Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation

Dean Chou; Salah Mahmoud Boulaaras; Ifrah Iqbal; Hamood Ur Rehman; Tsi-Li Li

doi:10.1515/nleng-2024-0033

Article Open Access

Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation

Dean Chou , Salah Mahmoud Boulaaras , Ifrah Iqbal , Hamood Ur Rehman and Tsi-Li Li

Published/Copyright: February 10, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

Ultrasound imaging stands as a cornerstone of modern medical diagnostics, revolutionising clinical practice with its non-invasive, real-time visualisation of internal structures. Central to this technique is the propagation of ultrasound waves and their intricate interplay with biological tissues, culminating in the generation of intricate and detailed images. This study delves into the symbiotic relationship between solitons and ultrasound imaging within the framework of the Westervelt equation, a fundamental model governing ultrasound propagation. Employing the generalised Riccati equation mapping method and the generalised exponential rational function method, a diverse array of soliton solutions is elucidated, encompassing dark, kink, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Visualisation of these solutions through 3D plots, contour plots, and 2D plots at varying time intervals offers a captivating insight into their dynamic nature. We provide a comparison of these solutions through 2D plots at different parameter values, highlighting their varying impacts. Central to this study is the exploration of how these soliton solutions can be harnessed to enhance the quality and accuracy of ultrasound images in medical imaging. Through meticulous analysis of their characteristics, this research seeks to illuminate their potential applications, paving the way for a new era of precision diagnostics in healthcare. By conducting thorough mathematical analyses and numerical simulations, we seek to elucidate the complex relationship between soliton theory and ultrasound imaging, connecting the theoretical aspects of nonlinear wave phenomena with their practical applications in medical diagnostics. An intensive literature review underscores the novelty of our work.

Keywords: ultrasound imaging; Westervelt equation; generalised Riccati equation mapping method; soliton solutions; nonlinear equations

1 Introduction

In recent years, the exploration of nonlinear partial differential equations (NLPDEs) has emerged as a cornerstone in advancing scientific inquiry and technological innovation across various disciplines. NLPDEs such as potential Kadomtsev–Petviashvili [1], complex coupled Kuralay system [2], nonlinear Schrödinger equation [3], Sasa–Satsuma equation [4], and stochastic Chen–Lee–Liu equation [5] all contribute to our understanding of complex dynamic systems. Within the realm of medical imaging, particularly in ultrasound imaging, NLPDEs have become indispensable tools, facilitating nuanced investigations into the complex phenomena of wave propagation and its intricate interactions with biological tissues. This interdisciplinary convergence underscores the profound impact of theoretical advancements on practical applications, demonstrating the transformative potential inherent in nonlinear ultrasound wave theory.

The elucidation of NLPDEs has been instrumental in deepening our understanding of complex physical systems, with far-reaching implications across diverse fields such as acoustics, optics, fluid dynamics, and so on [6–9]. In the context of medical imaging, these mathematical frameworks provide the foundation upon which sophisticated models for the propagation of nonlinear ultrasound waves are built, contributing significantly to diagnostic and therapeutic modalities within modern medical practice. Noteworthy advancements have been made in refining ultrasonic imaging techniques, particularly in tumour and organ imaging [10,11], reflecting the ongoing evolution of ultrasound technology in addressing clinical challenges.

Ultrasound imaging’s established utility lies in its non-invasive capability to generate high-resolution images of internal tissues, offering clinicians invaluable insights into anatomical structures and pathological conditions. This diagnostic modality relies on the emission and reception of high-frequency sound waves, whose propagation characteristics are intricately governed by nonlinear wave equations. Of particular interest within this domain are solitons, solitary wave solutions characterised by their unique stability properties, which remain undistorted during transmission through a medium. The inherent stability of solitons confers distinct advantages in imaging applications, notably enhancing resolution and facilitating precise spatial localisation of ultrasound energy delivery.

Central to the theoretical underpinnings of ultrasound propagation is the Westervelt equation [12], a seminal contribution introduced by Westervelt in 1963 [13]. This equation, incorporating both loss and inertial terms, provides a versatile framework for modelling ultrasound propagation phenomena across various media, including biological tissues [14], acoustic cavitation [15], underwater acoustics [16], and acoustic levitation [17]. Within the realm of medical studies, ultrasound imaging technology enables the visualisation of human body tissues [18], contributing to numerous industrial and medical applications. High-intensity focused ultrasound represents a technique employing high-energy sound waves directed at abnormal tissues to induce their necrosis. This method is employed in the treatment of conditions such as tumours, uterine fibroids, and tremors, using ultrasound or thermotherapy, welding, lithotripsy, and sonochemistry. Within the context of medical imaging, the acoustic nonlinearity parameter tomography [19] serves as a coefficient, facilitating the description of high-intensity ultrasound [20]. The pressure formulation of the Westervelt equation involves the acoustic pressure u , sound diffusivity b , a constant c , and K , where K is defined as β α λ and β α = 1 + C ∕ 2 D as the nonlinear parameter. This equation is expressed as follows [21]:

(1) 2 K ( u u t t + u t 2 ) + c 2 u x x + b u x x t − u t t = 0 .

