our future work, and we expect to have a richer understanding of the multi-lump solutions of DLWEs.

## Data availability

Enquiries about data availability should be directed to the authors.

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## Acknowledgements

We express our sincere thanks to each member of our discussion group for their suggestions. This work has been supported by the Shanxi Province Science Foundation under Grant No. 202303021221031, and the Fund Program for the Scientific Activities of Selected Returned Overseas Scholars in Shanxi Province under Grant No. 20220008.

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Yong-Ning An: Conceptualization (equal), Formal analysis (equal), investigation (equal), Methodology (equal), Software (equal),Validation (equal), Writing - original draft (lead); Yan-Nan Zhao: Formal analysis (equal), Software (equal); Hui-Qin Hao:Conceptualization (equal),Formal analysis (equal),Funding acquisition (lead), nvestigation (equal), Methodology(equal),Project administration (lead),Supervision (lead), Validation (equal),Writing- review & editing(lead).

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An, YN., Zhao, YN. & Hao, HQ. Analysis on construction and evolution dynamics for multi-lump solutions of the dispersive long wave equations.
*Nonlinear Dyn* **112**, 18473–18489 (2024). https://doi.org/10.1007/s11071-024-09975-0

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DOI: https://doi.org/10.1007/s11071-024-09975-0