Translating theoretical insights into practical applications necessitates developing analytical and numerical techniques for solving NLPDEs. A plethora of methodologies, ranging from classical approaches to innovative strategies, have been devised for this purpose, including the extended hyperbolic function method [22–26], Sardar sub-equation method [27,28], Kudryashov’s method [29,30], Bernoulli sub-ODE method [31,32], Jacobi’s elliptic function method [33,34], new extended direct algebraic method [35,36], mapping method [37–40], and ϕ 6 -expansion method [41]. Each methodology offers distinct advantages in addressing the complexities inherent in nonlinear wave equations, thereby facilitating the derivation of soliton solutions and enhancing understanding of wave propagation phenomena.

In this study, we employ the generalised Riccati equation mapping method (GREMM) and the generalised exponential rational function method (GERFM) to derive solutions for the Westervelt equation, elucidating the diverse soliton solutions encompassed therein. Through rigorous mathematical analyses and numerical simulations, our aim is to unravel the intricate interplay between soliton theory and ultrasound imaging, bridging the theoretical foundations of nonlinear wave phenomena with their practical applications in medical diagnostics. Subsequent sections will delineate the methodologies employed, the results obtained, and the ensuing discussion, culminating in a comprehensive analysis of our findings and their implications for advancing ultrasound technology in medical practice.

2 Overview of GREMM

This section provides a comprehensive elucidation of the GREMM. Let us consider a NLPDE of the form:

(2) Q ( u , u x , u t , u x x , … ) = 0 ,

where Q represents a polynomial in u ( x , t ) and its derivatives. Employing a wave transformation, we can convert the NLPDE into an ODE as follows:

(3) u ( x , t ) = u ( ζ ) and ζ = α x + ε t .

Subsequently, (3) reduces to

(4) P ( u , u ′ , u ″ , u ‴ , … ) = 0 ,

where P is a polynomial in u ( ζ ) and its derivatives. Assuming the solution of (4) takes the form:

(5) u ( ζ ) = ω 0 + ∑ i = 1 N ω i G ( ζ ) i ,

where G ( ζ ) satisfies the following ODE:

(6) G ′ = q 1 + q 2 G + q 3 G 2 ,

and ω 0 , ω i , q 1 , q 2 , q 3 ( i = 1 , 2 , … , N ) represent the constants that require evaluation. The positive integer N can be determined through a balancing procedure. Substituting (5) and (6) into (4), we obtain a system of algebraic equations. Solving this system yields the values of the constants. Now, employing these constants alongside the solutions of equation (6), we derive the following solutions:

Family 1: When q 2 2 − 4 q 1 q 3 > 0 ,

G 1 = − 1 2 q 3 q 2 + q 2 2 − 4 q 3 q 1 tanh q 2 2 − 4 q 3 q 1 2 ζ , G 2 = − 1 2 q 3 q 2 + q 2 2 − 4 q 3 q 1 coth q 2 2 − 4 q 3 q 1 2 ζ ,

G 3 = − 1 2 q 3 ( q 2 + q 2 2 − 4 q 3 q 1 ( tanh ( q 2 2 − 4 q 3 q 1 ζ ) ± s e c h ( q 2 2 − 4 q 3 q 1 ζ ) ) ) ,

G 4 = − 1 2 q 3 ( q 2 + q 2 2 − 4 q 3 q 1 ( coth ( q 2 2 − 4 q 3 q 1 ζ ) ± c s c h ( q 2 2 − 4 q 3 q 1 ζ ) ) ) ,

G 5 = − 1 4 q 3 2 q 2 + q 2 2 − 4 q 3 q 1 tanh q 2 2 − 4 q 3 q 1 4 ζ ± coth q 2 2 − 4 q 3 q 1 4 ζ ,

G 6 = 1 2 q 3 − q 2 + ( A 2 + B 2 ) ( q 2 2 − 4 q 3 q 1 ) − A q 2 2 − 4 q 3 q 1 cosh ( q 2 2 − 4 q 3 q 1 ζ ) A sinh ( q 2 2 − 4 q 3 q 1 ζ ) + B ,

G 7 = 1 2 q 3 − q 2 − ( A 2 + B 2 ) ( q 2 2 − 4 q 3 q 1 ) + A q 2 2 − 4 q 3 q 1 cosh ( q 2 2 − 4 q 3 q 1 ζ ) A sinh ( q 2 2 − 4 q 3 q 1 ζ ) + B ,

where A , B ≠ 0 represent the arbitrary constants satisfying the condition A 2 > B 2 ,

G 8 = 2 q 3 cosh q 2 2 − 4 q 3 q 1 2 ζ − q 2 cosh q 2 2 − 4 q 3 q 1 2 ζ + q 2 2 − 4 q 3 q 1 sinh q 2 2 − 4 q 3 q 1 2 ζ ,

G 9 = − 2 q 3 sinh q 2 2 − 4 q 3 q 1 2 ζ q 2 sinh q 2 2 − 4 q 3 q 1 2 ζ − q 2 2 − 4 q 3 q 1 cosh q 2 2 − 4 q 3 q 1 2 ζ ,

G 10 = 2 q 3 cosh q 2 2 − 4 q 3 q 1 2 ζ q 2 2 − 4 q 3 q 1 sinh ( q 2 2 − 4 q 3 q 1 ζ ) − q 2 cosh ( q 2 2 − 4 q 3 q 1 ζ ) ± ι q 2 2 − 4 q 3 q 1 ,

G 11 = 2 q 3 sinh q 2 2 − 4 q 3 q 1 2 ζ − q 2 sinh ( q 2 2 − 4 q 3 q 1 ζ ) + q 2 2 − 4 q 3 q 1 cosh ( q 2 2 − 4 q 3 q 1 ζ ) ± ι q 2 2 − 4 q 3 q 1 ,

G 12 = 4 q 3 sinh q 2 2 − 4 q 3 q 1 4 ζ cosh q 2 2 − 4 q 3 q 1 4 ζ − 2 q 2 sinh q 2 2 − 4 q 3 q 1 4 ζ cosh q 2 2 − 4 q 3 q 1 ζ 4 + 2 q 2 2 − 4 q 3 q 1 cosh 2 q 2 2 − 4 q 3 q 1 ζ 4 − q 2 2 − 4 q 3 q 1 .

Family 2: When q 2 2 − 4 q 1 q 3 < 0 ,

G 13 = 1 2 q 3 − q 2 + 4 q 3 q 1 − q 2 2 tan 4 q 3 q 1 − q 2 2 2 ζ ,

G 14 = − 1 2 q 3 q 2 + 4 q 3 q 1 − q 2 2 cot 4 q 3 q 1 − q 2 2 2 ζ ,

G 15 = 1 2 q 3 ( − q 2 + 4 q 3 q 1 − q 2 2 ( tan ( 4 q 3 q 1 − q 2 2 ζ ) ± sec ( 4 q 3 q 1 − q 2 2 ζ ) ) ) ,

G 16 = − 1 2 q 3 ( q 2 + 4 q 3 q 1 − q 2 2 ( cot ( 4 q 3 q 1 − q 2 2 ζ ) + csc ( 4 q 3 q 1 − q 2 2 ζ ) ) ) ,

G 17 = 1 4 q 3 − 2 q 2 + 4 q 3 q 1 − q 2 2 tan 4 q 3 q 1 − q 2 2 4 ζ − cot 4 q 3 q 1 − q 2 2 4 ζ ,

G 18 = 1 2 q 3 − q 2 + ± ( A 2 − B 2 ) ( 4 q 3 q 1 − q 2 2 ) − A 4 q 3 q 1 − q 2 2 cos ( 4 q 3 q 1 − q 2 2 ζ ) A sin ( 4 q 3 q 1 − q 2 2 ζ ) + B ,

G 19 = 1 2 q 3 − q 2 − ± ( A 2 − B 2 ) ( 4 q 3 q 1 − q 2 2 ) + A 4 q 3 q 1 − q 2 2 cos ( 4 q 3 q 1 − q 2 2 ζ ) A sin ( 4 q 3 q 1 − q 2 2 ζ ) + B ,

where A , B ≠ 0 represent the arbitrary constants and satisfy the condition A 2 > B 2 ,

G 20 = − 2 q 3 cos 4 q 3 q 1 − q 2 2 2 ζ q 1 c o s 4 q 3 q 1 − q 2 2 2 ζ + 4 q 3 q 1 − q 2 2 sin 4 q 3 q 1 − q 2 2 2 ζ ,

G 21 = 2 q 3 sin 4 q 3 q 1 − q 2 2 2 ζ q 1 sin 4 q 3 q 1 − q 2 2 2 ζ + 4 q 3 q 1 − q 2 2 cos 4 q 3 q 1 − q 2 2 2 ζ .

G 22 = − 2 q 3 cos 4 q 3 q 1 − q 2 2 2 ζ 4 q 3 q 1 − q 2 2 sin ( 4 q 3 q 1 − q 2 2 ζ ) + q 1 cos ( 4 q 3 q 1 − q 2 2 ζ ) ± ι 4 q 3 q 1 − q 2 2 ,

G 23 = 2 q 3 sin 1 2 4 q 3 q 1 − q 2 2 ζ − q 2 sin ( 4 q 3 q 1 − q 2 2 ζ ) − 4 q 3 q 1 − q 2 2 cos ( 4 q 3 q 1 − q 2 2 ζ ) ± ι 4 q 3 q 1 − q 2 2 ,

G 24 = 4 q 3 sin 4 q 3 q 1 − q 2 2 4 ζ cos 4 q 3 q 1 − q 2 2 4 ζ − 2 q 2 sin 4 q 3 q 1 − q 2 2 4 ζ cos 4 q 3 q 1 − q 2 2 ζ 4 + 2 4 q 3 q 1 − q 2 2 cos 2 4 q 3 q 1 − q 2 2 ζ 4 − 4 q 3 q 1 − q 2 2 .

Family 3: When q 3 = 0 and q 2 q 1 ≠ 0 ,

G 25 = − q 2 d q 1 ( d + cosh ( q 2 ζ ) − sinh ( q 2 ζ ) ) , G 26 = q 2 ( cosh ( q 2 ζ ) + sinh ( q 2 ζ ) ) q 1 ( d + cosh ( q 2 ζ ) + sinh ( q 2 ζ ) ) ,

where d is any arbitrary constant.

Family 4: When q 2 = 0 , q 3 = 0 and q 1 ≠ 0 ,

G 27 = − 1 c 1 + q 1 ζ ,

where c 1 is an arbitrary constant By inserting these cases along the values of constant in (5), we can obtain the solutions of (1).

2.1 Application of GREMM

By substituting (3) into (1), we obtain the following equation:

(7) ε 2 ( 2 K u u ″ + 2 K ( u ′ ) 2 − u ″ ) + α 2 ε b u ‴ + α 2 c 2 u ″ = 0 .

By applying the homogeneous balancing N = 1 on (7), we obtain the following solution:

(8) u ( ζ ) = ω 0 + ω 1 G ( ζ ) .

Now, by substituting (8) into (6), we obtain the following values of constant:

K = 1 2 , ω 1 = − 2 α 2 b q 3 ε , and ω 0 = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 .

Now, employing these constants alongside the solutions of equation (6), we derive the following solutions:

Family 1: When γ = q 2 2 − 4 q 1 q 3 > 0 ,

u 1 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b ε q 2 + γ tanh γ 2 ζ ,

u 2 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b ε q 2 + γ coth γ 2 ζ ,

u 3 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b q 2 ε q 3 ( q 2 + γ ( tanh ( γ ζ ) ± sech ( γ ζ ) ) ) ,

u 4 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b ε × ( q 2 + γ ( coth ( γ ζ ) ± csch ( γ ζ ) ) ) ,

u 5 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b 2 ε × 2 q 2 + γ tanh γ 4 ζ ± coth γ 4 ζ ,

u 6 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b 2 ε × − q 2 + ( A 2 + B 2 ) γ − A γ cosh ( γ ζ ) A sinh ( γ ζ ) + B ,

u 7 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b 2 ε × − q 2 − ( A 2 + B 2 ) γ + A γ cosh ( γ ζ ) A sinh ( γ ζ ) + B ,

where A , B ≠ 0 are the arbitrary constants satisfying the condition A 2 < B 2 . Additionally,

u 8 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 cosh γ 2 ζ − q 2 cosh γ 2 ζ + γ sinh γ 2 ζ ,

u 9 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × − 2 q 3 sinh γ 2 ζ q 2 sinh γ 2 ζ − γ cosh γ 2 ζ ,

u 10 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 cosh γ 2 ζ γ sinh ( γ ζ ) − q 2 cosh ( γ ζ ) ± ι γ ,

u 11 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 sinh γ 2 ζ − q 2 sinh ( γ ζ ) + γ cosh ( γ ζ ) ± ι γ ,

u 12 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 4 q 3 sinh γ 4 ζ cosh γ 4 ζ − 2 q 2 sinh γ 4 ζ cosh γ ζ 4 + 2 γ cosh 2 γ ζ 4 − γ .

Family 2: When γ = q 2 2 − 4 q 1 q 3 < 0 ,

u 13 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b ε × q 2 + − γ tan − γ 2 ζ ,

u 14 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b ε × q 2 + − γ cot − γ 2 ζ ,

u 15 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b ε × ( q 2 + − γ ( tan ( − γ ζ ) ± sec ( − γ ζ ) ) ) ,

u 16 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b ε × ( q 2 + − γ ( cot ( − γ ζ ) ± csc ( − γ ζ ) ) ) ,

u 17 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 + α 2 b 2 ε × 2 q 2 + − γ tan − γ 4 ζ ± cot − γ 4 ζ ,

u 18 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b 2 ε × − q 2 + − ( A 2 + B 2 ) γ − A − γ cos ( − γ ζ ) A sin ( − γ ζ ) + B ,

u 19 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − α 2 b 2 ε × − q 2 − − ( A 2 + B 2 ) γ + A − γ cos ( − γ ζ ) A sin ( − γ ζ ) + B ,

where A , B ≠ 0 are the arbitrary constants satisfying the condition A 2 > B 2 ,

u 20 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 cos − γ 2 ζ q 1 cosh − γ 2 ζ + − γ sin − γ 2 ζ ,

u 21 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 sin − γ 2 ζ − q 1 sin − γ 2 ζ + − γ cos − γ 2 ζ ,

u 22 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × − 2 q 3 cos − γ 2 ζ − γ sin ( − γ ζ ) − q 1 cos ( − γ ζ ) ± ι − γ ,

u 23 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 2 q 3 sin − γ 2 ζ − q 2 sin ( − γ ζ ) + − γ cos ( − γ ζ ) ± ι − γ ,

u 24 ( x , t ) = − α 2 b ε q 2 − α 2 c 2 + ε 2 ε 2 − 2 α 2 b q 3 ε × 4 q 3 sin − γ 4 ζ cos − γ 4 ζ − 2 q 2 sin − γ 4 ζ cos − γ ζ 4 + 2 − γ cos 2 − γ ζ 4 − − γ .

Special case: If we take q 2 = 0 and q 3 = 1 , then (6) changed into [42]

(9) G ′ = q 1 + G 2 .

Using (9), we have the following solutions:

ω 0 = ε 2 − α 2 c 2 ε 2 , ω 1 = − 2 α 2 b ε , K = 1 2

The obtained solutions from the aforementioned values of constants are

Case 1: If q 1 < 0 , then

u 1 , * ( x , t ) = ε 2 − α 2 c 2 ε 2 + 2 α 2 b ε ( − − q 1 tanh ( − q 1 ζ ) ) , u 2 , * ( x , t ) = ε 2 − α 2 c 2 ε 2 + 2 α 2 b ε ( − − q 1 coth ( − q 1 ζ ) ) .

Case 3: If q 1 > 0 , then

u 3 , * ( x , t ) = ε 2 − α 2 c 2 ε 2 + 2 α 2 b ε ( q 1 tan ( q 1 ζ ) ) , u 4 , * ( x , t ) = ε 2 − α 2 c 2 ε 2 + 2 α 2 b ε ( q 1 cot ( q 1 ζ ) ) .

Case 3: If q 1 = 0 , then

u 5 , * ( x , t ) = ε 2 − α 2 c 2 ε 2 + 2 α 2 b ε 1 ζ .

3 GERFM

Let us assume that (4) has the subsequent solution:

(10) u ( ζ ) = a 0 + ∑ i = 1 n a i F ′ ( ζ ) F ( ζ ) n + ∑ i = 1 n b i F ′ ( ζ ) F ( ζ ) − n ,

where a 0 , a i , b i (for i = 1 , 2 , … , n ), and F ( ζ ) satisfy the following ordinary differential equation:

(11) F ( ζ ) = g 1 exp ( h 1 ζ ) + g 2 exp ( h 2 ζ ) g 3 exp ( h 3 ζ ) + g 4 exp ( h 4 ζ ) .

To embark on this investigation, it is crucial to ascertain the values of several coefficients: a 0 , a i , b i (where i = 1 , 2 , … , n ), g j , and h j (where 1 ≤ j ≤ 4 ). These coefficients will be determined based on the assumed structures (10) and (11). To determine the value of the positive integer n , we employ the homogeneous balancing rule. By substituting (10) into (4), we obtain an algebraic system, and by solving this system, the values of the system can be obtained.

3.1 Application of GERFM

Using the homogeneous balancing rule, we find n = 1 . Let us assume the following solution:

(12) u ( ζ ) = a 0 + a 1 F ′ ( ζ ) F ( ζ ) + b 1 F ′ ( ζ ) F ( ζ ) − 1 .

Case 1: By setting [ g 1 , g 2 , g 3 , g 4 ] = [ 1 , − 1 , 2 , 0 ] and [ h 1 , h 2 , h 3 , h 4 ] = [ 1 , − 1 , 0 , 0 ] in (3), we obtain

(13) F ( ζ ) = sinh ( ζ ) .

Substituting (12) along with (13) into (7), we obtain the following sets of constants:

(14) Set 1 K = 1 2 , a 1 = 0 , b 1 = 2 α 2 b ε , a 0 = ε 2 − α 2 c 2 ε 2 .

(15) Set 2 K = 1 2 , a 1 = 2 α 2 b ε , b 1 = 2 α 2 b ε , a 0 = ε 2 − α 2 c 2 ε 2 .

(16) Set 3 K = 1 2 , a 1 = 0 , b 1 = 2 α 2 b ε , a 0 = ε 2 − α 2 c 2 ε 2 .

The solutions obtained from (14), (15), and (16) are given, respectively, as

u 2 , 1 ( x , t ) = 2 α 2 b coth ( ζ ) ε + ε 2 − α 2 c 2 ε 2 ,

u 2 , 2 ( x , t ) = 2 α 2 b ε tanh ( ζ ) + 2 α 2 b ε coth ( ζ ) − α 2 c 2 + ε 2 ε 2 ,

u 2 , 3 ( x , t ) = 2 α 2 b ε tanh ( ζ ) − α 2 c 2 + ε 2 ε 2 .

Case 2: By setting [ g 1 , g 2 , g 3 , g 4 ] = [ 2 , 0 , 1 , 1 ] and [ h 1 , h 2 , h 3 , h 4 ] = [ − 2 , 0 , 1 , − 1 ] in (3), we derive

(17) F ( ζ ) = sech ( ζ ) exp ( − 2 ζ ) .

Substituting (12) along with (17) into (7), we obtain the following sets of constants:

(18) Set 1 K = 1 2 , b 1 = 0 , a 1 = − 2 α 2 b ε , a 0 = − 4 α 2 b ε − α 2 c 2 + ε 2 ε 2 ,

(19) Set 2 K = 1 2 , a 1 = 0 , b 1 = 6 α 2 b ε , a 0 = 4 α 2 b ε − α 2 c 2 + ε 2 ε 2 .

The solutions obtained from (18) and (19) are given, respectively, as

u 2 , 4 ( x , t ) = 2 α 2 b ε tanh ( ζ ) − α 2 c 2 + ε 2 ε 2 ,

u 2 , 5 ( x , t ) = tanh ( ζ ) ( 4 α 2 b ε − α 2 c 2 + ε 2 ) + 2 ( α 2 b ε − α 2 c 2 + ε 2 ) ε 2 ( tanh ( ζ ) + 2 ) .

Case 3: By setting [ g 1 , g 2 , g 3 , g 4 ] = [ 2 , 0 , 1 , − 1 ] and [ h 1 , h 2 , h 3 , h 4 ] = [ 0 , 0 , 1 , − 1 ] in (3), we derive:

(20) F ( ζ ) = cosh ( ζ ) .

Substituting (12) along with (20) into (7), we obtain the following sets of constants:

(21) Set K = 1 2 , b 1 = 0 , a 1 = − 2 α 2 b ε , a 0 = ε 2 − α 2 , c 2 ε 2 .

The solution obtained from the aforementioned set is

u 2 , 6 ( x , t ) = 2 α 2 b ε coth ( ζ ) − α 2 c 2 + ε 2 ε 2 .

Case 4: By setting [ g 1 , g 2 , g 3 , g 4 ] = [ 1 , 1 , 2 , 0 ] and [ h 1 , h 2 , h 3 , h 4 ] = [ ι , − ι , 0 , 0 ] in (3), we find

(22) F ( ζ ) = cos ( ζ ) .

Substituting (12) into (22) and then into (7), we obtain the following sets of constants:

(23) Set 1 K = 1 2 , a 1 = 0 , b 1 = − 2 α 2 b ε , a 0 = ε 2 − α 2 c 2 ε 2 ,

(24) Set 2 K = 1 2 , a 1 = 2 α 2 b ε , b 1 = 0 , a 0 = ε 2 − α 2 c 2 ε 2 ,

(25) Set 3 K = 1 2 , a 1 = 2 α 2 b ε , b 1 = − 2 α 2 b ε , a 0 = ε 2 − α 2 c 2 ε 2 .

The solutions obtained from (23), (24), and (25) are given, respectively, as

u 2 , 7 ( x , t ) = 2 α 2 b ε cot ( ζ ) − α 2 c 2 + ε 2 ε 2 ,

u 2 , 8 ( x , t ) = − 2 α 2 b ε tan ( ζ ) − α 2 c 2 + ε 2 ε 2 ,

u 2 , 9 ( x , t ) = − 2 α 2 b ε tan ( ζ ) + 2 α 2 b g cot ( ζ ) − α 2 c 2 + ε 2 ε 2 .

Case 5: By setting [ g 1 , g 2 , g 3 , g 4 ] = [ 2 , 3 , 2 , 2 ] and [ h 1 , h 2 , h 3 , h 4 ] = [ 2 ∕ 5 , 0 , 0 , 0 ] in (3), we find

(26) F ( ζ ) = 1 4 2 e 2 ζ 5 + 3 .

Substituting (12) into (26) and then into (7), we obtain the following sets of constants:

(27) Set 1 K = − 1 2 , a 1 = 0 , a 0 = − 4 α 2 b ε + 10 α 2 c 2 − 25 ε 2 b 1 − 10 ε 2 10 ε 2 ,

(28) Set 2 K = 1 2 , a 1 = 2 α 2 b ε , b 1 = 0 , a 0 = − 2 α 2 b ε − 5 α 2 c 2 + 5 ε 2 5 ε 2 .

The solutions obtained from (27) and (28) are given, respectively, as

u 2 , 8 ( x , t ) = − 2 α 2 b 5 ε + α 2 c 2 ε 2 + 15 4 e − 2 ζ 5 b 1 − 1 ,

u 2 , 9 ( x , t ) = − 2 α 2 b 2 e 2 ζ 5 − 3 5 ε 2 e 2 ζ 5 + 3 − α 2 c 2 ε 2 + 1 .

4 Results and discussion

The Westervelt equation stands as a pivotal cornerstone in the realm of ultrasound imaging, offering profound insights into the physics governing ultrasound wave propagation and its interactions with biological tissues. Through the application of analytical methodologies such as GREMM and GERFM, a diverse array of soliton solutions has been unveiled. These soliton solutions, ranging from dark to periodic singular solitons, embody distinctive characteristics that hold promise for enhancing the accuracy and quality of ultrasound imaging. In the GERFM, we can obtain hyperbolic, rational, and trigonometric solutions by varying the constraint conditions, and there are no inherent limitations. After applying this method, we have determined whether it is applicable. In contrast, the GREMM provides 27 different types of solutions but fails to construct bright soliton solutions. Furthermore, neither of these methods yields algebraic solutions.

Our research delves into the relationship between soliton solutions and their potential applications in the field of medical imaging. By analysing the spatial and temporal behaviours of these solitons, we aim to decipher their role in detecting and characterising abnormalities or diseases within biological tissues. Through the visualisation of soliton solutions via 3D plots, contour plots, and 2D plots at different time intervals, we gain valuable insights into their dynamic nature and their potential utility in clinical practice.

The 2D plots, in particular, reveal intriguing observations regarding the spatial positioning of solitons over time. As the value of the temporal parameter changes, the position of the soliton undergoes shifts to either the left or the right along the x -axis. This dynamic behaviour underscores the potential for precise spatial localisation of ultrasound energy delivery, which is a crucial aspect in targeting specific regions of interest within biological tissues during imaging procedures.

Furthermore, the ability to manipulate the temporal parameter offers opportunities for optimising imaging techniques. By strategically adjusting the value of time, we can precisely position the soliton to interact with specific regions of interest within the imaged tissue, thereby enhancing the efficacy and accuracy of ultrasound imaging procedures.

Detailed analysis of various soliton solutions, including kink, combined dark–bright, combined dark-singular, periodic singular, and singular solitons, sheds light on their potential applications in ultrasound imaging. Each soliton type exhibits unique characteristics that could be leveraged to overcome existing challenges in medical imaging, such as improving resolution, reducing artefacts, and enhancing diagnostic accuracy.

Figure 1 illustrates the kink-type soliton for u 1 ( x , t ) , showcasing its distinct profile under specific parameter values. Similarly, Figures 2 and 3 depict the combined-dark bright and combined-dark singular solitons for u 3 ( x , t ) and u 5 ( x , t ) , respectively, offering insights into their spatial distributions and temporal behaviours. Moreover, Figure 4 presents the singular soliton solution for u 9 ( x , t ) , highlighting its unique characteristics under specified parameter conditions.

$Figure 1 Graphical representation of kink soliton for u 1 ( x , t ) {u}_{1}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 1

Graphical representation of kink soliton for u 1 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 2 Graphical representation of combined dark–bright soliton for u 3 ( x , t ) {u}_{3}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 2

Graphical representation of combined dark–bright soliton for u 3 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 3 Graphical representation of combined dark-singular soliton for u 5 ( x , t ) {u}_{5}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 3

Graphical representation of combined dark-singular soliton for u 5 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 4 Graphical representation of singular soliton for u 9 ( x , t ) {u}_{9}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 4

Graphical representation of singular soliton for u 9 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

Figures 5 and 6 provide visual representations of the periodic singular solitons for u 13 ( x , t ) and u 17 ( x , t ) , respectively, showcasing their periodic nature and spatial distributions over time. Additionally, Figures 7, 8, and 9 present dark, singular, and periodic singular soliton solutions for the solutions u 2 , 3 ( x , t ) , u 2 , 6 ( x , t ) , and u 2 , 7 ( x , t ) , respectively, under specified parameter values.

$Figure 5 Graphical representation of periodic singular soliton for u 13 ( x , t ) {u}_{13}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 5

Graphical representation of periodic singular soliton for u 13 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 6 Graphical representation of periodic singular soliton for u 17 ( x , t ) {u}_{17}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 6

Graphical representation of periodic singular soliton for u 17 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 7 Graphical representation of dark soliton for u 2 , 3 ( x , t ) {u}_{2,3}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 7

Graphical representation of dark soliton for u 2 , 3 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 8 Graphical representation of singular soliton for u 2 , 6 ( x , t ) {u}_{2,6}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 8

Graphical representation of singular soliton for u 2 , 6 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

$Figure 9 Graphical representation of periodic-singular soliton for u 2 , 7 ( x , t ) {u}_{2,7}\left(x,t) via 3D (a), contour (b), and 2D plots (c).$

Figure 9

Graphical representation of periodic-singular soliton for u 2 , 7 ( x , t ) via 3D (a), contour (b), and 2D plots (c).

Through these visualisations and analyses, we gain a deeper understanding of the complex interplay between soliton theory and ultrasound imaging. The elucidation of soliton solutions paves the way for the development of advanced imaging techniques, offering new avenues for improving diagnostic capabilities and enhancing patient care in medical practice.

5 Conclusion

Solitons, with their inherent robustness and stability, hold the promise of elevating image resolution while mitigating artefacts in medical ultrasound imaging. This study has embarked upon an exploration of the intricate relationship between soliton solutions and their relevance to ultrasound imaging within the framework of the Westervelt equation. Leveraging GREMM and GERFM, we have uncovered a diverse spectrum of soliton solutions – from the enigmatic dark solitons to the periodic singular entities – each offering profound insights into the realm of ultrasound propagation.

Our endeavour extends beyond theoretical musings, seeking to forge tangible connections between soliton theory and its application in medical diagnostics. Through the intricate dance of advanced analytical techniques, we have peeled back the layers of complexity surrounding the spatial and temporal dynamics of solitons, illuminating their potential utility in identifying and characterising anomalies within biological tissues.

The visual representation of soliton solutions, rendered through the intricate tapestry of 3D, contour, and 2D plots, has bestowed upon us a visual symphony, revealing the graceful cadence of solitons as they traverse the domain of ultrasound imaging. Through meticulous mathematical scrutiny and numerical simulations, we have unravelled the rich tapestry of interactions between soliton theory and ultrasound imaging, paving the way for a new dawn of enhanced imaging methodologies.

Looking ahead, our gaze is fixed upon the horizon of further exploration, where the crucible of experimental validation, optimisation of imaging parameters, and seamless integration of soliton-based techniques into existing ultrasound frameworks beckons us. Armed with the insights gleaned from this odyssey, we stand poised at the precipice of transformative innovation, poised to unleash the full potential of soliton theory in reshaping the landscape of medical diagnostics and heralding a new era of precision healthcare. In future, we suggest exploring the fractional and stochastic versions of the equation, and applying different techniques to obtain other types of solutions, such as multi-solitons, lump solitons, and breather solitons. In summation, the elucidation of soliton solutions heralds a paradigm shift in the realm of ultrasound technology, offering a tableau of possibilities for enhancing medical imaging capabilities. With each stride forward in research and innovation, we inch closer to unlocking the true potential of soliton theory, poised to chart new frontiers in medical diagnostics and redefine the boundaries of patient care with elegance and precision.

Acknowledgements

The authors would like to express their appreciation for the invaluable support from the National Science and Technology Council in Taiwan, funded by Grant Numbers 112-2115-M-006-002 and 112-2321-B-006-020. The generous backing of this distinguished council has been crucial in advancing the research presented here. Their steadfast commitment to fostering and advancing scientific knowledge is sincerely acknowledged and highly valued.

Funding information: This research was funded by the National Science and Technology Council in Taiwan through Grant Numbers 112-2115-M-006-002 and 112-2321-B-006-020.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. DC: Conceived the primary idea and contributed to the conceptualisation, methodology, formal analysis, supervision, editing, reviewing, and project management. SMB: Undertook data curation, validation, and formal analysis. II: Carried out validation, formal analysis, and reviewing. HUR: Contributed to validation, methodology, formal analysis, reviewing, editing, and supervision. TLL: Assisted with editing and investigation.
Conflict of interest: The authors declare that they have no conflicts of interest or competing interests related to this work.
Data availability statement: The datasets generated and/or analyzed during the current study are accessible within the manuscript.

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Received: 2024-06-19

Revised: 2024-08-19

Accepted: 2024-08-28

Published Online: 2025-02-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0033

Keywords for this article

ultrasound imaging; Westervelt equation; generalised Riccati equation mapping method; soliton solutions; nonlinear equations

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